\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\_2Etransc\_2EMCLAURIN (TMK1JbdML7KT5T9CvDkzbz9H55mJMgSBdU2)}
\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\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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}
\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: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: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}
\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: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: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 $ 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}
Let $ ty\_2Erealax\_2Ereal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erealax\_2Ereal}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgbgihpfcdfechfgbgmg}
\end{equation}
Let $ c\_2Ereal\_2Ereal\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ereal\_2Ereal\_\_of\_\_num} \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfechfgbgmgpfpfpgggpfpfogfhng}
\end{equation}
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\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\_2Erealax\_2Ereal\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Ereal\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeafpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfcffeaf}
We define $\mname{c\_2Erealax\_2Ereal\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})}}~{({{{ap}~{c\_2Erealax\_2Ereal\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfbgegeg}
\end{equation}
Let $ c\_2Erealax\_2Etreal\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpffgbh}
\end{equation}
Let $ c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS \in (ty\_2Erealax\_2Ereal^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbecedf}
We define $\mname{c\_2Erealax\_2Ereal\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})} .  {({{{ap}~{c\_2Erealax\_2Ereal\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfbgegeg}
We define $\mname{c\_2Erealax\_2Ereal\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_add}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_neg : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_neg \in ((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfogfghg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfogfghg}
We define $\mname{c\_2Erealax\_2Ereal\_\_neg}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_neg}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfechfgbgmgpfpfdhfhcg}
We define $\mname{c\_2Ereal\_2Ereal\_\_sub}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_add}}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_neg}}}~{V1y}})}})}}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_inv : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_inv \in ((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfjgoggh}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfejgoggh}
We define $\mname{c\_2Erealax\_2Einv}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_inv}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_mul \in (((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfngfhmg}
We define $\mname{c\_2Erealax\_2Ereal\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_mul}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfepfcdge}
We define $\mname{c\_2Ereal\_2E\_2F}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Einv}}}~{V1y}})}})}}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_lt : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_lt \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfmgeh}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfmgeh}
We define $\mname{c\_2Erealax\_2Ereal\_\_lt}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_lt}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfechfgbgmgpfpfmgehfg}
We define $\mname{c\_2Ereal\_2Ereal\_\_lte}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{V1y}})}}~{V0x}})}})}}}$.
\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:dgpfcdfechfgbgmgpfcdfebgcgdh}
We define $\mname{c\_2Ereal\_2Eabs}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2Ereal\_\_lte}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{V0x}})}})}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_neg}}}~{V0x}})}})}}$.
\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}
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:dgpfcdfeahbgjgchpfcdfeffoedeffcfcfjf}
We define $\mname{c\_2Epair\_2EUNCURRY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1v \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{V0f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}}}}}$.
\end{definition}
Let $ ty\_2Emetric\_2Emetric : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Emetric\_2Emetric}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfengfgehchjgdgpfcdfengfgehchjgdg}
\end{equation}
Let $ c\_2Emetric\_2Emetric : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Emetric\_2Emetric~A\_27a \in ((ty\_2Emetric\_2Emetric~\\
A\_27a)^{{({{ty\_2Erealax\_2Ereal}^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfengfgehchjgdgpfcdfengfgehchjgdg}
\end{equation}
\begin{definition}\label{def:dgpfcdfengfgehchjgdgpfcdfengchbd}
We define $\mname{c\_2Emetric\_2Emr1}$ to be ${({{{ap}~{({{c\_2Emetric\_2Emetric}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{{\mname{c\_2Epair\_2EUNCURRY}}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {({{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Ereal\_2Eabs}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2Ereal\_\_sub}}}~{V1y}})}}~{V0x}})}})}})}})}})}})}$.
\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\_2Enets\_2Etendsto : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Enets\_2Etendsto~A\_27a \in (((2^{{A\_27a}})^{{A\_27a}})^{{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Emetric\_2Emetric}~{A\_27a}})}}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeogfgehdhpfcdfeehfgogegdhehpg}
\end{equation}
Let $ c\_2Emetric\_2Edist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Emetric\_2Edist~A\_27a \in ((ty\_2Erealax\_2Ereal^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}})^{{({{ty\_2Emetric\_2Emetric}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfengfgehchjgdgpfcdfeegjgdheh}
\end{equation}
Let $ ty\_2Etopology\_2Etopology : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Etopology\_2Etopology}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeehpgahpgmgpghgjhpfcdfeehpgahpgmgpghgjh}
\end{equation}
Let $ c\_2Etopology\_2Etopology : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Etopology\_2Etopology~A\_27a \in \\
((ty\_2Etopology\_2Etopology~A\_27a)^{{({{2}^{{({{2}^{{A\_27a}}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehpgahpgmgpghgjhpfcdfeehpgahpgmgpghgjh}
\end{equation}
\begin{definition}\label{def:dgpfcdfengfgehchjgdgpfcdfengehpgah}
We define $\mname{c\_2Emetric\_2Emtop}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0m \in {({{ty\_2Emetric\_2Emetric}~{A\_27a}})} .  {({{{ap}~{({{c\_2Etopology\_2Etopology}~{A\_27a}})}}~{({{\uniddcgcg} V1S\_27 \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1S\_27}}~{V2x}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V3e \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{V3e}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{({{{ap}~{({{{ap}~{({{c\_2Emetric\_2Edist}~{A\_27a}})}}~{V0m}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{A\_27a}})}}~{V2x}})}}~{V4y}})}})}})}}~{V3e}})}})}}~{({{{ap}~{V1S\_27}}~{V4y}})}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
Let $ c\_2Enets\_2Etends : {{\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\_2Enets\_2Etends~\\
A\_27a~A\_27b \in (((2^{{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Etopology\_2Etopology}~{A\_27a}})}}~{({{({{2}^{{A\_27b}}})}^{{A\_27b}}})}})}})^{{A\_27a}})^{{({{A\_27a}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeogfgehdhpfcdfeehfgogegdh}
\end{equation}
\begin{definition}\label{def:dgpfcdfemgjgngpfcdfeehfgogegdhpfpfchfgbgmgpfpfchfgbgmg}
We define $\mname{c\_2Elim\_2Etends\_\_real\_\_real}$ to be ${{\uniddcgcg} V0f \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Erealax\_2Ereal}}})} .  {{\uniddcgcg} V1l \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V2x0 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{c\_2Enets\_2Etends}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}})}}~{V0f}})}}~{V1l}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{ty\_2Etopology\_2Etopology}~{ty\_2Erealax\_2Ereal}})}}~{({{({{2}^{{ty\_2Erealax\_2Ereal}}})}^{{ty\_2Erealax\_2Ereal}}})}})}}~{({{{ap}~{({{\mname{c\_2Emetric\_2Emtop}}~{ty\_2Erealax\_2Ereal}})}}~{\mname{c\_2Emetric\_2Emr1}}})}})}}~{({{{ap}~{({{c\_2Enets\_2Etendsto}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{ty\_2Emetric\_2Emetric}~{ty\_2Erealax\_2Ereal}})}}~{ty\_2Erealax\_2Ereal}})}}~{\mname{c\_2Emetric\_2Emr1}}})}}~{V2x0}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgjgngpfcdfeegjgggggmg}
We define $\mname{c\_2Elim\_2Ediffl}$ to be ${{\uniddcgcg} V0f \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Erealax\_2Ereal}}})} .  {{\uniddcgcg} V1l \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V2x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2Elim\_2Etends\_\_real\_\_real}}}~{({{\uniddcgcg} V3h \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2E\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2Ereal\_\_sub}}}~{({{{ap}~{V0f}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_add}}}~{V2x}})}}~{V3h}})}})}})}}~{({{{ap}~{V0f}}~{V2x}})}})}})}}~{V3h}})}})}})}}~{V1l}})}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgjgngpfcdfeegjgggggfgchfgogehjgbgcgmgfg}
We define $\mname{c\_2Elim\_2Edifferentiable}$ to be ${{\uniddcgcg} V0f \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Erealax\_2Ereal}}})} .  {{\uniddcgcg} V1x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V2l \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2Elim\_2Ediffl}}}~{V0f}})}}~{V2l}})}}~{V1x}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfemgjgngpfcdfedgpgogehmg}
We define $\mname{c\_2Elim\_2Econtl}$ to be ${{\uniddcgcg} V0f \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Erealax\_2Ereal}}})} .  {{\uniddcgcg} V1x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{\mname{c\_2Elim\_2Etends\_\_real\_\_real}}}~{({{\uniddcgcg} V2h \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{V0f}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_add}}}~{V1x}})}}~{V2h}})}})}})}})}}~{({{{ap}~{V0f}}~{V1x}})}})}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}}$.
\end{definition}
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}
Let $ c\_2Earithmetic\_2EFACT : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Earithmetic\_2EFACT} \in {({{ty\_2Enum\_2Enum}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfegebedeef}
\end{equation}
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}
Let $ c\_2Ereal\_2Epow : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ereal\_2Epow \in ((ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})^{{ty\_2Erealax\_2Ereal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfeahpghh}
\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 $ c\_2Ereal\_2Esum : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ereal\_2Esum \in ((ty\_2Erealax\_2Ereal^{{({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfedhfhng}
\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} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n)=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1n)~V0m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdfjfne}
\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 .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~V2p))=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))~V2p)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((V0m=\mname{c\_2Enum\_2E0}){\unicdcdcdid}({\unicdcdaddd} V1n \in \\
ty\_2Enum\_2Enum .  (V0m=(ap~\mname{c\_2Enum\_2ESUC}~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeogfhngpfpfdebedffedf}
\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\_2Eprim\_\_rec\_2E\_3C}~\\
V0m)~V1n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1n)~V2p))){\unicdbdjdcd}(p~(ap~(ap~\\
\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  (((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
\mname{c\_2Enum\_2E0})~V0m)=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V0m)~\mname{c\_2Enum\_2E0})=V0m)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Enum\_2ESUC}~V0m)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~V0m)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeebd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~(\\
ap~\mname{c\_2Enum\_2ESUC}~V0m))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))=V0m))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpfdfffcebd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1n)~V0m)){\unicdbdjdcd}({\unicdcdaddd} V2p \in ty\_2Enum\_2Enum .  \\
(V0m=(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V2p)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfbeeeeepfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0a \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1c \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0a)~\\
V1c))~V1c)=V0a)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdfffce}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0c \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~V0c)~\\
V0c)=\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpffebfffbemepfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(((ap~c\_2Earithmetic\_2EFACT~\mname{c\_2Enum\_2E0})=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}({\unicdcdadad} V0n \in \\
ty\_2Enum\_2Enum .  ((ap~c\_2Earithmetic\_2EFACT~(ap~\mname{c\_2Enum\_2ESUC}~\\
V0n))=(ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Enum\_2ESUC}~V0n))~(ap~\\
c\_2Earithmetic\_2EFACT~V0n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfegebedeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
(ap~c\_2Earithmetic\_2EFACT~V0n))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfegebedeefpfpfmefedfdf}
\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}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdcdcdhd}\\
((p~V1t2){\unicdcdcdhd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdcdcdhd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoekepfpfbedfdfpede}
\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{\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} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1l \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2m \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3x \in \\
ty\_2Erealax\_2Ereal .  (((p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V1l)~\\
V3x)){\unicdcdcdhd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V2m)~V3x))){\unicdbdjdcd}(V1l=V2m))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfffoejebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1l \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2x \in ty\_2Erealax\_2Ereal .  ((p~(ap~\\
(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V1l)~V2x)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Elim\_2Econtl}~\\
V0f)~V2x))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfdepeoeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0k \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1x \in ty\_2Erealax\_2Ereal .  \\
(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~({\uniddcgcg} V2x \in ty\_2Erealax\_2Ereal .  V0k))~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V1x))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfdepeoedfef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1g \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V2l \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V3m \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V4x \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V2l)~V4x)){\unicdcdcdhd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~\\
V1g)~V3m)~V4x))){\unicdbdjdcd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~({\uniddcgcg} V5x \in ty\_2Erealax\_2Ereal .  \\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(ap~V0f~V5x))~(ap~V1g~V5x))))~(ap~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V2l)~V3m))~V4x))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1g \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V2l \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V3m \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V4x \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V2l)~V4x)){\unicdcdcdhd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~\\
V1g)~V3m)~V4x))){\unicdbdjdcd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~({\uniddcgcg} V5x \in ty\_2Erealax\_2Ereal .  \\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~(ap~V0f~V5x))~(ap~V1g~V5x))))~(ap~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V2l)~\\
(ap~V1g~V4x)))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V3m)~(ap~V0f~V4x))))~\\
V4x))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfneffme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1g \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V2l \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V3m \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V4x \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V2l)~V4x)){\unicdcdcdhd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~\\
V1g)~V3m)~V4x))){\unicdbdjdcd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~({\uniddcgcg} V5x \in ty\_2Erealax\_2Ereal .  \\
(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~(ap~V0f~V5x))~(ap~V1g~V5x))))~(ap~(ap~\\
\mname{c\_2Ereal\_2Ereal\_\_sub}~V2l)~V3m))~V4x))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfdfffce}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1g \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V2l \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V3m \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V4x \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V2l)~(ap~V1g~V4x))){\unicdcdcdhd}(p~(ap~(\\
ap~(ap~\mname{c\_2Elim\_2Ediffl}~V1g)~V3m)~V4x))){\unicdbdjdcd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~\\
({\uniddcgcg} V5x \in ty\_2Erealax\_2Ereal .  (ap~V0f~(ap~V1g~V5x))))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V2l)~V3m))~V4x))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfdeiebejeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  (p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~\\
({\uniddcgcg} V1x \in ty\_2Erealax\_2Ereal .  V1x))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
V0x)))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfif}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1x \in ty\_2Erealax\_2Ereal .  \\
(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~({\uniddcgcg} V2x \in ty\_2Erealax\_2Ereal .  (\\
ap~(ap~c\_2Ereal\_2Epow~V2x)~V0n)))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V0n))~(ap~(ap~c\_2Ereal\_2Epow~V1x)~\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~V0n)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))))~V1x))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfafpehf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1g \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V2l \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V3m \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V4x \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~V2l)~V4x)){\unicdcdcdhd}((p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~\\
V1g)~V3m)~V4x)){\unicdcdcdhd}(\preoh ((ap~V1g~V4x)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))))){\unicdbdjdcd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~({\uniddcgcg} V5x \in ty\_2Erealax\_2Ereal .  \\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(ap~V0f~V5x))~(ap~V1g~V5x))))~(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~\\
(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V2l)~\\
(ap~V1g~V4x)))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V3m)~(ap~V0f~V4x))))~\\
(ap~(ap~c\_2Ereal\_2Epow~(ap~V1g~V4x))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))))~V4x))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfeejegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in ((ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V1f\_27 \in ((ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V2m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V3n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V4x \in ty\_2Erealax\_2Ereal .  (({\unicdcdadad} V5r \in ty\_2Enum\_2Enum .  \\
(((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V2m)~V5r)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V5r)~(ap~(ap~c\_2Earithmetic\_2E\_2B~V2m)~V3n)))){\unicdbdjdcd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~\\
({\uniddcgcg} V6x \in ty\_2Erealax\_2Ereal .  (ap~(ap~V0f~V5r)~V6x)))~(ap~(ap~V1f\_27~\\
V5r)~V4x))~V4x)))){\unicdbdjdcd}(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~({\uniddcgcg} V7x \in ty\_2Erealax\_2Ereal .  \\
(ap~(ap~c\_2Ereal\_2Esum~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~V2m)~V3n))~({\uniddcgcg} V8n \in ty\_2Enum\_2Enum .  (ap~(ap~V0f~\\
V8n)~V7x)))))~(ap~(ap~c\_2Ereal\_2Esum~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~V2m)~V3n))~({\uniddcgcg} V9r \in ty\_2Enum\_2Enum .  (ap~(ap~V1f\_27~\\
V9r)~V4x))))~V4x))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfeeejegegepfpfdfffne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  (((p~(\\
ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V1a)~V2b)){\unicdcdcdhd}(((ap~V0f~V1a)=(ap~V0f~\\
V2b)){\unicdcdcdhd}(({\unicdcdadad} V3x \in ty\_2Erealax\_2Ereal .  (((p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V1a)~V3x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~V3x)~V2b))){\unicdbdjdcd}(p~(ap~(\\
ap~\mname{c\_2Elim\_2Econtl}~V0f)~V3x)))){\unicdcdcdhd}({\unicdcdadad} V4x \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V1a)~V4x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V4x)~V2b))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Elim\_2Edifferentiable}~V0f)~V4x))))))){\unicdbdjdcd}\\
({\unicdcdaddd} V5z \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V1a)~V5z)){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V5z)~V2b)){\unicdcdcdhd}(p~(ap~\\
(ap~(ap~\mname{c\_2Elim\_2Ediffl}~V0f)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~\\
V5z)))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgngpfcdfecfpememefe}
\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 .  (\preoh (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V0n)~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfcffegeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0n)~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfdfffdepfpfcffegeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V1y)=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V1y)~V0x))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfdfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfmejeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2z \in ty\_2Erealax\_2Ereal .  (((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V0x)~V1y)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V1y)~V2z))){\unicdbdjdcd}(p~(ap~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V0x)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V1y)=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V1y)~V0x))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfdfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2z \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V0x)~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V1y)~V2z))=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V1y))~V2z)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(\\
ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~V0x)=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfmejeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((\preoh (V0x=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))){\unicdbdjdcd}((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~(ap~\mname{c\_2Erealax\_2Einv}~\\
V0x))~V0x)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfmejeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V0x)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfcfjeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V0x)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfcfjeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
(((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V1y)=V1y){\unicdbdjded}(V0x=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfmejeeepfpfffoejebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V0x)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfmekffecfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V0x)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfcfkffecfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V0x)~V1y)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfjeneafpfpfmefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2z \in ty\_2Erealax\_2Ereal .  (((p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V0x)~V1y)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~V1y)~V2z))){\unicdbdjdcd}(p~(ap~(\\
ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~V0x)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~\\
V0x)~V0x)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfdfffcepfpfcffegeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
(((ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~V0x)~V1y)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfdfffcepfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((\preoh (V0x=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))){\unicdbdjdcd}((ap~\mname{c\_2Erealax\_2Einv}~(ap~\mname{c\_2Erealax\_2Einv}~V0x))=\\
V0x)))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfjeoegfjeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V0x)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~(ap~\mname{c\_2Erealax\_2Einv}~\\
V0x)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfjeoegfpfpfafpedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V0m))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
((ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V0m)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V1n)){\unicdbdjded}(V0m=V1n))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfjeoeke}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V0m))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1n))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((ap~\mname{c\_2Erealax\_2Einv}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfjeoegfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(ap~\\
c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V0x)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfeejegfpfpfmekffecfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~\\
V0x)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfdfffcepfpfcfkffecfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2z \in ty\_2Erealax\_2Ereal .  ((V0x=(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~\\
V1y)~V2z)){\unicdbdjded}((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V2z)=V1y)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpffebfpfpfdfffcepfpfmebeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2z \in ty\_2Erealax\_2Ereal .  (((ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~\\
V0x)~V1y)=V2z){\unicdbdjded}(V0x=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V2z)~V1y))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpffebfpfpfdfffcepfpfcfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
(((\preoh (V0x=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}(\preoh (V1y=\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0})))){\unicdbdjdcd}((ap~\mname{c\_2Erealax\_2Einv}~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V1y))=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~\mname{c\_2Erealax\_2Einv}~V0x))~(ap~\mname{c\_2Erealax\_2Einv}~V1y))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfjeoegfpfpfneffme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V0x)){\unicdbdjdcd}(\preoh (V0x=(ap~\\
c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0})))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfafpedfpfpfoekf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (\preoh ((ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~c\_2Earithmetic\_2EFACT~V0n))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfgebedeefpfpfoekf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~c\_2Ereal\_2Epow~V0x)~\\
\mname{c\_2Enum\_2E0})=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))){\unicdcdcdhd}({\unicdcdadad} V1x \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Ereal\_2Epow~\\
V1x)~(ap~\mname{c\_2Enum\_2ESUC}~V2n))=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V1x)~\\
(ap~(ap~c\_2Ereal\_2Epow~V1x)~V2n))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfeahpghh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Ereal\_2Epow~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~(ap~\mname{c\_2Enum\_2ESUC}~V0n))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfeafpehfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0c \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  \\
((\preoh (V0c=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))){\unicdbdjdcd}(\preoh ((ap~\\
(ap~c\_2Ereal\_2Epow~V0c)~V1n)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfeafpehfpfpfoekf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~c\_2Ereal\_2Epow~V0x)~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))=\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V0x)))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfeafpehfpfpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1f \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  \\
((ap~(ap~c\_2Ereal\_2Esum~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~V0n)~\mname{c\_2Enum\_2E0}))~V1f)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))){\unicdcdcdhd}({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V3m \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V4f \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  \\
((ap~(ap~c\_2Ereal\_2Esum~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~V2n)~(ap~\mname{c\_2Enum\_2ESUC}~V3m)))~V4f)=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
(ap~(ap~c\_2Ereal\_2Esum~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~V2n)~V3m))~V4f))~(ap~V4f~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V2n)~V3m))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfedhfhng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Ereal\_2Esum~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~V1n)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~V0f)=(\\
ap~V0f~V1n))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfedfffnepfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1n \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V2k \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Ereal\_2Esum~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\mname{c\_2Enum\_2E0})~\\
V1n))~({\uniddcgcg} V3m \in ty\_2Enum\_2Enum .  (ap~V0f~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V3m)~V2k))))=(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~(ap~(ap~c\_2Ereal\_2Esum~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\mname{c\_2Enum\_2E0})~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~V2k)))~V0f))~(ap~(ap~c\_2Ereal\_2Esum~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\mname{c\_2Enum\_2E0})~\\
V2k))~V0f))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfedfffnepfpfpegegedffeef}
\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\_2Ereal\_2Esum~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~V0m)~V1n))~({\uniddcgcg} V2r \in ty\_2Enum\_2Enum .  (ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfedfffnepfpfad}
\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){\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}
\begin{theorem}\label{thm:ehigngpfcdfeehchbgogdhdgpfcdfenedemebeffcfjeoe}
$$\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1diff \in \\
((ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}})^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdadad} V2h \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3n \in ty\_2Enum\_2Enum .  \\
(((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V2h)){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
V3n)){\unicdcdcdhd}(((ap~V1diff~\mname{c\_2Enum\_2E0})=V0f){\unicdcdcdhd}({\unicdcdadad} V4m \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V5t \in ty\_2Erealax\_2Ereal .  (((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V4m)~V3n)){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V5t)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~V5t)~V2h)))){\unicdbdjdcd}\\
(p~(ap~(ap~(ap~\mname{c\_2Elim\_2Ediffl}~(ap~V1diff~V4m))~(ap~(ap~V1diff~(ap~\\
\mname{c\_2Enum\_2ESUC}~V4m))~V5t))~V5t)))))))){\unicdbdjdcd}({\unicdcdaddd} V6t \in ty\_2Erealax\_2Ereal .  \\
((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V6t)){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V6t)~V2h)){\unicdcdcdhd}\\
((ap~V0f~V2h)=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(ap~(ap~c\_2Ereal\_2Esum~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\mname{c\_2Enum\_2E0})~\\
V3n))~({\uniddcgcg} V7m \in ty\_2Enum\_2Enum .  (ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(ap~(ap~V1diff~V7m)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~c\_2Earithmetic\_2EFACT~\\
V7m))))~(ap~(ap~c\_2Ereal\_2Epow~V2h)~V7m)))))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~(ap~(ap~V1diff~V3n)~V6t))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
(ap~c\_2Earithmetic\_2EFACT~V3n))))~(ap~(ap~c\_2Ereal\_2Epow~V2h)~\\
V3n))))))))))))\end{array}$$
\end{theorem}
\end{document}
