\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\_2ELIMPT\_\_OF\_\_OPEN\_\_IN (TMYzbodmZFdYN3pg7PT2DvPmFDUKuvx5qAv)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
\begin{definition}\label{def: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:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\begin{definition}\label{def: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:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\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\_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 $ 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: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: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\_\_inv : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_inv \in ((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfjgoggh}
\end{equation}
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:dgpfcdfechfgbgmgbgihpfcdfejgoggh}
We define $\mname{c\_2Erealax\_2Einv}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_inv}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}})}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_mul \in (((ty\_2Epair\_2Eprod~ty\_2Ehreal\_2Ehreal~\\
ty\_2Ehreal\_2Ehreal)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfngfhmg}
We define $\mname{c\_2Erealax\_2Ereal\_\_mul}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_mul}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfcdfepfcdge}
We define $\mname{c\_2Ereal\_2E\_2F}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_mul}}}~{V0x}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Einv}}}~{V1y}})}})}}}$.
\end{definition}
Let $ c\_2Erealax\_2Etreal\_\_lt : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Erealax\_2Etreal\_\_lt \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Ehreal\_2Ehreal}}~{ty\_2Ehreal\_2Ehreal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgbgihpfcdfeehchfgbgmgpfpfmgeh}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgbgihpfcdfechfgbgmgpfpfmgeh}
We define $\mname{c\_2Erealax\_2Ereal\_\_lt}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1T2 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{c\_2Erealax\_2Etreal\_\_lt}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_REP}}}~{V1T2}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
\begin{definition}\label{def: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: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:dgpfcdfechfgbgmgpfcdfengjgog}
We define $\mname{c\_2Ereal\_2Emin}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ereal\_2Ereal\_\_lte}}}~{V0x}})}}~{V1y}})}})}}~{V0x}})}}~{V1y}})}}}$.
\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: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}
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\_2Ereal\_\_topology\_2EDist : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ereal\_\_topology\_2EDist \in (ty\_2Erealax\_2Ereal^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfeeejgdheh}
\end{equation}
Let $ c\_2Ereal\_2Ereal\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ereal\_2Ereal\_\_of\_\_num} \in {({{ty\_2Erealax\_2Ereal}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfcdfechfgbgmgpfpfpgggpfpfogfhng}
\end{equation}
\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:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfepeahfgog}
We define $\mname{c\_2Ereal\_\_topology\_2EOpen}$ to be ${{\uniddcgcg} V0s \in {({{2}^{{ty\_2Erealax\_2Ereal}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V1x \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{ty\_2Erealax\_2Ereal}})}}~{V1x}})}}~{V0s}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V2e \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{V2e}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V3x\_27 \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{({{{ap}~{c\_2Ereal\_\_topology\_2EDist}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}})}}~{V3x\_27}})}}~{V1x}})}})}})}}~{V2e}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{ty\_2Erealax\_2Ereal}})}}~{V3x\_27}})}}~{V0s}})}})}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ ty\_2Etopology\_2Etopology : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Etopology\_2Etopology}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeehpgahpgmgpghgjhpfcdfeehpgahpgmgpghgjh}
\end{equation}
Let $ c\_2Etopology\_2Etopology : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Etopology\_2Etopology~A\_27a \in \\
((ty\_2Etopology\_2Etopology~A\_27a)^{{({{2}^{{({{2}^{{A\_27a}}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehpgahpgmgpghgjhpfcdfeehpgahpgmgpghgjh}
\end{equation}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfefgfhdgmgjgegfgbgog}
We define $\mname{c\_2Ereal\_\_topology\_2Eeuclidean}$ to be ${({{{ap}~{({{c\_2Etopology\_2Etopology}~{ty\_2Erealax\_2Ereal}})}}~{\mname{c\_2Ereal\_\_topology\_2EOpen}}})}$.
\end{definition}
Let $ c\_2Etopology\_2Eopen\_\_in : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Etopology\_2Eopen\_\_in~A\_27a \in \\
((2^{{({{2}^{{A\_27a}}})}})^{{({{ty\_2Etopology\_2Etopology}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeehpgahpgmgpghgjhpfcdfepgahfgogpfpfjgog}
\end{equation}
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:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoeeffecf}
We define $\mname{c\_2Epred\_\_set\_2EINTER}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfedhfhcgehpgahpgmgpghgjh}
We define $\mname{c\_2Ereal\_\_topology\_2Esubtopology}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0top \in {({{ty\_2Etopology\_2Etopology}~{A\_27a}})} .  {{\uniddcgcg} V1u \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{c\_2Etopology\_2Etopology}~{A\_27a}})}}~{({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{2}^{{A\_27a}}})}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{2}^{{A\_27a}}})}}~{2}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINTER}}~{A\_27a}})}}~{V2s}})}}~{V1u}})}})}}~{({{{ap}~{({{{ap}~{({{c\_2Etopology\_2Eopen\_\_in}~{A\_27a}})}}~{V0top}})}}~{V2s}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfemgjgngjgehpfpfahpgjgogehpfpfpggg}
We define $\mname{c\_2Ereal\_\_topology\_2Elimit\_\_point\_\_of}$ to be ${{\uniddcgcg} V0x \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V1s \in {({{2}^{{ty\_2Erealax\_2Ereal}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{ty\_2Erealax\_2Ereal}}})}})}}~{({{\uniddcgcg} V2t \in {({{2}^{{ty\_2Erealax\_2Ereal}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{ty\_2Erealax\_2Ereal}})}}~{V0x}})}}~{V2t}})}})}}~{({{{ap}~{\mname{c\_2Ereal\_\_topology\_2EOpen}}}~{V2t}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V3y \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Erealax\_2Ereal}})}}~{V3y}})}}~{V0x}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{ty\_2Erealax\_2Ereal}})}}~{V3y}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{ty\_2Erealax\_2Ereal}})}}~{V3y}})}}~{V2t}})}})}})}})}})}})}})}})}}}$.
\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}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} 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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0b \in 2 .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V2g \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V3x \in A\_27a .  ((ap~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(A\_27b^{{A\_27a}}))~\\
V0b)~V1f)~V2g)~V3x)=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27b)~V0b)~(ap~\\
V1f~V3x))~(ap~V2g~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfcfbeefpecf}
\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 (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1b \in 2 .  ({\unicdcdadad} V2x \in A\_27a .  \\
({\unicdcdadad} V3y \in A\_27a .  ((ap~V0f~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\\
V1b)~V2x)~V3y))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27b)~V1b)~(ap~V0f~\\
V2x))~(ap~V0f~V3y)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfcfbeoeee}
\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 .  (((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} V0d \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~\\
V0d)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))))){\unicdbdjded}(p~(\\
ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~\\
V0d))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfiebemegebd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0d \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V0d)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))))~V0d)){\unicdbdjded}\\
(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V0d))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfiebemegecd}
\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~(ap~\mname{c\_2Ereal\_2Emin}~V0x)~V1y))~\\
V0x))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfnejeoepfpfmefebd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0u \in (2^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1s \in (2^{{ty\_2Erealax\_2Ereal}}) .  \\
((p~(ap~(ap~(c\_2Etopology\_2Eopen\_\_in~ty\_2Erealax\_2Ereal)~(ap~\\
(ap~(\mname{c\_2Ereal\_\_topology\_2Esubtopology}~ty\_2Erealax\_2Ereal)~\\
\mname{c\_2Ereal\_\_topology\_2Eeuclidean})~V0u))~V1s)){\unicdbdjded}((p~(ap~(ap~(\mname{c\_2Epred\_\_set\_2ESUBSET}~\\
ty\_2Erealax\_2Ereal)~V1s)~V0u)){\unicdcdcdhd}({\unicdcdadad} V2x \in ty\_2Erealax\_2Ereal .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~V2x)~V1s)){\unicdbdjdcd}({\unicdcdaddd} V3e \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\\
\mname{c\_2Enum\_2E0}))~V3e)){\unicdcdcdhd}({\unicdcdadad} V4x\_27 \in ty\_2Erealax\_2Ereal .  ((\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~V4x\_27)~V0u)){\unicdcdcdhd}(\\
p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~c\_2Ereal\_\_topology\_2EDist~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal)~\\
V4x\_27)~V2x)))~V3e))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~\\
V4x\_27)~V1s))))))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfepgahfgogpfpfjgog}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1s \in (2^{{ty\_2Erealax\_2Ereal}}) .  \\
((p~(ap~(ap~\mname{c\_2Ereal\_\_topology\_2Elimit\_\_point\_\_of}~V0x)~V1s)){\unicdbdjded}\\
({\unicdcdadad} V2e \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V2e)){\unicdbdjdcd}({\unicdcdaddd} V3x\_27 \in \\
ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~\\
V3x\_27)~V1s)){\unicdcdcdhd}((\preoh (V3x\_27=V0x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_\_topology\_2EDist~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~\\
ty\_2Erealax\_2Ereal)~V3x\_27)~V0x)))~V2e))))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfemejeneafefpfpfbeafafcfpebedeiebecemefe}
\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}
\begin{array}{c}({\unicdcdadad} V0p \in 2 .  ({\unicdcdadad} V1q \in 2 .  ({\unicdcdadad} V2r \in 2 .  ({\unicdcdadad} V3s \in \\
2 .  (((p~V0p){\unicdbdjded}(p~(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~2)~V1q)~V2r)~V3s))){\unicdbdjded}\\
(((p~V0p){\unicdcdcdid}((p~V1q){\unicdcdcdid}(\preoh (p~V3s)))){\unicdcdcdhd}(((p~V0p){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V1q)))){\unicdcdcdhd}\\
(((p~V0p){\unicdcdcdid}((\preoh (p~V2r)){\unicdcdcdid}(\preoh (p~V3s)))){\unicdcdcdhd}(((\preoh (p~V1q)){\unicdcdcdid}((p~V2r){\unicdcdcdid}(\preoh (\\
p~V0p)))){\unicdcdcdhd}((p~V1q){\unicdcdcdid}((p~V3s){\unicdcdcdid}(\preoh (p~V0p)))))))))))))\end{array}
\label{hyp:ehigngpfcdfedhbgehpfcdfeegdgpfpfdgpgogeg}
\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:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfemejeneafefpfpfpegepfpfpeaffeoepfpfjeoe}
$$\begin{array}{c}({\unicdcdadad} V0s \in (2^{{ty\_2Erealax\_2Ereal}}) .  ({\unicdcdadad} V1t \in (2^{{ty\_2Erealax\_2Ereal}}) .  \\
({\unicdcdadad} V2x \in ty\_2Erealax\_2Ereal .  (((p~(ap~(ap~(c\_2Etopology\_2Eopen\_\_in~\\
ty\_2Erealax\_2Ereal)~(ap~(ap~(\mname{c\_2Ereal\_\_topology\_2Esubtopology}~\\
ty\_2Erealax\_2Ereal)~\mname{c\_2Ereal\_\_topology\_2Eeuclidean})~V0s))~\\
V1t)){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Ereal\_\_topology\_2Elimit\_\_point\_\_of}~V2x)~\\
V0s)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Erealax\_2Ereal)~V2x)~V1t)))){\unicdbdjdcd}\\
(p~(ap~(ap~\mname{c\_2Ereal\_\_topology\_2Elimit\_\_point\_\_of}~V2x)~V1t))))))\end{array}$$
\end{theorem}
\end{document}
