\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\_2Ehreal\_2EHREAL\_\_SUB\_\_ISACUT (TMSWLLkdKUz5E9qnr5ULErF1UVeweyLDcdR)}
\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}
\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: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 $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\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\_2Ehrat\_2Ehrat : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehrat\_2Ehrat}}\label{hyp:ogfgpfehjhpfcdfeigchbgehpfcdfeigchbgeh}
\end{equation}
Let $ c\_2Ehrat\_2Ehrat\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Ehrat\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Ehrat\_2Ehrat}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeigchbgehpfpfcffeafpfpfdemebedfdf}
\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:dgpfcdfeigchbgehpfcdfeigchbgehpfpfcffeaf}
We define $\mname{c\_2Ehrat\_2Ehrat\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Ehrat\_2Ehrat\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Ehrat\_2Etrat\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Etrat\_\_add \in (((ty\_2Epair\_2Eprod~ty\_2Enum\_2Enum~ty\_2Enum\_2Enum)^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeehchbgehpfpfbgegeg}
\end{equation}
Let $ c\_2Ehrat\_2Etrat\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Etrat\_\_eq \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeehchbgehpfpffgbh}
\end{equation}
Let $ c\_2Ehrat\_2Ehrat\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ehrat\_2Ehrat\_\_ABS\_\_CLASS \in (ty\_2Ehrat\_2Ehrat^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeigchbgehpfcdfeigchbgehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfeigchbgehpfcdfeigchbgehpfpfbecedf}
We define $\mname{c\_2Ehrat\_2Ehrat\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Ehrat\_2Ehrat\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Ehrat\_2Etrat\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeigchbgehpfcdfeigchbgehpfpfbgegeg}
We define $\mname{c\_2Ehrat\_2Ehrat\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Ehrat\_2Ehrat} .  {{\uniddcgcg} V1T2 \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Ehrat\_2Etrat\_\_add}}~{({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_REP}}}~{V1T2}})}})}})}}}$.
\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:dgpfcdfeigchfgbgmgpfcdfeigchbgehpfpfmgeh}
We define $\mname{c\_2Ehreal\_2Ehrat\_\_lt}$ to be ${{\uniddcgcg} V0x \in {ty\_2Ehrat\_2Ehrat} .  {{\uniddcgcg} V1y \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V2d \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Ehrat\_2Ehrat}})}}~{V1y}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ehrat\_2Ehrat\_\_add}}}~{V0x}})}}~{V2d}})}})}})}})}}}$.
\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: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:dgpfcdfeigchfgbgmgpfcdfejgdhbgdgfheh}
We define $\mname{c\_2Ehreal\_2Eisacut}$ to be ${{\uniddcgcg} V0C \in {({{2}^{{ty\_2Ehrat\_2Ehrat}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V1x \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{V0C}}~{V1x}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V2x \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V0C}}~{V2x}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V3x \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V4y \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V0C}}~{V3x}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ehreal\_2Ehrat\_\_lt}}}~{V4y}})}}~{V3x}})}})}})}}~{({{{ap}~{V0C}}~{V4y}})}})}})}})}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V5x \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V0C}}~{V5x}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V6y \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V0C}}~{V6y}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ehreal\_2Ehrat\_\_lt}}}~{V5x}})}}~{V6y}})}})}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ ty\_2Ehreal\_2Ehreal : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ehreal\_2Ehreal}}\label{hyp:ogfgpfehjhpfcdfeigchfgbgmgpfcdfeigchfgbgmg}
\end{equation}
Let $ c\_2Ehreal\_2Ecut : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Ehreal\_2Ecut} \in {({{({{2}^{{ty\_2Ehrat\_2Ehrat}}})}^{{ty\_2Ehreal\_2Ehreal}}})}}\label{hyp:ngfgngpfdgpfcdfeigchfgbgmgpfcdfedgfheh}
\end{equation}
\begin{definition}\label{def:dgpfcdfeigchfgbgmgpfcdfeigchfgbgmgpfpfmgeh}
We define $\mname{c\_2Ehreal\_2Ehreal\_\_lt}$ to be ${{\uniddcgcg} V0X \in {ty\_2Ehreal\_2Ehreal} .  {{\uniddcgcg} V1Y \in {ty\_2Ehreal\_2Ehreal} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{ty\_2Ehreal\_2Ehreal}})}}~{V0X}})}}~{V1Y}})}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Ehrat\_2Ehrat}})}}~{({{\uniddcgcg} V2x \in {ty\_2Ehrat\_2Ehrat} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{c\_2Ehreal\_2Ecut}}~{V0X}})}}~{V2x}})}})}}~{({{{ap}~{({{{ap}~{c\_2Ehreal\_2Ecut}}~{V1Y}})}}~{V2x}})}})}})}})}})}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} 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} V0h \in ty\_2Ehrat\_2Ehrat .  ({\unicdcdadad} V1i \in ty\_2Ehrat\_2Ehrat .  \\
({\unicdcdadad} V2j \in ty\_2Ehrat\_2Ehrat .  ((ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~\\
V0h)~(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~V1i)~V2j))=(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~\\
(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~V0h)~V1i))~V2j)))))\end{array}
\label{hyp:ehigngpfcdfeigchbgehpfcdfeiecfbeefpfpfbeeeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Ehrat\_2Ehrat .  ({\unicdcdadad} V1y \in ty\_2Ehrat\_2Ehrat .  \\
(p~(ap~(ap~\mname{c\_2Ehreal\_2Ehrat\_\_lt}~V0x)~(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~\\
V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfeiecfbeefpfpfmeefpfpfbeeeeeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Ehrat\_2Ehrat .  ({\unicdcdadad} V1y \in ty\_2Ehrat\_2Ehrat .  \\
(p~(ap~(ap~\mname{c\_2Ehreal\_2Ehrat\_\_lt}~V1y)~(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~\\
V0x)~V1y)))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfeiecfbeefpfpfmeefpfpfbeeeeecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in ty\_2Ehrat\_2Ehrat .  ({\unicdcdadad} V1y \in ty\_2Ehrat\_2Ehrat .  \\
({\unicdcdadad} V2z \in ty\_2Ehrat\_2Ehrat .  ((p~(ap~(ap~\mname{c\_2Ehreal\_2Ehrat\_\_lt}~\\
(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~V2z)~V0x))~(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~\\
V2z)~V1y))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Ehreal\_2Ehrat\_\_lt}~V0x)~V1y))))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfeiecfbeefpfpfmeefpfpfmebeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdaddd} V1x \in ty\_2Ehrat\_2Ehrat .  \\
(\preoh (p~(ap~(ap~c\_2Ehreal\_2Ecut~V0X)~V1x)))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfedeffefpfpfcepeffoeeefeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1x \in ty\_2Ehrat\_2Ehrat .  \\
({\unicdcdadad} V2y \in ty\_2Ehrat\_2Ehrat .  (((p~(ap~(ap~c\_2Ehreal\_2Ecut~V0X)~\\
V1x)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ehreal\_2Ehrat\_\_lt}~V2y)~V1x))){\unicdbdjdcd}(p~(ap~(ap~c\_2Ehreal\_2Ecut~\\
V0X)~V2y))))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfedeffefpfpfeepehfoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1x \in ty\_2Ehrat\_2Ehrat .  \\
((p~(ap~(ap~c\_2Ehreal\_2Ecut~V0X)~V1x)){\unicdbdjdcd}({\unicdcdaddd} V2y \in ty\_2Ehrat\_2Ehrat .  \\
((p~(ap~(ap~c\_2Ehreal\_2Ecut~V0X)~V2y)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ehreal\_2Ehrat\_\_lt}~\\
V1x)~V2y)))))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfedeffefpfpfffaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1x \in ty\_2Ehrat\_2Ehrat .  \\
({\unicdcdadad} V2y \in ty\_2Ehrat\_2Ehrat .  (((\preoh (p~(ap~(ap~c\_2Ehreal\_2Ecut~\\
V0X)~V1x))){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Ehreal\_2Ehrat\_\_lt}~V1x)~V2y))){\unicdbdjdcd}(\preoh (p~(\\
ap~(ap~c\_2Ehreal\_2Ecut~V0X)~V2y)))))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfedeffefpfpfffcepeffoeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1Y \in ty\_2Ehreal\_2Ehreal .  \\
((p~(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_lt}~V0X)~V1Y)){\unicdbdjdcd}({\unicdcdaddd} V2x \in ty\_2Ehrat\_2Ehrat .  \\
((\preoh (p~(ap~(ap~c\_2Ehreal\_2Ecut~V0X)~V2x))){\unicdcdcdhd}(p~(ap~(ap~c\_2Ehreal\_2Ecut~\\
V1Y)~V2x)))))))\end{array}
\label{hyp:ehigngpfcdfeigchfgbgmgpfcdfeiecffebemepfpfmeefpfpfmefenenebe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeigchfgbgmgpfcdfeiecffebemepfpfdfffcepfpfjedfbedeffef}
$$\begin{array}{c}({\unicdcdadad} V0X \in ty\_2Ehreal\_2Ehreal .  ({\unicdcdadad} V1Y \in ty\_2Ehreal\_2Ehreal .  \\
((p~(ap~(ap~\mname{c\_2Ehreal\_2Ehreal\_\_lt}~V0X)~V1Y)){\unicdbdjdcd}(p~(ap~\mname{c\_2Ehreal\_2Eisacut}~\\
({\uniddcgcg} V2w \in ty\_2Ehrat\_2Ehrat .  (ap~(\mname{c\_2Ebool\_2E\_3F}~ty\_2Ehrat\_2Ehrat)~\\
({\uniddcgcg} V3x \in ty\_2Ehrat\_2Ehrat .  (ap~(ap~\mname{c\_2Ebool\_2E\_2F\_5C}~(ap~\mname{c\_2Ebool\_2E\_7E}~\\
(ap~(ap~c\_2Ehreal\_2Ecut~V0X)~V3x)))~(ap~(ap~c\_2Ehreal\_2Ecut~V1Y)~\\
(ap~(ap~\mname{c\_2Ehrat\_2Ehrat\_\_add}~V3x)~V2w)))))))))))\end{array}$$
\end{theorem}
\end{document}
