\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{thm\_2Etransc\_2EDIVISION\_\_APPEND (TMaRHcDnDDW4QtGZLcV7dnHc4U4rYiqAgyd)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Earithmetic\_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}
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:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
Let $ ty\_2Erealax\_2Ereal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Erealax\_2Ereal}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgbgihpfcdfechfgbgmg}
\end{equation}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Etransc\_2Etdiv : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Etransc\_2Etdiv \in ((2^{{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}}~{({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehchbgogdhdgpfcdfeehegjggh}
\end{equation}
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\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\_\_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}
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: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\_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}
\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: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\_2Etransc\_2Efine : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Etransc\_2Efine \in ((2^{{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}}~{({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}})}})^{{({{ty\_2Erealax\_2Ereal}^{{ty\_2Erealax\_2Ereal}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehchbgogdhdgpfcdfeggjgogfg}
\end{equation}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfddfe}
We define $\mname{c\_2Earithmetic\_2E\_3E}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V1n}})}}~{V0m}})}}}$.
\end{definition}
\begin{definition}\label{def: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}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfepfdddepfddee}
We define $\mname{c\_2Earithmetic\_2E\_3C\_3D}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V0m}})}}~{V1n}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Enum\_2Enum}})}}~{V0m}})}}~{V1n}})}})}}}$.
\end{definition}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
Let $ 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:dgpfcdfeehchbgogdhdgpfcdfeegdhjgkhfg}
We define $\mname{c\_2Etransc\_2Edsize}$ to be ${{\uniddcgcg} V0D \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1N \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2n}})}}~{V1N}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{({{{ap}~{V0D}}~{V2n}})}})}}~{({{{ap}~{V0D}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V2n}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Earithmetic\_2E\_3E\_3D}}}~{V3n}})}}~{V1N}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{V0D}}~{V3n}})}})}}~{({{{ap}~{V0D}}~{V1N}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Epair\_2EABS\_\_prod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Etransc\_2Edivision : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Etransc\_2Edivision \in ((2^{{({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehchbgogdhdgpfcdfeegjgghjgdhjgpgog}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
\mname{c\_2Enum\_2E0})~V0m)=\mname{c\_2Enum\_2E0})){\unicdcdcdhd}({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V2n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~(\\
ap~\mname{c\_2Enum\_2ESUC}~V1m))~V2n)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~ty\_2Enum\_2Enum)~\\
(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1m)~V2n))~\mname{c\_2Enum\_2E0})~(ap~\mname{c\_2Enum\_2ESUC}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~V1m)~V2n)))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffce}
\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 .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  (((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V0m)~V1n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1n)~V2p))){\unicdbdjdcd}(p~(ap~(ap~\\
\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\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} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0m))~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfpecf}
\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 .  ((\mname{c\_2Enum\_2E0}=V0m){\unicdcdcdid}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V0m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfadpfpfdebedffedf}
\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\_2Earithmetic\_2E\_3C\_3D}~V0m)~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfbeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~(ap~\mname{c\_2Enum\_2ESUC}~V0m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfdfffdepfpfcffegeme}
\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 .  (((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
\mname{c\_2Enum\_2E0})~V0m)=\mname{c\_2Enum\_2E0}){\unicdcdcdhd}((ap~(ap~c\_2Earithmetic\_2E\_2D~\\
V0m)~\mname{c\_2Enum\_2E0})=V0m)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(\\
ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V2p))~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1n)~V2p))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V1n))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfnepeoepepfpfbeeeeepfpffebf}
\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}~\\
V0m)~V1n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1n)~V2p))){\unicdbdjdcd}(p~(ap~(ap~\\
\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0m)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfmefedfdfpfpfefcfbeoedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~V0m)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpfcffegeme}
\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)){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfjeneafpfpfmefedfdfpfpfpecfpfpffebf}
\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}~V0n)~V1m)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1m)~V0n))){\unicdbdjdcd}(V0n=V1m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfffbemepfpfbeoeefjedfjfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0a \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1c \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~(ap~(ap~c\_2Earithmetic\_2E\_2B~V0a)~\\
V1c))~V1c)=V0a)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfdfffce}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0c \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~V0c)~\\
V0c)=\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpffebfffbemepfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0n)~V1m)){\unicdbdjdcd}({\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  \\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0n)~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1m)~V2p)))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfjeneafpfpfmefedfdfpfpfbeeeee}
\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\_3E\_3D}~V0n)~V1m)){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V1m)~V0n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfehecffebeeffecfpfpffebf}
\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\_2Earithmetic\_2E\_3C\_3D}~V0m)~V1n)){\unicdbdjded}({\unicdcdaddd} V2p \in ty\_2Enum\_2Enum .  \\
(V1n=(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V2p))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpffebfpfpffeifjedfefdf}
\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)~(ap~(ap~c\_2Earithmetic\_2E\_2D~V1n)~V2p))){\unicdbdjded}((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V2p))~V1n)){\unicdcdcdid}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~\\
V0m)~\mname{c\_2Enum\_2E0})))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpfmefegeefpfpfmefedfdfpfpffebf}
\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\_3E\_3D}~\\
V0m)~(ap~(ap~c\_2Earithmetic\_2E\_2D~V1n)~V2p))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V2p))~V1n))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffcepfpfmefegeefpfpfhecffebeeffecfpfpffebf}
\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} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} 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}}) .  ({\unicdcdadad} V1x \in \\
A\_27a .  (({\unicdcdadad} V2y \in A\_27a .  ((p~(ap~V0P~V2y)){\unicdbdjded}(V2y=V1x))){\unicdbdjdcd}((ap~\\
(\mname{c\_2Emin\_2E\_40}~A\_27a)~V0P)=V1x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedffemefedeefpfpfffoejebffffe}
\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} 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} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0n)~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfdfffdepfpfcffegeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  (\preoh (p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V0x)~V0x))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfcffegeme}
\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} V0a \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1b \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V3p \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ((p~(ap~(ap~c\_2Etransc\_2Etdiv~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal)~\\
V0a)~V1b))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V2D)~V3p))){\unicdbdjded}((p~(ap~(ap~\\
c\_2Etransc\_2Edivision~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V0a)~V1b))~V2D)){\unicdcdcdhd}({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~V2D~V4n))~(ap~V3p~V4n))){\unicdcdcdhd}(p~\\
(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~V3p~V4n))~(ap~V2D~(ap~\mname{c\_2Enum\_2ESUC}~\\
V4n))))))))))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeehegjggh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0g \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1D \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V2p \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  \\
((p~(ap~(ap~c\_2Etransc\_2Efine~V0g)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V1D)~V2p))){\unicdbdjded}({\unicdcdadad} V3n \in \\
ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V3n)~(ap~\mname{c\_2Etransc\_2Edsize}~\\
V1D))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_sub}~\\
(ap~V1D~(ap~\mname{c\_2Enum\_2ESUC}~V3n)))~(ap~V1D~V3n)))~(ap~V0g~(ap~V2p~V3n))))))))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeggjgogfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ((p~(ap~\\
(ap~c\_2Etransc\_2Edivision~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V1a)~V2b))~V0D)){\unicdbdjdcd}((ap~V0D~\mname{c\_2Enum\_2E0})=V1a)))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpfmeiedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ((p~(ap~\\
(ap~c\_2Etransc\_2Edivision~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V1a)~V2b))~V0D)){\unicdbdjded}(((ap~V0D~\mname{c\_2Enum\_2E0})=\\
V1a){\unicdcdcdhd}(({\unicdcdadad} V3n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V3n)~(ap~\mname{c\_2Etransc\_2Edsize}~V0D))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~V0D~V3n))~(ap~V0D~(ap~\mname{c\_2Enum\_2ESUC}~V3n)))))){\unicdcdcdhd}({\unicdcdadad} V4n \in \\
ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Earithmetic\_2E\_3E\_3D}~V4n)~(ap~\mname{c\_2Etransc\_2Edsize}~\\
V0D))){\unicdbdjdcd}((ap~V0D~V4n)=V2b)))))))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ((p~(ap~\\
(ap~c\_2Etransc\_2Edivision~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V1a)~V2b))~V0D)){\unicdbdjdcd}((ap~V0D~(ap~\mname{c\_2Etransc\_2Edsize}~\\
V0D))=V2b)))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpfcfiedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3m \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  (((p~(ap~(ap~c\_2Etransc\_2Edivision~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal)~\\
V1a)~V2b))~V0D)){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V3m)~V4n)){\unicdcdcdhd}(p~\\
(ap~(ap~\mname{c\_2Earithmetic\_2E\_3C\_3D}~V4n)~(ap~\mname{c\_2Etransc\_2Edsize}~V0D))))){\unicdbdjdcd}\\
(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~V0D~V3m))~(ap~V0D~V4n)))))))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpfmeefpfpfhefeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ((p~(ap~\\
(ap~c\_2Etransc\_2Edivision~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V1a)~V2b))~V0D)){\unicdbdjdcd}((V1a=V2b){\unicdbdjded}((ap~\mname{c\_2Etransc\_2Edsize}~\\
V0D)=\mname{c\_2Enum\_2E0}))))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ((p~(ap~\\
(ap~c\_2Etransc\_2Edivision~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V1a)~V2b))~V0D)){\unicdbdjdcd}({\unicdcdadad} V3r \in ty\_2Enum\_2Enum .  \\
(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~V1a)~(ap~V0D~V3r))))))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpfmecepeffoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0D \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3n \in \\
ty\_2Enum\_2Enum .  (((p~(ap~(ap~c\_2Etransc\_2Edivision~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal)~V1a)~V2b))~V0D)){\unicdcdcdhd}(p~(\\
ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V3n)~(ap~\mname{c\_2Etransc\_2Edsize}~V0D)))){\unicdbdjdcd}\\
(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~V0D~V3n))~V2b)))))))\end{array}
\label{hyp:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpfffcepeffoeeepfpfmeef}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeehchbgogdhdgpfcdfeeejegfjedfjepeoepfpfbeafaffeoeee}
$$\begin{array}{c}({\unicdcdadad} V0g \in (ty\_2Erealax\_2Ereal^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1a \in \\
ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2b \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3c \in \\
ty\_2Erealax\_2Ereal .  ((({\unicdcdaddd} V4D1 \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  \\
({\unicdcdaddd} V5p1 \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ((p~(ap~(\\
ap~c\_2Etransc\_2Etdiv~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V1a)~V2b))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V4D1)~V5p1))){\unicdcdcdhd}(p~(ap~(ap~\\
c\_2Etransc\_2Efine~V0g)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V4D1)~V5p1)))))){\unicdcdcdhd}({\unicdcdaddd} V6D2 \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdaddd} V7p2 \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  \\
((p~(ap~(ap~c\_2Etransc\_2Etdiv~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V2b)~V3c))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V6D2)~V7p2))){\unicdcdcdhd}(p~(ap~(ap~\\
c\_2Etransc\_2Efine~V0g)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V6D2)~V7p2))))))){\unicdbdjdcd}({\unicdcdaddd} V8D \in \\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  ({\unicdcdaddd} V9p \in (ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}) .  \\
((p~(ap~(ap~c\_2Etransc\_2Etdiv~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V1a)~V3c))~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V8D)~V9p))){\unicdcdcdhd}(p~(ap~(ap~c\_2Etransc\_2Efine~\\
V0g)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}})~\\
(ty\_2Erealax\_2Ereal^{{ty\_2Enum\_2Enum}}))~V8D)~V9p)))))))))))\end{array}$$
\end{theorem}
\end{document}
