\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\_2EquantHeuristics\_2EGUESS\_\_RULES\_\_DISJ (TMYGZD8Q2aipcmqj6329mRVeSbZi6Hig2jA)}
\maketitle
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\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:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def: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:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def: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:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfgepecfbememepfpfhebeaf}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1P}}~{V2v}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V2v}})}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpffeifjedfefdfpfpfhebeaf}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V1P}}~{V2v}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V2v}})}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfgepecfbememepfpfafpejeoeef}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2fv \in {A\_27a} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V2fv}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpffeifjedfefdfpfpfafpejeoeef}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2fv \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V2fv}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfgepecfbememe}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{V1P}}~{V2v}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpffeifjedfefdf}
We define $\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0i \in {({{A\_27b}^{{A\_27a}}})} .  {{\uniddcgcg} V1P \in {({{2}^{{A\_27b}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27b}})}}~{({{\uniddcgcg} V2v \in {A\_27b} .  {({{{ap}~{V1P}}~{V2v}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{({{\uniddcgcg} V3fv \in {A\_27a} .  {({{{ap}~{V1P}}~{({{{ap}~{V0i}}~{V3fv}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
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}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdaddd} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfdfjeneaf}
\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 .  ({\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} V0A \in 2 .  ({\unicdcdadad} V1B \in 2 .  (((\preoh ((p~V0A){\unicdcdcdhd}(p~V1B))){\unicdbdjded}((\preoh (\\
p~V0A)){\unicdcdcdid}(\preoh (p~V1B)))){\unicdcdcdhd}((\preoh ((p~V0A){\unicdcdcdid}(p~V1B))){\unicdbdjded}((\preoh (p~V0A)){\unicdcdcdhd}(\preoh (p~V1B)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeefepfpfnepecfhebeoepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} 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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0i \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1P \in (2^{{A\_27b}}) .  ((((p~(ap~(\\
ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~A\_27a~A\_27b)~V0i)~V1P)){\unicdbdjded}\\
({\unicdcdadad} V2v \in A\_27b .  ((p~(ap~V1P~V2v)){\unicdbdjdcd}({\unicdcdaddd} V3fv \in A\_27a .  (p~(\\
ap~V1P~(ap~V0i~V3fv))))))){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~\\
A\_27a~A\_27b)~V0i)~V1P)){\unicdbdjded}({\unicdcdadad} V4v \in A\_27b .  ((\preoh (p~(ap~V1P~V4v))){\unicdbdjdcd}\\
({\unicdcdaddd} V5fv \in A\_27a .  (\preoh (p~(ap~V1P~(ap~V0i~V5fv))))))))){\unicdcdcdhd}(({\unicdcdadad} V6i \in \\
(A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V7P \in (2^{{A\_27b}}) .  ((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V6i)~V7P)){\unicdbdjded}({\unicdcdadad} V8fv \in A\_27a .  (p~(ap~V7P~(ap~V6i~V8fv))))))){\unicdcdcdhd}\\
(({\unicdcdadad} V9i \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V10P \in (2^{{A\_27b}}) .  ((p~(ap~\\
(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~A\_27a~A\_27b)~\\
V9i)~V10P)){\unicdbdjded}({\unicdcdadad} V11fv \in A\_27a .  (\preoh (p~(ap~V10P~(ap~V9i~V11fv)))))))){\unicdcdcdhd}\\
(({\unicdcdadad} V12i \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V13P \in (2^{{A\_27b}}) .  ((p~(ap~\\
(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~A\_27a~A\_27b)~\\
V12i)~V13P)){\unicdbdjded}({\unicdcdadad} V14v \in A\_27b .  ((p~(ap~V13P~V14v)){\unicdbdjdcd}({\unicdcdaddd} V15fv \in \\
A\_27a .  (V14v=(ap~V12i~V15fv)))))))){\unicdcdcdhd}({\unicdcdadad} V16i \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V17P \in (2^{{A\_27b}}) .  ((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~\\
A\_27a~A\_27b)~V16i)~V17P)){\unicdbdjded}({\unicdcdadad} V18v \in A\_27b .  ((\preoh (p~(ap~V17P~V18v))){\unicdbdjdcd}\\
({\unicdcdaddd} V19fv \in A\_27a .  (V18v=(ap~V16i~V19fv))))))))))))))\end{array}
\label{hyp:ehigngpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfcffehfcfjeeffedf}
\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}
\begin{theorem}\label{thm:ehigngpfcdfebhfhbgogehiefgfhchjgdhehjgdgdhpfcdfehefffedfdfpfpfcfffmefedfpfpfeejedfke}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0i \in (A\_27b^{{A\_27a}}) .  ({\unicdcdadad} V1P \in (2^{{A\_27b}}) .  ({\unicdcdadad} V2Q \in \\
(2^{{A\_27b}}) .  ({\unicdcdadad} V3iK \in A\_27b .  ({\unicdcdadad} V4q \in 2 .  ({\unicdcdadad} V5p \in \\
2 .  (((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V6x \in A\_27b .  (ap~V1P~V6x)))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V7x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(ap~\\
V1P~V7x))~(ap~V2Q~V7x)))))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V8x \in A\_27b .  (ap~V2Q~V8x)))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V9x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(ap~\\
V1P~V9x))~(ap~V2Q~V9x)))))){\unicdcdcdhd}((((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V10x \in A\_27b .  (ap~V1P~V10x)))){\unicdcdcdhd}(p~(ap~(ap~(\\
\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~A\_27a~A\_27b)~V0i)~({\uniddcgcg} V11x \in \\
A\_27b .  (ap~V2Q~V11x))))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V12x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(\\
ap~V1P~V12x))~(ap~V2Q~V12x)))))){\unicdcdcdhd}((((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V13x \in A\_27b .  (ap~V1P~V13x)))){\unicdcdcdhd}(p~(ap~(ap~(\\
\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~A\_27a~A\_27b)~V0i)~\\
({\uniddcgcg} V14x \in A\_27b .  (ap~V2Q~V14x))))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_EXISTS\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V15x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(\\
ap~V1P~V15x))~(ap~V2Q~V15x)))))){\unicdcdcdhd}((((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~\\
ty\_2Eone\_2Eone~A\_27b)~({\uniddcgcg} V16xxx \in ty\_2Eone\_2Eone .  V3iK))~({\uniddcgcg} V17x \in \\
A\_27b .  (ap~V1P~V17x)))){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~\\
ty\_2Eone\_2Eone~A\_27b)~({\uniddcgcg} V18xxx \in ty\_2Eone\_2Eone .  V3iK))~({\uniddcgcg} V19x \in \\
A\_27b .  (ap~V2Q~V19x))))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~\\
ty\_2Eone\_2Eone~A\_27b)~({\uniddcgcg} V20xxx \in ty\_2Eone\_2Eone .  V3iK))~({\uniddcgcg} V21x \in \\
A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(ap~V1P~V21x))~(ap~V2Q~V21x)))))){\unicdcdcdhd}\\
(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~A\_27a~A\_27b)~\\
V0i)~({\uniddcgcg} V22x \in A\_27b .  (ap~V1P~V22x)))){\unicdbdjdcd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V23x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(\\
ap~V1P~V23x))~V4q))))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V24x \in A\_27b .  (ap~V2Q~V24x)))){\unicdbdjdcd}(p~(ap~(ap~(\\
\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL}~A\_27a~A\_27b)~V0i)~({\uniddcgcg} V25x \in \\
A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~V5p)~(ap~V2Q~V25x)))))){\unicdcdcdhd}((((p~\\
(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~A\_27a~A\_27b)~\\
V0i)~({\uniddcgcg} V26x \in A\_27b .  (ap~V1P~V26x)))){\unicdcdcdhd}(p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V27x \in A\_27b .  (ap~V2Q~V27x))))){\unicdbdjdcd}(p~(ap~(ap~\\
(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_POINT}~A\_27a~A\_27b)~V0i)~\\
({\uniddcgcg} V28x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(ap~V1P~V28x))~(ap~V2Q~\\
V28x)))))){\unicdcdcdhd}(((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V29x \in A\_27b .  (ap~V1P~V29x)))){\unicdbdjdcd}(p~(ap~(ap~(\\
\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~A\_27a~A\_27b)~V0i)~\\
({\uniddcgcg} V30x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(ap~V1P~V30x))~(ap~V2Q~\\
V30x)))))){\unicdcdcdhd}((p~(ap~(ap~(\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~\\
A\_27a~A\_27b)~V0i)~({\uniddcgcg} V31x \in A\_27b .  (ap~V2Q~V31x)))){\unicdbdjdcd}(p~(ap~(ap~(\\
\mname{c\_2EquantHeuristics\_2EGUESS\_\_FORALL\_\_GAP}~A\_27a~A\_27b)~V0i)~\\
({\uniddcgcg} V32x \in A\_27b .  (ap~(ap~\mname{c\_2Ebool\_2E\_5C\_2F}~(ap~V1P~V32x))~(ap~V2Q~\\
V32x)))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
