\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\_2ELIM\_\_ADD (TMQkuSWpugwkccaQouxhXN6BJxoy5XvXw21)}
\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\_\_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}
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}
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 $ ty\_2Ereal\_\_topology\_2Enet : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ereal\_\_topology\_2Enet}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfeogfgeh}
\end{equation}
Let $ c\_2Ereal\_\_topology\_2Enetord : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ereal\_\_topology\_2Enetord~A\_27a \in \\
(((2^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Ereal\_\_topology\_2Enet}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfeogfgehpgcheg}
\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}
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: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:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfeehchjgghjgbgmgpfpfmgjgngjgeh}
We define $\mname{c\_2Ereal\_\_topology\_2Etrivial\_\_limit}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0net \in {({{ty\_2Ereal\_\_topology\_2Enet}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1a \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2b \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1a}})}}~{V2b}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3a \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V4b \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3a}})}}~{V4b}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ereal\_\_topology\_2Enetord}~{A\_27a}})}}~{V0net}})}}~{V5x}})}}~{V3a}})}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ereal\_\_topology\_2Enetord}~{A\_27a}})}}~{V0net}})}}~{V5x}})}}~{V4b}})}})}})}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfefgghfgogehfhbgmgmgjh}
We define $\mname{c\_2Ereal\_\_topology\_2Eeventually}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0p \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1net \in {({{ty\_2Ereal\_\_topology\_2Enet}~{A\_27a}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{\mname{c\_2Ereal\_\_topology\_2Etrivial\_\_limit}}~{A\_27a}})}}~{V1net}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ereal\_\_topology\_2Enetord}~{A\_27a}})}}~{V1net}})}}~{V3x}})}}~{V2y}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{{ap}~{({{c\_2Ereal\_\_topology\_2Enetord}~{A\_27a}})}}~{V1net}})}}~{V4x}})}}~{V2y}})}})}}~{({{{ap}~{V0p}}~{V4x}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfepfcdeepfcdeepfddfe}
We define $\mname{c\_2Ereal\_\_topology\_2E\_2D\_2D\_3E}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0f \in {({{ty\_2Erealax\_2Ereal}^{{A\_27a}}})} .  {{\uniddcgcg} V1l \in {ty\_2Erealax\_2Ereal} .  {{\uniddcgcg} V2net \in {({{ty\_2Ereal\_\_topology\_2Enet}~{A\_27a}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Erealax\_2Ereal}})}}~{({{\uniddcgcg} V3e \in {ty\_2Erealax\_2Ereal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{\mname{c\_2Enum\_2E0}}})}})}}~{V3e}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ereal\_\_topology\_2Eeventually}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Erealax\_2Ereal\_\_lt}}}~{({{{ap}~{c\_2Ereal\_\_topology\_2EDist}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Erealax\_2Ereal}}~{ty\_2Erealax\_2Ereal}})}}~{({{{ap}~{V0f}}~{V4x}})}})}}~{V1l}})}})}})}}~{V3e}})}})}})}}~{V2net}})}})}})}})}}}}}$.
\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}
{({{\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} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
(2^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdcdcdhd}(p~(ap~V1Q~V2x)))){\unicdbdjded}\\
(({\unicdcdadad} V3x \in A\_27a .  (p~(ap~V0P~V3x))){\unicdcdcdhd}({\unicdcdadad} V4x \in A\_27a .  (p~(\\
ap~V1Q~V4x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfbeoeeepfpfefiene}
\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} 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} V0w \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1x \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2y \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3z \in ty\_2Erealax\_2Ereal .  \\
(((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~V0w)~V1x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
V2y)~V3z))){\unicdbdjdcd}(p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V0w)~V2y))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V1x)~V3z))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfmeefpfpfbeeeeecd}
\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} V0x \in ty\_2Erealax\_2Ereal .  ((ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
(ap~(ap~\mname{c\_2Ereal\_2E\_2F}~V0x)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))))~(ap~(ap~\\
\mname{c\_2Ereal\_2E\_2F}~V0x)~(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))))=V0x))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfcdfecffebemepfpfiebemegepfpfeepeffcemefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V1x\_27 \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V2y \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V3y\_27 \in ty\_2Erealax\_2Ereal .  \\
(p~(ap~(ap~\mname{c\_2Ereal\_2Ereal\_\_lte}~(ap~c\_2Ereal\_\_topology\_2EDist~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal)~\\
(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V0x)~V2y))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~\\
V1x\_27)~V3y\_27))))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(ap~c\_2Ereal\_\_topology\_2EDist~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal)~\\
V0x)~V1x\_27)))~(ap~c\_2Ereal\_\_topology\_2EDist~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
ty\_2Erealax\_2Ereal~ty\_2Erealax\_2Ereal)~V2y)~V3y\_27)))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfeeejedfefpfpfefcfjebeoehemefepfpfbeeeee}
\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^{{A\_27a}}) .  ({\unicdcdadad} V2net \in (ty\_2Ereal\_\_topology\_2Enet~A\_27a) .  \\
((({\unicdcdaddd} V3a \in A\_27a .  (({\unicdcdaddd} V4x \in A\_27a .  (p~(ap~(ap~(ap~(c\_2Ereal\_\_topology\_2Enetord~\\
A\_27a)~V2net)~V4x)~V3a))){\unicdcdcdhd}({\unicdcdadad} V5x \in A\_27a .  ((p~(ap~(ap~(ap~(c\_2Ereal\_\_topology\_2Enetord~\\
A\_27a)~V2net)~V5x)~V3a)){\unicdbdjdcd}(p~(ap~V0P~V5x)))))){\unicdcdcdhd}({\unicdcdaddd} V6b \in A\_27a .  \\
(({\unicdcdaddd} V7x \in A\_27a .  (p~(ap~(ap~(ap~(c\_2Ereal\_\_topology\_2Enetord~\\
A\_27a)~V2net)~V7x)~V6b))){\unicdcdcdhd}({\unicdcdadad} V8x \in A\_27a .  ((p~(ap~(ap~(ap~(c\_2Ereal\_\_topology\_2Enetord~\\
A\_27a)~V2net)~V8x)~V6b)){\unicdbdjdcd}(p~(ap~V1Q~V8x))))))){\unicdbdjdcd}({\unicdcdaddd} V9c \in A\_27a .  \\
(({\unicdcdaddd} V10x \in A\_27a .  (p~(ap~(ap~(ap~(c\_2Ereal\_\_topology\_2Enetord~\\
A\_27a)~V2net)~V10x)~V9c))){\unicdcdcdhd}({\unicdcdadad} V11x \in A\_27a .  ((p~(ap~(ap~(ap~\\
(c\_2Ereal\_\_topology\_2Enetord~A\_27a)~V2net)~V11x)~V9c)){\unicdbdjdcd}((p~(\\
ap~V0P~V11x)){\unicdcdcdhd}(p~(ap~V1Q~V11x)))))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfeoefeefpfpfeejemefenenebe}
\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} V1l \in ty\_2Erealax\_2Ereal .  ({\unicdcdadad} V2net \in (ty\_2Ereal\_\_topology\_2Enet~\\
A\_27a) .  ((p~(ap~(ap~(ap~(\mname{c\_2Ereal\_\_topology\_2E\_2D\_2D\_3E}~A\_27a)~\\
V0f)~V1l)~V2net)){\unicdbdjded}((p~(ap~(\mname{c\_2Ereal\_\_topology\_2Etrivial\_\_limit}~\\
A\_27a)~V2net)){\unicdcdcdid}({\unicdcdadad} V3e \in ty\_2Erealax\_2Ereal .  ((p~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_lt}~\\
(ap~c\_2Ereal\_2Ereal\_\_of\_\_num~\mname{c\_2Enum\_2E0}))~V3e)){\unicdbdjdcd}({\unicdcdaddd} V4y \in \\
A\_27a .  (({\unicdcdaddd} V5x \in A\_27a .  (p~(ap~(ap~(ap~(c\_2Ereal\_\_topology\_2Enetord~\\
A\_27a)~V2net)~V5x)~V4y))){\unicdcdcdhd}({\unicdcdadad} V6x \in A\_27a .  ((p~(ap~(ap~(ap~(c\_2Ereal\_\_topology\_2Enetord~\\
A\_27a)~V2net)~V6x)~V4y)){\unicdbdjdcd}(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)~\\
(ap~V0f~V6x))~V1l)))~V3e)))))))))))))\end{array}
\label{hyp:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfemejene}
\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:ehigngpfcdfechfgbgmgpfpfehpgahpgmgpghgjhpfcdfemejenepfpfbeeeee}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0net \in (ty\_2Ereal\_\_topology\_2Enet~\\
A\_27a) .  ({\unicdcdadad} V1f \in (ty\_2Erealax\_2Ereal^{{A\_27a}}) .  ({\unicdcdadad} V2g \in \\
(ty\_2Erealax\_2Ereal^{{A\_27a}}) .  ({\unicdcdadad} V3l \in ty\_2Erealax\_2Ereal .  \\
({\unicdcdadad} V4m \in ty\_2Erealax\_2Ereal .  (((p~(ap~(ap~(ap~(\mname{c\_2Ereal\_\_topology\_2E\_2D\_2D\_3E}~\\
A\_27a)~V1f)~V3l)~V0net)){\unicdcdcdhd}(p~(ap~(ap~(ap~(\mname{c\_2Ereal\_\_topology\_2E\_2D\_2D\_3E}~\\
A\_27a)~V2g)~V4m)~V0net))){\unicdbdjdcd}(p~(ap~(ap~(ap~(\mname{c\_2Ereal\_\_topology\_2E\_2D\_2D\_3E}~\\
A\_27a)~({\uniddcgcg} V5x \in A\_27a .  (ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~(ap~V1f~\\
V5x))~(ap~V2g~V5x))))~(ap~(ap~\mname{c\_2Erealax\_2Ereal\_\_add}~V3l)~V4m))~\\
V0net))))))))\end{array}$$
\end{theorem}
\end{document}
