\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\_2Emeasure\_2EIN\_\_MEASURABLE\_\_BOREL\_\_LT (TMPLg6a3BVqMGwF4naH5VXTwNNHuVjyeK6A)}
\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:dgpfcdfedgpgngcgjgogpfcdfedf}
We define $\mname{c\_2Ecombin\_2ES}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1g \in {({{A\_27b}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2x}})}}~{({{{ap}~{V1g}}~{V2x}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfede}
We define $\mname{c\_2Ecombin\_2EC}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {({{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {({{\uniddcgcg} V1x \in {A\_27b} .  {({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{V0f}}~{V2y}})}}~{V1x}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
Let $ ty\_2Eextreal\_2Eextreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eextreal\_2Eextreal}}\label{hyp:ogfgpfehjhpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmg}
\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\_2Eextreal\_2ENormal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Eextreal\_2ENormal} \in {({{ty\_2Eextreal\_2Eextreal}^{{ty\_2Erealax\_2Ereal}}})}}\label{hyp:ngfgngpfdgpfcdfefgihehchfgbgmgpfcdfeoepgchngbgmg}
\end{equation}
Let $ c\_2Eextreal\_2EPosInf : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Eextreal\_2EPosInf} \in {ty\_2Eextreal\_2Eextreal}}\label{hyp:ngfgngpfdgpfcdfefgihehchfgbgmgpfcdfeafpgdhjeoggg}
\end{equation}
Let $ c\_2Eextreal\_2ENegInf : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Eextreal\_2ENegInf} \in {ty\_2Eextreal\_2Eextreal}}\label{hyp:ngfgngpfdgpfcdfefgihehchfgbgmgpfcdfeoefghgjeoggg}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfejeoe}
We define $\mname{c\_2Ebool\_2EIN}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1f \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V1f}}~{V0x}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def: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 $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejegf}
We define $\mname{c\_2Epred\_\_set\_2EUNIV}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}}$.
\end{definition}
\begin{definition}\label{def: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:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoeke}
We define $\mname{c\_2Epred\_\_set\_2EINJ}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V2t \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V3x}})}})}}~{V2t}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V4x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V5y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V4x}})}}~{V1s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V5y}})}}~{V1s}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V4x}})}})}}~{({{{ap}~{V0f}}~{V5y}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V4x}})}}~{V5y}})}})}})}})}})}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfedgpgfhogehbgcgmgfg}
We define $\mname{c\_2Epred\_\_set\_2Ecountable}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{ty\_2Enum\_2Enum}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1f \in {({{ty\_2Enum\_2Enum}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epred\_\_set\_2EINJ}}~{A\_27a}}~{ty\_2Enum\_2Enum}})}}~{V1f}})}}~{V0s}})}}~{({{\mname{c\_2Epred\_\_set\_2EUNIV}}~{ty\_2Enum\_2Enum}})}})}})}})}}}$.
\end{definition}
Let $ ty\_2Epair\_2Eprod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Emeasure\_2Esubsets : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Emeasure\_2Esubsets~A\_27a \in (\\
(2^{{({{2}^{{A\_27a}}})}})^{{({{{ty\_2Epair\_2Eprod}~{({{2}^{{A\_27a}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfengfgbgdhfhchfgpfcdfedhfhcgdhfgehdh}
\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}
Let $ c\_2Epair\_2EABS\_\_prod : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
Let $ c\_2Epred\_\_set\_2EGSPEC : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b \in ((2^{{A\_27a}})^{{({{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{2}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffede}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeffoejepeoe}
We define $\mname{c\_2Epred\_\_set\_2EUNION}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Emeasure\_2Espace : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Emeasure\_2Espace~A\_27a \in ((2^{{A\_27a}})^{{({{{ty\_2Epair\_2Eprod}~{({{2}^{{A\_27a}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfengfgbgdhfhchfgpfcdfedhahbgdgfg}
\end{equation}
\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: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:dgpfcdfeahchfgegpfpfdhfgehpfcdfeeejegege}
We define $\mname{c\_2Epred\_\_set\_2EDIFF}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0s \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V0s}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1t}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfefeneafefjf}
We define $\mname{c\_2Epred\_\_set\_2EEMPTY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2EF}}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengfgbgdhfhchfgpfcdfedhfhcgdhfgehpfpfdgmgbgdhdh}
We define $\mname{c\_2Emeasure\_2Esubset\_\_class}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0sp \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1sts \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2x \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{V2x}})}}~{V1sts}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{A\_27a}})}}~{V2x}})}}~{V0sp}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengfgbgdhfhchfgpfcdfebgmghgfgcgchbg}
We define $\mname{c\_2Emeasure\_2Ealgebra}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0a \in {({{{ty\_2Epair\_2Eprod}~{({{2}^{{A\_27a}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emeasure\_2Esubset\_\_class}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Emeasure\_2Espace}~{A\_27a}})}}~{V0a}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{({{\mname{c\_2Epred\_\_set\_2EEMPTY}}~{A\_27a}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{V1s}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EDIFF}}~{A\_27a}})}}~{({{{ap}~{({{c\_2Emeasure\_2Espace}~{A\_27a}})}}~{V0a}})}})}}~{V1s}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V3t \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{V2s}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{V3t}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{A\_27a}})}}~{V2s}})}}~{V3t}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}})}})}})}})}})}})}})}}}$.
\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:dgpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfpgggpfpfogfhng}
We define $\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Eextreal\_2ENormal}}~{({{{ap}~{c\_2Ereal\_2Ereal\_\_of\_\_num}}~{V0n}})}})}}$.
\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 $ 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}
Let $ c\_2Eextreal\_2Eextreal\_\_le : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Eextreal\_2Eextreal\_\_le \in ((2^{{ty\_2Eextreal\_2Eextreal}})^{{ty\_2Eextreal\_2Eextreal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfmgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfmgeh}
We define $\mname{c\_2Eextreal\_2Eextreal\_\_lt}$ to be ${{\uniddcgcg} V0x \in {ty\_2Eextreal\_2Eextreal} .  {{\uniddcgcg} V1y \in {ty\_2Eextreal\_2Eextreal} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{c\_2Eextreal\_2Eextreal\_\_le}}~{V1y}})}}~{V0x}})}})}}}$.
\end{definition}
Let $ c\_2Eextreal\_2Eextreal\_\_inv : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Eextreal\_2Eextreal\_\_inv \in (ty\_2Eextreal\_2Eextreal^{{ty\_2Eextreal\_2Eextreal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfjgoggh}
\end{equation}
Let $ c\_2Eextreal\_2Eextreal\_\_mul : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Eextreal\_2Eextreal\_\_mul \in ((ty\_2Eextreal\_2Eextreal^{{ty\_2Eextreal\_2Eextreal}})^{{ty\_2Eextreal\_2Eextreal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfngfhmg}
\end{equation}
\begin{definition}\label{def:dgpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfegjggh}
We define $\mname{c\_2Eextreal\_2Eextreal\_\_div}$ to be ${{\uniddcgcg} V0x \in {ty\_2Eextreal\_2Eextreal} .  {{\uniddcgcg} V1y \in {ty\_2Eextreal\_2Eextreal} .  {({{{ap}~{({{{ap}~{c\_2Eextreal\_2Eextreal\_\_mul}}~{V0x}})}}~{({{{ap}~{c\_2Eextreal\_2Eextreal\_\_inv}}~{V1y}})}})}}}$.
\end{definition}
Let $ c\_2Eextreal\_2Eextreal\_\_ainv : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Eextreal\_2Eextreal\_\_ainv \in (ty\_2Eextreal\_2Eextreal^{{ty\_2Eextreal\_2Eextreal}})\end{array}
\label{hyp:ngfgngpfdgpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfbgjgoggh}
\end{equation}
\begin{definition}\label{def:dgpfcdfefgihehchfgbgmgpfcdfebfpfpfdhfgeh}
We define $\mname{c\_2Eextreal\_2EQ\_\_set}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EUNION}}~{ty\_2Eextreal\_2Eextreal}})}}~{({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{ty\_2Eextreal\_2Eextreal}}~{ty\_2Eextreal\_2Eextreal}})}}~{({{\uniddcgcg} V0x \in {ty\_2Eextreal\_2Eextreal} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Eextreal\_2Eextreal}}~{2}})}}~{V0x}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V1a \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2b \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Eextreal\_2Eextreal}})}}~{V0x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_div}}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{V1a}})}})}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{V2b}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_lt}}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{V2b}})}})}})}})}})}})}})}})}})}})}})}}~{({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{ty\_2Eextreal\_2Eextreal}}~{ty\_2Eextreal\_2Eextreal}})}}~{({{\uniddcgcg} V3x \in {ty\_2Eextreal\_2Eextreal} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Eextreal\_2Eextreal}}~{2}})}}~{V3x}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V4a \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V5b \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Eextreal\_2Eextreal}})}}~{V3x}})}}~{({{{ap}~{c\_2Eextreal\_2Eextreal\_\_ainv}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_div}}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{V4a}})}})}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{V5b}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_lt}}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{\mname{c\_2Enum\_2E0}}})}})}}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_of\_\_num}}}~{V5b}})}})}})}})}})}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfecejeheffoejepeoe}
We define $\mname{c\_2Epred\_\_set\_2EBIGUNION}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0P \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V1x}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{V2s}})}}~{V0P}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V1x}})}}~{V2s}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengfgbgdhfhchfgpfcdfedhjghgngbgpfpfbgmghgfgcgchbg}
We define $\mname{c\_2Emeasure\_2Esigma\_\_algebra}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0a \in {({{{ty\_2Epair\_2Eprod}~{({{2}^{{A\_27a}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Emeasure\_2Ealgebra}}~{A\_27a}})}}~{V0a}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V1c \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2Ecountable}}~{({{2}^{{A\_27a}}})}})}}~{V1c}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{({{2}^{{A\_27a}}})}})}}~{V1c}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EBIGUNION}}~{A\_27a}})}}~{V1c}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfejenebehefe}
We define $\mname{c\_2Epred\_\_set\_2EIMAGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27b}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27b}}~{2}})}}~{({{{ap}~{V0f}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2x}})}}~{V1s}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfecejehejeoeeffecf}
We define $\mname{c\_2Epred\_\_set\_2EBIGINTER}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0P \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V1x}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{V2s}})}}~{V0P}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V1x}})}}~{V2s}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengfgbgdhfhchfgpfcdfedhjghgngbg}
We define $\mname{c\_2Emeasure\_2Esigma}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0sp \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1st \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{2}^{{A\_27a}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}}~{V0sp}})}}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EBIGINTER}}~{({{2}^{{A\_27a}}})}})}}~{({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{2}^{{({{2}^{{A\_27a}}})}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}}~{({{\uniddcgcg} V2s \in {({{2}^{{({{2}^{{A\_27a}}})}}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}}~{2}})}}~{V2s}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2ESUBSET}}~{({{2}^{{A\_27a}}})}})}}~{V1st}})}}~{V2s}})}})}}~{({{{ap}~{({{\mname{c\_2Emeasure\_2Esigma\_\_algebra}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{2}^{{A\_27a}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})}}~{V0sp}})}}~{V2s}})}})}})}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengfgbgdhfhchfgpfcdfecepgchfgmg}
We define $\mname{c\_2Emeasure\_2EBorel}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emeasure\_2Esigma}}~{ty\_2Eextreal\_2Eextreal}})}}~{({{\mname{c\_2Epred\_\_set\_2EUNIV}}~{ty\_2Eextreal\_2Eextreal}})}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epred\_\_set\_2EIMAGE}}~{ty\_2Eextreal\_2Eextreal}}~{({{2}^{{ty\_2Eextreal\_2Eextreal}}})}})}}~{({{\uniddcgcg} V0a \in {ty\_2Eextreal\_2Eextreal} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{ty\_2Eextreal\_2Eextreal}}~{ty\_2Eextreal\_2Eextreal}})}}~{({{\uniddcgcg} V1x \in {ty\_2Eextreal\_2Eextreal} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{ty\_2Eextreal\_2Eextreal}}~{2}})}}~{V1x}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eextreal\_2Eextreal\_\_lt}}}~{V1x}})}}~{V0a}})}})}})}})}})}})}}~{({{\mname{c\_2Epred\_\_set\_2EUNIV}}~{ty\_2Eextreal\_2Eextreal}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahchfgegpfpfdhfgehpfcdfeafcffejenebehefe}
We define $\mname{c\_2Epred\_\_set\_2EPREIMAGE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2x}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{({{{ap}~{V0f}}~{V2x}})}})}}~{V1s}})}})}})}})}}}}}$.
\end{definition}
\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:dgpfcdfeahchfgegpfpfdhfgehpfcdfegeffoedffeef}
We define $\mname{c\_2Epred\_\_set\_2EFUNSET}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {{\uniddcgcg} V1Q \in {({{2}^{{A\_27b}}})} .  {({{\uniddcgcg} V2f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V3x}})}}~{V0P}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27b}})}}~{({{{ap}~{V2f}}~{V3x}})}})}}~{V1Q}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengfgbgdhfhchfgpfcdfengfgbgdhfhchbgcgmgfg}
We define $\mname{c\_2Emeasure\_2Emeasurable}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0a \in {({{{ty\_2Epair\_2Eprod}~{({{2}^{{A\_27a}}})}}~{({{2}^{{({{2}^{{A\_27a}}})}}})}})} .  {{\uniddcgcg} V1b \in {({{{ty\_2Epair\_2Eprod}~{({{2}^{{A\_27b}}})}}~{({{2}^{{({{2}^{{A\_27b}}})}}})}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{({{A\_27b}^{{A\_27a}}})}}~{({{A\_27b}^{{A\_27a}}})}})}}~{({{\uniddcgcg} V2f \in {({{A\_27b}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{({{A\_27b}^{{A\_27a}}})}}~{2}})}}~{V2f}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Emeasure\_2Esigma\_\_algebra}}~{A\_27a}})}}~{V0a}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Emeasure\_2Esigma\_\_algebra}}~{A\_27b}})}}~{V1b}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{A\_27b}^{{A\_27a}}})}})}}~{V2f}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epred\_\_set\_2EFUNSET}}~{A\_27a}}~{A\_27b}})}}~{({{{ap}~{({{c\_2Emeasure\_2Espace}~{A\_27a}})}}~{V0a}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Espace}~{A\_27b}})}}~{V1b}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{({{2}^{{A\_27b}}})}})}}~{({{\uniddcgcg} V3s \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27b}}})}})}}~{V3s}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27b}})}}~{V1b}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{({{2}^{{A\_27a}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Epred\_\_set\_2EINTER}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Epred\_\_set\_2EPREIMAGE}}~{A\_27a}}~{A\_27b}})}}~{V2f}})}}~{V3s}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Espace}~{A\_27a}})}}~{V0a}})}})}})}}~{({{{ap}~{({{c\_2Emeasure\_2Esubsets}~{A\_27a}})}}~{V0a}})}})}})}})}})}})}})}})}})}})}})}}}}}$.
\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} 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}
{({{\unicdcdadad} V0t \in {2} .  {({{({\preoh{{({{p}~{V0t}})}}})}{\unicdbdjdcd}{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepfpfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} A\_27a .  {{{nonempty}~{A\_27a}}{\unicdbdjdcd}{({{\unicdcdadad} V0x \in {A\_27a} .  {({{V0x}={V0x}})}})}}}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfcffegeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1Q \in \\
2 .  ((({\unicdcdadad} V2x \in A\_27a .  (p~(ap~V0P~V2x))){\unicdcdcdhd}(p~V1Q)){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~(ap~V0P~V3x)){\unicdcdcdhd}(p~V1Q))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfbeoeeepfpfgepecfbememepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in 2 .  ({\unicdcdadad} V1Q \in (\\
2^{{A\_27a}}) .  (((p~V0P){\unicdcdcdhd}({\unicdcdadad} V2x \in A\_27a .  (p~(ap~V1Q~V2x)))){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~V0P){\unicdcdcdhd}(p~(ap~V1Q~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfbeoeeepfpfgepecfbememepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0Q \in 2 .  ({\unicdcdadad} V1P \in (\\
2^{{A\_27a}}) .  (({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~V1P~V2x)){\unicdcdcdid}(p~V0Q))){\unicdbdjded}(({\unicdcdadad} V3x \in \\
A\_27a .  (p~(ap~V1P~V3x))){\unicdcdcdid}(p~V0Q)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfgepecfbememepfpfpecfpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  ({\unicdcdadad} V2C \in 2 .  ((((p~V1B){\unicdcdcdhd}\\
(p~V2C)){\unicdcdcdid}(p~V0A)){\unicdbdjded}(((p~V1B){\unicdcdcdid}(p~V0A)){\unicdcdcdhd}((p~V2C){\unicdcdcdid}(p~V0A)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpfpecfpfpfpegffecfpfpfbeoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1x\_27 \in 2 .  ({\unicdcdadad} V2y \in 2 .  ({\unicdcdadad} V3y\_27 \in \\
2 .  ((((p~V0x){\unicdbdjded}(p~V1x\_27)){\unicdcdcdhd}((p~V1x\_27){\unicdbdjdcd}((p~V2y){\unicdbdjded}(p~V3y\_27)))){\unicdbdjdcd}\\
(((p~V0x){\unicdbdjdcd}(p~V2y)){\unicdbdjded}((p~V1x\_27){\unicdbdjdcd}(p~V3y\_27))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Eextreal\_2Eextreal .  ((V0x=c\_2Eextreal\_2ENegInf){\unicdcdcdid}\\
((V0x=c\_2Eextreal\_2EPosInf){\unicdcdcdid}({\unicdcdaddd} V1r \in ty\_2Erealax\_2Ereal .  \\
(V0x=(ap~c\_2Eextreal\_2ENormal~V1r))))))\end{array}
\label{hyp:ehigngpfcdfefgihehchfgbgmgpfcdfefgihehchfgbgmgpfpfdgbgdhfgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Eextreal\_2Eextreal .  ({\unicdcdadad} V1y \in ty\_2Eextreal\_2Eextreal .  \\
({\unicdcdadad} V2z \in ty\_2Eextreal\_2Eextreal .  (((p~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~\\
V0x)~V1y)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~V1y)~V2z))){\unicdbdjdcd}(\\
p~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~V0x)~V2z))))))\end{array}
\label{hyp:ehigngpfcdfefgihehchfgbgmgpfcdfemgehpfpfehchbgogdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0r \in ty\_2Eextreal\_2Eextreal .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Eextreal\_2Eextreal)~V0r)~\mname{c\_2Eextreal\_2EQ\_\_set})){\unicdbdjdcd}((\preoh (V0r=\\
c\_2Eextreal\_2ENegInf)){\unicdcdcdhd}(\preoh (V0r=c\_2Eextreal\_2EPosInf)))))\end{array}
\label{hyp:ehigngpfcdfefgihehchfgbgmgpfcdfechbgehpfpfogpgehpfpfjgogggehjh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Eextreal\_2Eextreal .  ({\unicdcdadad} V1y \in ty\_2Eextreal\_2Eextreal .  \\
((p~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~V0x)~V1y)){\unicdbdjdcd}({\unicdcdaddd} V2r \in \\
ty\_2Eextreal\_2Eextreal .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Eextreal\_2Eextreal)~\\
V2r)~\mname{c\_2Eextreal\_2EQ\_\_set})){\unicdcdcdhd}((p~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~\\
V0x)~V2r)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~V2r)~V1y))))))))\end{array}
\label{hyp:ehigngpfcdfefgihehchfgbgmgpfcdfebfpfpfeefeoedffepfpfjeoepfpfcf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in (ty\_2Epair\_2Eprod~\\
(2^{{A\_27a}})~(2^{{({{2}^{{A\_27a}}})}})) .  ({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2t \in \\
(2^{{A\_27a}}) .  (((p~(ap~(\mname{c\_2Emeasure\_2Ealgebra}~A\_27a)~V0a)){\unicdcdcdhd}((p~(\\
ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~V1s)~(ap~(c\_2Emeasure\_2Esubsets~\\
A\_27a)~V0a))){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~V2t)~(ap~(c\_2Emeasure\_2Esubsets~\\
A\_27a)~V0a))))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~V1s)~V2t))~(ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V0a)))))))\end{array}
\label{hyp:ehigngpfcdfengfgbgdhfhchfgpfcdfebemehefececfbepfpfjeoeeffecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in (ty\_2Epair\_2Eprod~\\
(2^{{A\_27a}})~(2^{{({{2}^{{A\_27a}}})}})) .  ({\unicdcdadad} V1f \in ((2^{{A\_27a}})^{{ty\_2Eextreal\_2Eextreal}}) .  \\
(((p~(ap~(\mname{c\_2Emeasure\_2Esigma\_\_algebra}~A\_27a)~V0a)){\unicdcdcdhd}(p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~((2^{{A\_27a}})^{{ty\_2Eextreal\_2Eextreal}}))~V1f)~(ap~\\
(ap~(\mname{c\_2Epred\_\_set\_2EFUNSET}~ty\_2Eextreal\_2Eextreal~(2^{{A\_27a}}))~\\
\mname{c\_2Eextreal\_2EQ\_\_set})~(ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V0a))))){\unicdbdjdcd}\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(\mname{c\_2Epred\_\_set\_2EBIGUNION}~\\
A\_27a)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~ty\_2Eextreal\_2Eextreal~\\
(2^{{A\_27a}}))~V1f)~\mname{c\_2Eextreal\_2EQ\_\_set})))~(ap~(c\_2Emeasure\_2Esubsets~\\
A\_27a)~V0a))))))\end{array}
\label{hyp:ehigngpfcdfengfgbgdhfhchfgpfcdfecejeheffoejepeoepfpfjenebehefepfpfbf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (ty\_2Eextreal\_2Eextreal^{{A\_27a}}) .  \\
({\unicdcdadad} V1a \in (ty\_2Epair\_2Eprod~(2^{{A\_27a}})~(2^{{({{2}^{{A\_27a}}})}})) .  ((\\
p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Eextreal\_2Eextreal^{{A\_27a}}))~V0f)~\\
(ap~(ap~(\mname{c\_2Emeasure\_2Emeasurable}~A\_27a~ty\_2Eextreal\_2Eextreal)~\\
V1a)~\mname{c\_2Emeasure\_2EBorel}))){\unicdbdjded}((p~(ap~(\mname{c\_2Emeasure\_2Esigma\_\_algebra}~\\
A\_27a)~V1a)){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Eextreal\_2Eextreal^{{A\_27a}}))~\\
V0f)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EFUNSET}~A\_27a~ty\_2Eextreal\_2Eextreal)~\\
(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a))~(\mname{c\_2Epred\_\_set\_2EUNIV}~\\
ty\_2Eextreal\_2Eextreal)))){\unicdcdcdhd}({\unicdcdadad} V2c \in ty\_2Eextreal\_2Eextreal .  \\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V3x \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V3x)~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~\\
(ap~V0f~V3x))~V2c)))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(\\
ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V1a)))))))))\end{array}
\label{hyp:ehigngpfcdfengfgbgdhfhchfgpfcdfejeoepfpfnefebedfffcfbecemefepfpfcepecffeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (ty\_2Eextreal\_2Eextreal^{{A\_27a}}) .  \\
({\unicdcdadad} V1a \in (ty\_2Epair\_2Eprod~(2^{{A\_27a}})~(2^{{({{2}^{{A\_27a}}})}})) .  ((\\
p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Eextreal\_2Eextreal^{{A\_27a}}))~V0f)~\\
(ap~(ap~(\mname{c\_2Emeasure\_2Emeasurable}~A\_27a~ty\_2Eextreal\_2Eextreal)~\\
V1a)~\mname{c\_2Emeasure\_2EBorel}))){\unicdbdjdcd}(({\unicdcdadad} V2c \in ty\_2Eextreal\_2Eextreal .  \\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V3x \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V3x)~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~\\
(ap~V0f~V3x))~V2c)))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(\\
ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V1a)))){\unicdcdcdhd}(({\unicdcdadad} V4c \in ty\_2Eextreal\_2Eextreal .  \\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V5x \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V5x)~(ap~(ap~c\_2Eextreal\_2Eextreal\_\_le~\\
V4c)~(ap~V0f~V5x))))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(\\
ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V1a)))){\unicdcdcdhd}(({\unicdcdadad} V6c \in ty\_2Eextreal\_2Eextreal .  \\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V7x \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V7x)~(ap~(ap~c\_2Eextreal\_2Eextreal\_\_le~\\
(ap~V0f~V7x))~V6c)))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(\\
ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V1a)))){\unicdcdcdhd}(({\unicdcdadad} V8c \in ty\_2Eextreal\_2Eextreal .  \\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V9x \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V9x)~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~\\
V8c)~(ap~V0f~V9x))))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(\\
ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V1a)))){\unicdcdcdhd}(({\unicdcdadad} V10c \in ty\_2Eextreal\_2Eextreal .  \\
({\unicdcdadad} V11d \in ty\_2Eextreal\_2Eextreal .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27a)~({\uniddcgcg} V12x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~\\
V12x)~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~\\
V10c)~(ap~V0f~V12x)))~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~(ap~V0f~\\
V12x))~V11d))))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(ap~(\\
c\_2Emeasure\_2Esubsets~A\_27a)~V1a))))){\unicdcdcdhd}(({\unicdcdadad} V13c \in ty\_2Eextreal\_2Eextreal .  \\
({\unicdcdadad} V14d \in ty\_2Eextreal\_2Eextreal .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27a)~({\uniddcgcg} V15x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~\\
V15x)~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~(ap~c\_2Eextreal\_2Eextreal\_\_le~\\
V13c)~(ap~V0f~V15x)))~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~(ap~V0f~\\
V15x))~V14d))))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(ap~(\\
c\_2Emeasure\_2Esubsets~A\_27a)~V1a))))){\unicdcdcdhd}(({\unicdcdadad} V16c \in ty\_2Eextreal\_2Eextreal .  \\
({\unicdcdadad} V17d \in ty\_2Eextreal\_2Eextreal .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27a)~({\uniddcgcg} V18x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~\\
V18x)~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~\\
V16c)~(ap~V0f~V18x)))~(ap~(ap~c\_2Eextreal\_2Eextreal\_\_le~(ap~V0f~\\
V18x))~V17d))))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(ap~(\\
c\_2Emeasure\_2Esubsets~A\_27a)~V1a))))){\unicdcdcdhd}(({\unicdcdadad} V19c \in ty\_2Eextreal\_2Eextreal .  \\
({\unicdcdadad} V20d \in ty\_2Eextreal\_2Eextreal .  (p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27a)~({\uniddcgcg} V21x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~\\
V21x)~(ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~(ap~c\_2Eextreal\_2Eextreal\_\_le~\\
V19c)~(ap~V0f~V21x)))~(ap~(ap~c\_2Eextreal\_2Eextreal\_\_le~(ap~V0f~\\
V21x))~V20d))))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(ap~(\\
c\_2Emeasure\_2Esubsets~A\_27a)~V1a))))){\unicdcdcdhd}(({\unicdcdadad} V22c \in ty\_2Eextreal\_2Eextreal .  \\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V23x \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V23x)~(ap~\mname{c\_2Ebool\_2E\_7E}~(ap~(\\
ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Eextreal\_2Eextreal)~(ap~V0f~V23x))~V22c))))))~\\
(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~(ap~(c\_2Emeasure\_2Esubsets~\\
A\_27a)~V1a)))){\unicdcdcdhd}({\unicdcdadad} V24c \in ty\_2Eextreal\_2Eextreal .  (p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~\\
A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~A\_27a~A\_27a)~({\uniddcgcg} V25x \in A\_27a .  \\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~V25x)~(ap~(ap~(\mname{c\_2Emin\_2E\_3D}~ty\_2Eextreal\_2Eextreal)~\\
(ap~V0f~V25x))~V24c)))))~(ap~(c\_2Emeasure\_2Espace~A\_27a)~V1a)))~\\
(ap~(c\_2Emeasure\_2Esubsets~A\_27a)~V1a))))))))))))))))\end{array}
\label{hyp:ehigngpfcdfengfgbgdhfhchfgpfcdfejeoepfpfnefebedfffcfbecemefepfpfcepecffemepfpfbememe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ({\unicdcdadad} V2a \in A\_27a .  ({\unicdcdadad} V3b \in \\
A\_27b .  (((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y)=(ap~(ap~\\
(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2a)~V3b)){\unicdbdjded}((V0x=V2a){\unicdcdcdhd}(V1y=V3b)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedemepedffeeepfpfafbejecfpfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ((V0s=V1t){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V0s)){\unicdbdjded}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfefeifeffeoedfjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0f \in ((ty\_2Epair\_2Eprod~A\_27a~2)^{{A\_27b}}) .  ({\unicdcdadad} V1v \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V1v)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27b)~V0f))){\unicdbdjded}({\unicdcdaddd} V2x \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
A\_27a~2)~V1v)~\mname{c\_2Ebool\_2ET})=(ap~V0f~V2x))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfehedfaffedejegejedebeefjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1t \in \\
(2^{{A\_27a}}) .  ({\unicdcdadad} V2x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~\\
V2x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfjeoeeffecf}
\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} V1P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2Q \in \\
(2^{{A\_27b}}) .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(A\_27b^{{A\_27a}}))~V0f)~(ap~(ap~\\
(\mname{c\_2Epred\_\_set\_2EFUNSET}~A\_27a~A\_27b)~V1P)~V2Q))){\unicdbdjded}({\unicdcdadad} V3x \in \\
A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V3x)~V1P)){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27b)~(ap~V0f~V3x))~V2Q))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfgeffoedffeef}
\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 ((2^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1s \in (2^{{A\_27a}}) .  ({\unicdcdadad} V2y \in \\
A\_27b .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V2y)~(ap~(\mname{c\_2Epred\_\_set\_2EBIGUNION}~\\
A\_27b)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EIMAGE}~A\_27a~(2^{{A\_27b}}))~V0f)~V1s)))){\unicdbdjded}\\
({\unicdcdaddd} V3x \in A\_27a .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V3x)~V1s)){\unicdcdcdhd}\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27b)~V2y)~(ap~V0f~V3x)))))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfcejeheffoejepeoepfpfjenebehefe}
\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:ehigngpfcdfengfgbgdhfhchfgpfcdfejeoepfpfnefebedfffcfbecemefepfpfcepecffemepfpfmeef}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f \in (ty\_2Eextreal\_2Eextreal^{{A\_27a}}) .  \\
({\unicdcdadad} V1g \in (ty\_2Eextreal\_2Eextreal^{{A\_27a}}) .  ({\unicdcdadad} V2a \in (\\
ty\_2Epair\_2Eprod~(2^{{A\_27a}})~(2^{{({{2}^{{A\_27a}}})}})) .  (((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
(ty\_2Eextreal\_2Eextreal^{{A\_27a}}))~V0f)~(ap~(ap~(\mname{c\_2Emeasure\_2Emeasurable}~\\
A\_27a~ty\_2Eextreal\_2Eextreal)~V2a)~\mname{c\_2Emeasure\_2EBorel}))){\unicdcdcdhd}\\
(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(ty\_2Eextreal\_2Eextreal^{{A\_27a}}))~V1g)~\\
(ap~(ap~(\mname{c\_2Emeasure\_2Emeasurable}~A\_27a~ty\_2Eextreal\_2Eextreal)~\\
V2a)~\mname{c\_2Emeasure\_2EBorel})))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~(2^{{A\_27a}}))~\\
(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINTER}~A\_27a)~(ap~(c\_2Epred\_\_set\_2EGSPEC~\\
A\_27a~A\_27a)~({\uniddcgcg} V3x \in A\_27a .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~2)~\\
V3x)~(ap~(ap~\mname{c\_2Eextreal\_2Eextreal\_\_lt}~(ap~V0f~V3x))~(ap~V1g~V3x))))))~\\
(ap~(c\_2Emeasure\_2Espace~A\_27a)~V2a)))~(ap~(c\_2Emeasure\_2Esubsets~\\
A\_27a)~V2a)))))))\end{array}$$
\end{theorem}
\end{document}
