\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\_2Ereal\_\_topology\_2EBOUNDED\_\_PARTIAL\_\_SUMS (TMKhPv2iqZ3Y8qa8NEWncckFwvNVFLA8qoV)}
\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: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:dgpfcdfeahchfgegpfpfdhfgehpfcdfefeneafefjf}
We define $\mname{c\_2Epred\_\_set\_2EEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfejeoe}
We define $\mname{c\_2Ebool\_2EIN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1f \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V1f}}~{V0x}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def: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:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Epair\_2EABS\_\_prod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Epred\_\_set\_2EGSPEC : {{\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\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b \in ((2^{{A\_27a}})^{{({{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{2}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffede}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoedffecfef}
We define $\mname{c\_2Epred\_\_set\_2EINSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2y}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2y}})}}~{V1s}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfegejeoejeeffe}
We define $\mname{c\_2Epred\_\_set\_2EFINITE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V1P \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V1P}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1P}}~{V2s}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3e \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINSERT}}~{A\_27a}})}}~{V3e}})}}~{V2s}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{V1P}}~{V0s}})}})}})}})}}}$.
\end{definition}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
\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}
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: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: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}
\begin{definition}\label{def:dgpfcdfejgehfgchbgehfgpfcdfepfcdfepfcdfe}
We define $\mname{c\_2Eiterate\_2E\_2E\_2E}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2x \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Enum\_2Enum}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3C\_3D}}}~{V0m}})}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3C\_3D}}}~{V2x}})}}~{V1n}})}})}})}})}})}}}$.
\end{definition}
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\end{equation}
Let $ ty\_2Erealax\_2Ereal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erealax\_2Ereal}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgbgihpfcdfechfgbgmg}
\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}
\begin{definition}\label{def:dgpfcdfejgehfgchbgehfgpfcdfeogfgfhehchbgmg}
We define $\mname{c\_2Eiterate\_2Eneutral}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0op \in {({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{V0op}}~{V1x}})}}~{V2y}})}})}}~{V2y}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{V0op}}~{V2y}})}}~{V1x}})}})}}~{V2y}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgehfgchbgehfgpfcdfedhfhahahpgcheh}
We define $\mname{c\_2Eiterate\_2Esupport}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0op \in {({{({{A\_27b}^{{A\_27b}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V1f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V2s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V3x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x}})}}~{V2s}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V1f}}~{V3x}})}})}}~{({{{ap}~{({{\mname{c\_2Eiterate\_2Eneutral}}~{A\_27b}})}}~{V0op}})}})}})}})}})}})}})}}}}}}$.
\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:dgpfcdfejgehfgchbgehfgpfcdfejeefdffeef}
We define $\mname{c\_2Eiterate\_2EITSET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27a}^{{A\_27a}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27b}}})} .  {{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27a}^{{({{2}^{{A\_27b}}})}}})}})}}~{({{\uniddcgcg} V3g \in {({{A\_27a}^{{({{2}^{{A\_27b}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{V3g}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27b}})}})}})}}~{V2b}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V4x \in {A\_27b} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27b}}})}})}}~{({{\uniddcgcg} V5s \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EFINITE}}~{A\_27b}})}}~{V5s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{V3g}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINSERT}}~{A\_27b}})}}~{V4x}})}}~{V5s}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{V4x}})}}~{V5s}})}})}}~{({{{ap}~{V3g}}~{V5s}})}})}}~{({{{ap}~{({{{ap}~{V0f}}~{V4x}})}}~{({{{ap}~{V3g}}~{V5s}})}})}})}})}})}})}})}})}})}})}})}})}}~{V1s}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgehfgchbgehfgpfcdfejgehfgchbgehfg}
We define $\mname{c\_2Eiterate\_2Eiterate}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0op \in {({{({{A\_27b}^{{A\_27b}}})}^{{A\_27b}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{A\_27b}})}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EFINITE}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Eiterate\_2Esupport}}~{A\_27a}}~{A\_27b}})}}~{V0op}})}}~{V2f}})}}~{V1s}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Eiterate\_2EITSET}}~{A\_27b}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{\uniddcgcg} V4a \in {A\_27b} .  {({{{ap}~{({{{ap}~{V0op}}~{({{{ap}~{V2f}}~{V3x}})}})}}~{V4a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Eiterate\_2Esupport}}~{A\_27a}}~{A\_27b}})}}~{V0op}})}}~{V2f}})}}~{V1s}})}})}}~{({{{ap}~{({{\mname{c\_2Eiterate\_2Eneutral}}~{A\_27b}})}}~{V0op}})}})}})}}~{({{{ap}~{({{\mname{c\_2Eiterate\_2Eneutral}}~{A\_27b}})}}~{V0op}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgehfgchbgehfgpfcdfedffhng}
We define $\mname{c\_2Eiterate\_2ESum}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{{\mname{c\_2Eiterate\_2Eiterate}}~{A\_27a}}~{ty\_2Erealax\_2Ereal}})}}~{\mname{c\_2Erealax\_2Ereal\_\_add}}})}}$.
\end{definition}
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: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: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: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}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgjgdfffde}
We define $\mname{c\_2Enumeral\_2EiiSUC}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V0n}})}})}}$.
\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:dgpfcdfeogfhngfgchbgmgpfcdfejgeeffce}
We define $\mname{c\_2Enumeral\_2EiDUB}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0x}})}}~{V0x}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgkf}
We define $\mname{c\_2Enumeral\_2EiZ}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejegf}
We define $\mname{c\_2Epred\_\_set\_2EUNIV}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeeejegege}
We define $\mname{c\_2Epred\_\_set\_2EDIFF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejenebehefe}
We define $\mname{c\_2Epred\_\_set\_2EIMAGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27b}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{2}})}}~{({{{ap}~{V0f}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}})}})}}}}}$.
\end{definition}
\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\_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}
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\_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:dgpfcdfeahchfgegpfpfdhfgehpfcdfedfffcedffeef}
We define $\mname{c\_2Epred\_\_set\_2ESUBSET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejepeoe}
We define $\mname{c\_2Epred\_\_set\_2EUNION}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}}}}$.
\end{definition}
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 $ 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: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}
\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\_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}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfecgpgfhogegfgegpfpfegfggg}
We define $\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}$ to be ${{\uniddcgcg} V0s \in {({{2}^{{ty\_2Erealax\_2Ereal}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V1a \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V2x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{ty\_2Erealax\_2Ereal}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2Ereal\_\_lte}}}~{({{{ap}~{\mname{c\_2Ereal\_2Eabs}}}~{V2x}})}})}}~{V1a}})}})}})}})}})}})}}$.
\end{definition}
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 .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n)){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebf}
\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 .  (\\
(\preoh (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1n)~V0m)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfmefedfdf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(\preoh (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V1n)~V0m)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfmefedfdfpfpffebfffbeme}
\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 .  (\\
((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n)=\mname{c\_2Enum\_2E0}){\unicdbdjded}((V0m=\\
\mname{c\_2Enum\_2E0}){\unicdcdcdhd}(V1n=\mname{c\_2Enum\_2E0})))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpffebfpfpfad}
\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 .  (\\
{\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} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(\preoh (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V1n))~V0m)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfmefebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(\preoh (V0m=V1n)){\unicdbdjded}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Enum\_2ESUC}~\\
V0m))~V1n)){\unicdcdcdid}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1n))~V0m))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeoepeefpfpfoeffnepfpffebf}
\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}
\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0n)~\mname{c\_2Enum\_2E0})){\unicdbdjded}(V0n=\mname{c\_2Enum\_2E0}))){\unicdcdcdhd}({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1m)~(ap~\mname{c\_2Enum\_2ESUC}~V2n))){\unicdbdjded}((V1m=(ap~\mname{c\_2Enum\_2ESUC}~V2n)){\unicdcdcdid}\\
(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1m)~V2n)))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefe}
\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}
\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}
{({{\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}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{({{\unicdcdadad} V0x \in {A\_27a} .  {({{V0x}={V0x}})}})}}}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfcffegeme}
\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} 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}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~\\
V0t1)~V1t2)=V1t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ((\preoh ({\unicdcdaddd} V1x \in \\
A\_27a .  (p~(ap~V0P~V1x)))){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  (\preoh (p~(ap~V0P~V2x))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpffeifjedfefdfpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
2 .  ((({\unicdcdadad} V2x \in A\_27a .  (p~(ap~V0P~V2x))){\unicdcdcdhd}(p~V1Q)){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~(ap~V0P~V3x)){\unicdcdcdhd}(p~V1Q))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfbeoeeepfpfgepecfbememepfpfefiene}
\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^{{A\_27a}}) .  (((p~V0P){\unicdcdcdhd}({\unicdcdadad} V2x \in A\_27a .  (p~(ap~V1Q~V2x)))){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~V0P){\unicdcdcdhd}(p~(ap~V1Q~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfbeoeeepfpfgepecfbememepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0Q \in 2 .  ({\unicdcdadad} V1P \in (\\
2^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V1P~V2x)){\unicdcdcdid}(p~V0Q))){\unicdbdjded}(({\unicdcdadad} V3x \in \\
A\_27a .  (p~(ap~V1P~V3x))){\unicdcdcdid}(p~V0Q)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfgepecfbememepfpfpecfpfpfefiene}
\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^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~V0P){\unicdcdcdid}(p~(ap~V1Q~V2x)))){\unicdbdjded}((p~\\
V0P){\unicdcdcdid}({\unicdcdadad} V3x \in A\_27a .  (p~(ap~V1Q~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfgepecfbememepfpfpecfpfpfefiene}
\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}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  (((p~V0A){\unicdcdcdid}(\\
(p~V1B){\unicdcdcdhd}(p~V2C))){\unicdbdjded}(((p~V0A){\unicdcdcdid}(p~V1B)){\unicdcdcdhd}((p~V0A){\unicdcdcdid}(p~V2C)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfpecfpfpfpegffecfpfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((p~V0A){\unicdbdjdcd}(p~V1B)){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdid}\\
(p~V1B)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfeejedfkepfpfefiene}
\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} 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^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  (({\unicdcdadad} V2x \in A\_27a .  ((V2x=V1v){\unicdbdjdcd}(p~(ap~V0f~V2x)))){\unicdbdjded}(p~(\\
ap~V0f~V1v)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfgepecfbememepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
2 .  ((({\unicdcdaddd} V2x \in A\_27a .  (p~(ap~V0P~V2x))){\unicdbdjdcd}(p~V1Q)){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~(ap~V0P~V3x)){\unicdbdjdcd}(p~V1Q))))))\end{array}
\label{hyp:ehigngpfcdfedgbgchegjgogbgmgpfcdfemefegeefpfpfjeneafpfpffeifjedfefdfpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~\\
ty\_2Enum\_2Enum)~(ap~(c\_2Epred\_\_set\_2EGSPEC~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)~\\
({\uniddcgcg} V1m \in ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Enum\_2Enum~\\
2)~V1m)~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1m)~V0n)))))))\end{array}
\label{hyp:ehigngpfcdfedgbgchegjgogbgmgpfcdfegejeoejeeffepfpfoeffnedffehepfpfmeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~ty\_2Enum\_2Enum)~(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~\\
V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfegejeoejeeffepfpfoeffnedffehe}
\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\_2Ebool\_2EIN}~ty\_2Enum\_2Enum)~\\
V2p)~(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~V0m)~V1n))){\unicdbdjded}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~V2p)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V2p)~V1n)))))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfejeoepfpfoeffnedffehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(c\_2Epred\_\_set\_2EGSPEC~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum)~({\uniddcgcg} V1x \in ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Enum\_2Enum~2)~V1x)~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1x)~V0n))))=\\
(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(2^{{ty\_2Enum\_2Enum}}))~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~\\
ty\_2Enum\_2Enum)~V0n)~\mname{c\_2Enum\_2E0}))~(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Enum\_2Enum))~\\
(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~\mname{c\_2Enum\_2E0})~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V0n)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfeoeffnedffehepfpfmeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{A\_27a}}) .  \\
({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ((p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~\\
V1s)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~\mname{c\_2Ereal\_2Eabs}~(ap~(\\
ap~(\mname{c\_2Eiterate\_2ESum}~A\_27a)~V1s)~V0f)))~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~\\
A\_27a)~V1s)~({\uniddcgcg} V2x \in A\_27a .  (ap~\mname{c\_2Ereal\_2Eabs}~(ap~V0f~V2x)))))))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfedfffnepfpfbecedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{A\_27a}}) .  \\
({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1s)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~(ap~V0f~V2x))))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~\\
A\_27a)~V1s)~V0f))))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfedfffnepfpfafpedfpfpfmefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0u \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1v \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2f \in (ty\_2Erealax\_2Ereal^{{A\_27a}}) .  (((p~(ap~\\
(\mname{c\_2Epred\_\_set\_2EFINITE}~A\_27a)~V0u)){\unicdcdcdhd}((p~(ap~(\mname{c\_2Epred\_\_set\_2EFINITE}~\\
A\_27a)~V1v)){\unicdcdcdhd}(({\unicdcdadad} V3x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V3x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDIFF}~A\_27a)~V0u)~V1v))){\unicdbdjdcd}(p~(ap~(ap~\\
\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~V2f~V3x))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))))){\unicdcdcdhd}({\unicdcdadad} V4x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V4x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDIFF}~A\_27a)~V1v)~V0u))){\unicdbdjdcd}(\\
p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~\\
(ap~V2f~V4x)))))))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~\\
A\_27a)~V0u)~V2f))~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~A\_27a)~V1v)~V2f)))))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfedfffnepfpfdfffcedffeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1m \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V2n)~V1m)){\unicdbdjdcd}((ap~(ap~(\mname{c\_2Eiterate\_2ESum}~ty\_2Enum\_2Enum)~(ap~(ap~\\
\mname{c\_2Eiterate\_2E\_2E\_2E}~V1m)~V2n))~V0f)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfedfffnepfpfefcfjegfpfpfoeffnedffehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1m \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V3p \in ty\_2Enum\_2Enum .  \\
(((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~V2n)){\unicdcdcdhd}((p~(ap~(ap~\\
\mname{c\_2Earithmetic\_2E\_3C\_3D}~V1m)~V2n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V2n)~(ap~(ap~c\_2Earithmetic\_2E\_2B~V3p)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))))){\unicdbdjdcd}((\\
ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~ty\_2Enum\_2Enum)~\\
(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~V1m)~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V2n)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))))~V0f))~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~\\
ty\_2Enum\_2Enum)~(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~V2n)~V3p))~V0f))=\\
(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~ty\_2Enum\_2Enum)~(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~\\
V1m)~V3p))~V0f)))))))\end{array}
\label{hyp:ehigngpfcdfejgehfgchbgehfgpfcdfedfffnepfpfdepenecejeoefepfpfme}
\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 .  (\\
((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Earithmetic\_2EZERO})~\\
V0n))=V0n){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~\mname{c\_2Earithmetic\_2EZERO}))=V0n){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiZ}~(\\
ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))~(\\
ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~(\\
ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}\\
(((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Earithmetic\_2EZERO})~\\
V0n))=(ap~\mname{c\_2Enum\_2ESUC}~V0n)){\unicdcdcdhd}(((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~\mname{c\_2Earithmetic\_2EZERO}))=(ap~\mname{c\_2Enum\_2ESUC}~V0n)){\unicdcdcdhd}(((ap~\\
\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}(\\
((ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V1m)))){\unicdcdcdhd}\\
(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\mname{c\_2Earithmetic\_2EZERO})~\\
V0n))=(ap~\mname{c\_2Enumeral\_2EiiSUC}~V0n)){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~\mname{c\_2Earithmetic\_2EZERO}))=(\\
ap~\mname{c\_2Enumeral\_2EiiSUC}~V0n)){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(\\
ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V1m)))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))){\unicdcdcdhd}(((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V1m)))=\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))){\unicdcdcdhd}((ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V1m)))=\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Enumeral\_2EiiSUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0n)~V1m)))))))))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfbgegeg}
\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} V0n \in ty\_2Enum\_2Enum .  (((ap~\mname{c\_2Enumeral\_2EiDUB}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Enumeral\_2EiDUB}~V0n))){\unicdcdcdhd}\\
(((ap~\mname{c\_2Enumeral\_2EiDUB}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V0n))=(ap~\\
\mname{c\_2Earithmetic\_2EBIT2}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))){\unicdcdcdhd}((ap~\\
\mname{c\_2Enumeral\_2EiDUB}~\mname{c\_2Earithmetic\_2EZERO})=\mname{c\_2Earithmetic\_2EZERO}))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfejgeeffcepfpfchfgngpgghbgmg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
((ap~(ap~c\_2Earithmetic\_2E\_2A~\mname{c\_2Earithmetic\_2EZERO})~V0n)=\mname{c\_2Earithmetic\_2EZERO}){\unicdcdcdhd}\\
(((ap~(ap~c\_2Earithmetic\_2E\_2A~V0n)~\mname{c\_2Earithmetic\_2EZERO})=\\
\mname{c\_2Earithmetic\_2EZERO}){\unicdcdcdhd}(((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
V0n))~V1m)=(ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~\mname{c\_2Enumeral\_2EiDUB}~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0n)~V1m)))~\\
V1m))){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))~V1m)=(ap~\mname{c\_2Enumeral\_2EiDUB}~(ap~\mname{c\_2Enumeral\_2EiZ}~(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0n)~V1m))~\\
V1m)))))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfngfhmgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedemepedffeeepfpfafbejecfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1x \in \\
A\_27a .  ({\unicdcdadad} V2y \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~\\
A\_27b~A\_27c)~V0f)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V1x)~V2y))=\\
(ap~(ap~V0f~V1x)~V2y)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeffoedeffcfcfjfpfpfeefege}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0P \in (2^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  (({\unicdcdaddd} V1p \in \\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  (p~(ap~V0P~V1p))){\unicdbdjded}({\unicdcdaddd} V2p\_\_1 \in \\
A\_27a .  ({\unicdcdaddd} V3p\_\_2 \in A\_27b .  (p~(ap~V0P~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~A\_27b)~V2p\_\_1)~V3p\_\_2)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfefeifjedfefdfpfpfafcfpeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in ((ty\_2Epair\_2Eprod~A\_27a~2)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1v)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~2)~V1v)~\mname{c\_2Ebool\_2ET})=(ap~V0f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedejegejedebeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EUNIV}~A\_27a))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EDIFF}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~(\\
ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdhd}(\preoh (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfeejegege}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1f \in (A\_27a^{{A\_27b}}) .  ({\unicdcdadad} V2s \in \\
(2^{{A\_27b}}) .  (({\unicdcdadad} V3y \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V3y)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~A\_27b~A\_27a)~V1f)~V2s))){\unicdbdjdcd}(\\
p~(ap~V0P~V3y)))){\unicdbdjded}({\unicdcdadad} V4x \in A\_27b .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27b)~V4x)~V2s)){\unicdbdjdcd}(p~(ap~V0P~(ap~V1f~V4x)))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfegepecfbememepfpfjeoepfpfjenebehefe}
\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)~\mname{c\_2Enum\_2E0}))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfeoepeefpfpfmefedfdfpfpfad}
\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 .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2z \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V0x)~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V1y)~V2z))=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V1y))~V2z)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfbedfdfpede}
\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 .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V0x))~V0x)=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfmejeoegf}
\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 .  \\
({\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 .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V0x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V1y))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V1y))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfneffme}
\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\_\_add}~\\
V0x)~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V0x))=(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfcfjeoegf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
((ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~\\
V1y))=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
V0x))~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V1y)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfoefehepfpfbeeeee}
\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 .  \\
({\unicdcdadad} V2z \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V1y))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V0x)~V2z))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V1y)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfmebeeeee}
\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\_2Ereal\_2Ereal\_\_lte}~V0x)~V1y)){\unicdcdcdid}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V1y)~V0x)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfefpeefbeme}
\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\_2Ereal\_2Ereal\_\_lte}~V1y)~V2z))){\unicdbdjdcd}(p~(ap~(\\
ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V0x)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeeffepfpfefcfbeoedf}
\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\_2Erealax\_2Ereal\_\_lt}~V1y)~V2z))){\unicdbdjdcd}(p~(ap~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V0x)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefeefpfpfefcfbeoedf}
\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 .  ({\unicdcdadad} V1y \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~V0x)~V1y)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V1y)~V0x))){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfbeoeefjedfjfne}
\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\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V0x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V1y))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V1y))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfneffme}
\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} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(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\_2B~V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  (p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~(ap~\mname{c\_2Ereal\_2Eabs}~\\
V0x))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfebecedfpfpfafpedf}
\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)~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
V1y))=(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfcfoefehe}
\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}~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V0x))~\\
V1y)=(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfneffmepfpfmeoefehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0y \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1x \in ty\_2Erealax\_2Ereal .  \\
((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V1x)~V0y)){\unicdbdjded}(\preoh (p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V0y)~V1x))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfechfgbgmgpfpfmgeh}
\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}~\\
V1y)~V2z)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V0x)~V1y))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V2z)))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfmebeeeeepfpfjeneaf}
\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\_2Ereal\_2Ereal\_\_lte}~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V0x))~\\
V1y)){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V1y))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfmeoefehe}
\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\_2Ereal\_2Ereal\_\_lte}~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V0x))~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V1y))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~\\
V1y)~V0x)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfoefehecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ((ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~V0x))=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfoefehepfpfoefehe}
\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\_2Ereal\_2Ereal\_\_lte}~V0x)~(ap~\mname{c\_2Erealax\_2Ereal\_\_neg}~\\
V1y))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V0x)~V1y))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmefepfpfcfoefehe}
\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\_\_add}~V1y)~V2z))=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V1y))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V0x)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfmeeejedfefcfjece}
\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}~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V1y))~V2z)=(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~V0x)~V2z))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_mul}~\\
V1y)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfbeeeeepfpfcfeejedfefcfjece}
\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\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
V0m))~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfpegepfpfoeffnepfpfmefe}
\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:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfpegepfpfoeffnepfpfneffme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0s \in (2^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1t \in (2^{{ty\_2Erealax\_2Ereal}}) .  \\
(((p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~V1t)){\unicdcdcdhd}(p~(ap~(ap~(\\
\mname{c\_2Epred\_\_set\_2ESUBSET}~ty\_2Erealax\_2Ereal)~V0s)~V1t))){\unicdbdjdcd}(p~(\\
ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~V0s)))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfecepeffoeeefeeepfpfdfffcedffeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0s \in (2^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1t \in (2^{{ty\_2Erealax\_2Ereal}}) .  \\
((p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~\\
ty\_2Erealax\_2Ereal)~V0s)~V1t))){\unicdbdjded}((p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~\\
V0s)){\unicdcdcdhd}(p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~V1t))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfecepeffoeeefeeepfpfffoejepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0s \in (2^{{ty\_2Erealax\_2Ereal}}) .  ((p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~\\
V0s)){\unicdbdjded}({\unicdcdaddd} V1b \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V1b)){\unicdcdcdhd}({\unicdcdadad} V2x \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~\\
V2x)~V0s)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~\mname{c\_2Ereal\_2Eabs}~\\
V2x))~V1b))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfecepeffoeeefeeepfpfafpedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0s \in (2^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1t \in (2^{{ty\_2Erealax\_2Ereal}}) .  \\
(((p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~V0s)){\unicdcdcdhd}(p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~\\
V1t))){\unicdbdjdcd}(p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
ty\_2Erealax\_2Ereal~(ty\_2Epair\_2Eprod~ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal))~\\
(ap~(\mname{c\_2Epair\_2EUNCURRY}~ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal~\\
(ty\_2Epair\_2Eprod~ty\_2Erealax\_2Ereal~2))~({\uniddcgcg} V2x \in ty\_2Erealax\_2Ereal .  \\
({\uniddcgcg} V3y \in ty\_2Erealax\_2Ereal .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
2)~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~V2x)~V3y))~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~V2x)~V0s))~(ap~(ap~(\\
\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~V3y)~V1t))))))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfecepeffoeeefeeepfpfeejegegedf}
\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:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfecepeffoeeefeeepfpfafbecfefjebemepfpfdfffnedf}
$$\begin{array}{c}({\unicdcdadad} V0f \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1k \in \\
ty\_2Enum\_2Enum .  ((p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~(\\
ap~(c\_2Epred\_\_set\_2EGSPEC~ty\_2Erealax\_2Ereal~ty\_2Enum\_2Enum)~\\
({\uniddcgcg} V2n \in ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
2)~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~ty\_2Enum\_2Enum)~(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~\\
V1k)~V2n))~V0f))~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Enum\_2Enum)~V2n)~(\mname{c\_2Epred\_\_set\_2EUNIV}~\\
ty\_2Enum\_2Enum))))))){\unicdbdjdcd}(p~(ap~\mname{c\_2Ereal\_\_topology\_2Ebounded\_\_def}~\\
(ap~(c\_2Epred\_\_set\_2EGSPEC~ty\_2Erealax\_2Ereal~(ty\_2Epair\_2Eprod~\\
ty\_2Enum\_2Enum~ty\_2Enum\_2Enum))~(ap~(\mname{c\_2Epair\_2EUNCURRY}~ty\_2Enum\_2Enum~\\
ty\_2Enum\_2Enum~(ty\_2Epair\_2Eprod~ty\_2Erealax\_2Ereal~2))~({\uniddcgcg} V3m \in \\
ty\_2Enum\_2Enum .  ({\uniddcgcg} V4n \in ty\_2Enum\_2Enum .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Erealax\_2Ereal~2)~(ap~(ap~(\mname{c\_2Eiterate\_2ESum}~ty\_2Enum\_2Enum)~\\
(ap~(ap~\mname{c\_2Eiterate\_2E\_2E\_2E}~V3m)~V4n))~V0f))~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~\\
(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Enum\_2Enum)~V3m)~(\mname{c\_2Epred\_\_set\_2EUNIV}~\\
ty\_2Enum\_2Enum)))~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Enum\_2Enum)~V4n)~\\
(\mname{c\_2Epred\_\_set\_2EUNIV}~ty\_2Enum\_2Enum)))))))))))))\end{array}$$
\end{theorem}
\end{document}
