\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\_2EDeepSyntax\_2Ein\_\_aset (TMWkhLkh1nvzNL5sgkVS8btcQJCZvhdDsTo)}
\maketitle
Let $ ty\_2EDeepSyntax\_2Edeep\_\_form : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2EDeepSyntax\_2Edeep\_\_form}}\label{hyp:ogfgpfehjhpfcdfeeefgfgahdfjhogehbgihpfcdfeegfgfgahpfpfggpgchng}
\end{equation}
Let $ ty\_2Einteger\_2Eint : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Einteger\_2Eint}}\label{hyp:ogfgpfehjhpfcdfejgogehfghgfgchpfcdfejgogeh}
\end{equation}
Let $ c\_2EDeepSyntax\_2ExDivided : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2ExDivided \in ((ty\_2EDeepSyntax\_2Edeep\_\_form^{{ty\_2Einteger\_2Eint}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfeiheejgghjgegfgeg}
\end{equation}
Let $ c\_2EDeepSyntax\_2ExEQ : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2ExEQ \in (ty\_2EDeepSyntax\_2Edeep\_\_form^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfeihfebf}
\end{equation}
Let $ c\_2Enum\_2EZERO\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EZERO\_\_REP} \in {omega}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfekffecfpepfpfcffeaf}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def: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:dgpfcdfeogfhngpfcdfead}
We define $\mname{c\_2Enum\_2E0}$ to be ${({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{c\_2Enum\_2EZERO\_\_REP}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
Let $ c\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
\begin{definition}\label{def: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:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefbd}
We define $\mname{c\_2Earithmetic\_2EBIT1}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_2Einteger\_2Eint\_\_of\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Einteger\_2Eint\_\_of\_\_num} \in {({{ty\_2Einteger\_2Eint}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfpgggpfpfogfhng}
\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 $ c\_2Einteger\_2Eint\_\_REP\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_REP\_\_CLASS \in ((2^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeafpfpfdemebedfdf}
\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:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfcffeaf}
We define $\mname{c\_2Einteger\_2Eint\_\_REP}$ to be ${{\uniddcgcg} V0a \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}})}}~{({{{ap}~{c\_2Einteger\_2Eint\_\_REP\_\_CLASS}}~{V0a}})}})}}$.
\end{definition}
Let $ c\_2Einteger\_2Etint\_\_add : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_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:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpfbgegeg}
\end{equation}
Let $ c\_2Einteger\_2Etint\_\_eq : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Etint\_\_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:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfeehjgogehpfpffgbh}
\end{equation}
Let $ c\_2Einteger\_2Eint\_\_ABS\_\_CLASS : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Einteger\_2Eint\_\_ABS\_\_CLASS \in (ty\_2Einteger\_2Eint^{{({{2}^{{({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedfpfpfdemebedfdf}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbecedf}
We define $\mname{c\_2Einteger\_2Eint\_\_ABS}$ to be ${{\uniddcgcg} V0r \in {({{{ty\_2Epair\_2Eprod}~{ty\_2Enum\_2Enum}}~{ty\_2Enum\_2Enum}})} .  {({{{ap}~{c\_2Einteger\_2Eint\_\_ABS\_\_CLASS}}~{({{{ap}~{c\_2Einteger\_2Etint\_\_eq}}~{V0r}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogehfghgfgchpfcdfejgogehpfpfbgegeg}
We define $\mname{c\_2Einteger\_2Eint\_\_add}$ to be ${{\uniddcgcg} V0T1 \in {ty\_2Einteger\_2Eint} .  {{\uniddcgcg} V1T2 \in {ty\_2Einteger\_2Eint} .  {({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_ABS}}}~{({{{ap}~{({{{ap}~{c\_2Einteger\_2Etint\_\_add}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V0T1}})}})}}~{({{{ap}~{\mname{c\_2Einteger\_2Eint\_\_REP}}}~{V1T2}})}})}})}}}$.
\end{definition}
Let $ c\_2EDeepSyntax\_2ELTx : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2ELTx \in (ty\_2EDeepSyntax\_2Edeep\_\_form^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfemeefih}
\end{equation}
Let $ c\_2EDeepSyntax\_2ExLT : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2ExLT \in (ty\_2EDeepSyntax\_2Edeep\_\_form^{{ty\_2Einteger\_2Eint}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfeihmeef}
\end{equation}
Let $ c\_2EDeepSyntax\_2EUnrelatedBool : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2EDeepSyntax\_2EUnrelatedBool} \in {({{ty\_2EDeepSyntax\_2Edeep\_\_form}^{{2}}})}}\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfeffogchfgmgbgehfgegcepgpgmg}
\end{equation}
Let $ c\_2EDeepSyntax\_2ENegn : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2ENegn \in (ty\_2EDeepSyntax\_2Edeep\_\_form^{{ty\_2EDeepSyntax\_2Edeep\_\_form}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfeoefghgog}
\end{equation}
Let $ c\_2EDeepSyntax\_2EDisjn : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2EDisjn \in ((ty\_2EDeepSyntax\_2Edeep\_\_form^{{ty\_2EDeepSyntax\_2Edeep\_\_form}})^{{ty\_2EDeepSyntax\_2Edeep\_\_form}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfeeejgdhkgog}
\end{equation}
Let $ c\_2EDeepSyntax\_2EConjn : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2EConjn \in ((ty\_2EDeepSyntax\_2Edeep\_\_form^{{ty\_2EDeepSyntax\_2Edeep\_\_form}})^{{ty\_2EDeepSyntax\_2Edeep\_\_form}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfedepgogkgog}
\end{equation}
Let $ c\_2EDeepSyntax\_2EAset : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2EDeepSyntax\_2EAset \in (((2^{{ty\_2Einteger\_2Eint}})^{{ty\_2EDeepSyntax\_2Edeep\_\_form}})^{{2}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeeefgfgahdfjhogehbgihpfcdfebedhfgeh}
\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: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:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def: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: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: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}
\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:dgpfcdfeahchfgegpfpfdhfgehpfcdfejeoedffecfef}
We define $\mname{c\_2Epred\_\_set\_2EINSERT}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1s \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{c\_2Epred\_\_set\_2EGSPEC}~{A\_27a}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{({{{\mname{c\_2Epair\_2E\_2C}}~{A\_27a}}~{2}})}}~{V2y}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_5C\_2F}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2y}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2EIN}}~{A\_27a}})}}~{V2y}})}}~{V1s}})}})}})}})}})}}}}$.
\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}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0pos \in 2 .  ({\unicdcdadad} V1f1 \in ty\_2EDeepSyntax\_2Edeep\_\_form .  \\
({\unicdcdadad} V2f2 \in ty\_2EDeepSyntax\_2Edeep\_\_form .  ((ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V0pos)~(ap~(ap~c\_2EDeepSyntax\_2EConjn~V1f1)~V2f2))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~\\
ty\_2Einteger\_2Eint)~(ap~(ap~c\_2EDeepSyntax\_2EAset~V0pos)~V1f1))~\\
(ap~(ap~c\_2EDeepSyntax\_2EAset~V0pos)~V2f2)))))){\unicdcdcdhd}(({\unicdcdadad} V3pos \in \\
2 .  ({\unicdcdadad} V4f1 \in ty\_2EDeepSyntax\_2Edeep\_\_form .  ({\unicdcdadad} V5f2 \in \\
ty\_2EDeepSyntax\_2Edeep\_\_form .  ((ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V3pos)~(ap~(ap~c\_2EDeepSyntax\_2EDisjn~V4f1)~V5f2))=(ap~(ap~(\mname{c\_2Epred\_\_set\_2EUNION}~\\
ty\_2Einteger\_2Eint)~(ap~(ap~c\_2EDeepSyntax\_2EAset~V3pos)~V4f1))~\\
(ap~(ap~c\_2EDeepSyntax\_2EAset~V3pos)~V5f2)))))){\unicdcdcdhd}(({\unicdcdadad} V6pos \in \\
2 .  ({\unicdcdadad} V7f \in ty\_2EDeepSyntax\_2Edeep\_\_form .  ((ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V6pos)~(ap~c\_2EDeepSyntax\_2ENegn~V7f))=(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
(ap~\mname{c\_2Ebool\_2E\_7E}~V6pos))~V7f)))){\unicdcdcdhd}(({\unicdcdadad} V8pos \in 2 .  ({\unicdcdadad} V9b \in \\
2 .  ((ap~(ap~c\_2EDeepSyntax\_2EAset~V8pos)~(ap~c\_2EDeepSyntax\_2EUnrelatedBool~\\
V9b))=(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Einteger\_2Eint)))){\unicdcdcdhd}(({\unicdcdadad} V10pos \in \\
2 .  ({\unicdcdadad} V11i \in ty\_2Einteger\_2Eint .  ((ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V10pos)~(ap~c\_2EDeepSyntax\_2ExLT~V11i))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
(2^{{ty\_2Einteger\_2Eint}}))~V10pos)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
ty\_2Einteger\_2Eint)~V11i)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Einteger\_2Eint)))~\\
(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Einteger\_2Eint))))){\unicdcdcdhd}(({\unicdcdadad} V12pos \in \\
2 .  ({\unicdcdadad} V13i \in ty\_2Einteger\_2Eint .  ((ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V12pos)~(ap~c\_2EDeepSyntax\_2ELTx~V13i))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~\\
(2^{{ty\_2Einteger\_2Eint}}))~V12pos)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Einteger\_2Eint))~\\
(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~ty\_2Einteger\_2Eint)~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~\\
V13i)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))))~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
ty\_2Einteger\_2Eint)))))){\unicdcdcdhd}(({\unicdcdadad} V14pos \in 2 .  ({\unicdcdadad} V15i \in \\
ty\_2Einteger\_2Eint .  ((ap~(ap~c\_2EDeepSyntax\_2EAset~V14pos)~(\\
ap~c\_2EDeepSyntax\_2ExEQ~V15i))=(ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~(\\
2^{{ty\_2Einteger\_2Eint}}))~V14pos)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
ty\_2Einteger\_2Eint)~(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V15i)~(ap~\\
c\_2Einteger\_2Eint\_\_of\_\_num~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))))~(\mname{c\_2Epred\_\_set\_2EEMPTY}~\\
ty\_2Einteger\_2Eint)))~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~ty\_2Einteger\_2Eint)~\\
V15i)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Einteger\_2Eint)))))){\unicdcdcdhd}({\unicdcdadad} V16pos \in \\
2 .  ({\unicdcdadad} V17i1 \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V18i2 \in ty\_2Einteger\_2Eint .  \\
((ap~(ap~c\_2EDeepSyntax\_2EAset~V16pos)~(ap~(ap~c\_2EDeepSyntax\_2ExDivided~\\
V17i1)~V18i2))=(\mname{c\_2Epred\_\_set\_2EEMPTY}~ty\_2Einteger\_2Eint))))))))))))\end{array}
\label{hyp:ehigngpfcdfeeefgfgahdfjhogehbgihpfcdfebedhfgehpfpfegfggg}
\end{equation}
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} .  {({{({{({{p}~{V0t}})}{\unicdbdjdcd}{False}})}{\unicdbdjdcd}{({\preoh{{({{p}~{V0t}})}}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfge}
\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{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0t \in 2 .  ((\preoh (\preoh (p~V0t))){\unicdbdjded}(p~V0t))){\unicdcdcdhd}(((\preoh True){\unicdbdjded}False){\unicdcdcdhd}\\
((\preoh False){\unicdbdjded}True)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t1 \in A\_27a .  ({\unicdcdadad} V1t2 \in \\
A\_27a .  (((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2ET})~V0t1)~\\
V1t2)=V0t1){\unicdcdcdhd}((ap~(ap~(ap~(\mname{c\_2Ebool\_2ECOND}~A\_27a)~\mname{c\_2Ebool\_2EF})~\\
V0t1)~V1t2)=V1t2))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfedepeoeeepfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ((\preoh ({\unicdcdaddd} V1x \in \\
A\_27a .  (p~(ap~V0P~V1x)))){\unicdbdjded}({\unicdcdadad} V2x \in A\_27a .  (\preoh (p~(ap~V0P~V2x))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeoepeefpfpffeifjedfefdfpfpfefiene}
\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}}) .  (({\unicdcdaddd} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdcdcdid}(p~(ap~V1Q~V2x)))){\unicdbdjded}\\
(({\unicdcdaddd} V3x \in A\_27a .  (p~(ap~V0P~V3x))){\unicdcdcdid}({\unicdcdaddd} V4x \in A\_27a .  (p~(\\
ap~V1Q~V4x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfpecfpfpfefiene}
\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 .  (({\unicdcdaddd} V2x \in A\_27a .  ((p~(ap~V0P~V2x)){\unicdcdcdhd}(p~V1Q))){\unicdbdjded}(({\unicdcdaddd} V3x \in \\
A\_27a .  (p~(ap~V0P~V3x))){\unicdcdcdhd}(p~V1Q)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpffeifjedfefdfpfpfbeoeeepfpfefiene}
\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}}) .  (({\unicdcdaddd} V2x \in A\_27a .  ((p~V0P){\unicdcdcdhd}(p~(ap~V1Q~V2x)))){\unicdbdjded}((p~\\
V0P){\unicdcdcdhd}({\unicdcdaddd} V3x \in A\_27a .  (p~(ap~V1Q~V3x)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecfjeheieefpfpffeifjedfefdfpfpfbeoeeepfpfefiene}
\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} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  (\preoh (p~(ap~(ap~\\
(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfeoepeefpfpfjeoepfpffeneafefjf}
\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\_2EUNION}~A\_27a)~V0s)~V1t))){\unicdbdjded}((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V2x)~V0s)){\unicdcdcdid}(p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
A\_27a)~V2x)~V1t)))))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfffoejepeoe}
\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 .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~A\_27a)~V0x)~(ap~(ap~(\mname{c\_2Epred\_\_set\_2EINSERT}~\\
A\_27a)~V1y)~(\mname{c\_2Epred\_\_set\_2EEMPTY}~A\_27a)))){\unicdbdjded}(V0x=V1y))))\end{array}
\label{hyp:ehigngpfcdfeahchfgegpfpfdhfgehpfcdfejeoepfpfdfjeoehe}
\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}
\begin{theorem}\label{thm:ehigngpfcdfeeefgfgahdfjhogehbgihpfcdfejgogpfpfbgdhfgeh}
$$\begin{array}{c}({\unicdcdadad} V0pos \in 2 .  ({\unicdcdadad} V1f1 \in ty\_2EDeepSyntax\_2Edeep\_\_form .  \\
({\unicdcdadad} V2f2 \in ty\_2EDeepSyntax\_2Edeep\_\_form .  ({\unicdcdadad} V3P \in (\\
2^{{ty\_2Einteger\_2Eint}}) .  ({\unicdcdadad} V4f \in ty\_2EDeepSyntax\_2Edeep\_\_form .  \\
({\unicdcdadad} V5a0 \in 2 .  ({\unicdcdadad} V6i \in ty\_2Einteger\_2Eint .  ({\unicdcdadad} V7i1 \in \\
ty\_2Einteger\_2Eint .  ({\unicdcdadad} V8i2 \in ty\_2Einteger\_2Eint .  ((({\unicdcdaddd} V9a \in \\
ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~\\
V9a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~V0pos)~(ap~(ap~c\_2EDeepSyntax\_2EConjn~\\
V1f1)~V2f2)))){\unicdcdcdhd}(p~(ap~V3P~V9a)))){\unicdbdjded}(({\unicdcdaddd} V10a \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V10a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V0pos)~V1f1))){\unicdcdcdhd}(p~(ap~V3P~V10a)))){\unicdcdcdid}({\unicdcdaddd} V11a \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V11a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V0pos)~V2f2))){\unicdcdcdhd}(p~(ap~V3P~V11a)))))){\unicdcdcdhd}((({\unicdcdaddd} V12a \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V12a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V0pos)~(ap~(ap~c\_2EDeepSyntax\_2EDisjn~V1f1)~V2f2)))){\unicdcdcdhd}(p~(ap~V3P~\\
V12a)))){\unicdbdjded}(({\unicdcdaddd} V13a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V13a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~V0pos)~\\
V1f1))){\unicdcdcdhd}(p~(ap~V3P~V13a)))){\unicdcdcdid}({\unicdcdaddd} V14a \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V14a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V0pos)~V2f2))){\unicdcdcdhd}(p~(ap~V3P~V14a)))))){\unicdcdcdhd}((({\unicdcdaddd} V15a \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V15a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
\mname{c\_2Ebool\_2ET})~(ap~c\_2EDeepSyntax\_2ENegn~V4f)))){\unicdcdcdhd}(p~(ap~V3P~V15a)))){\unicdbdjded}\\
({\unicdcdaddd} V16a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V16a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\mname{c\_2Ebool\_2EF})~\\
V4f))){\unicdcdcdhd}(p~(ap~V3P~V16a))))){\unicdcdcdhd}((({\unicdcdaddd} V17a \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V17a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
\mname{c\_2Ebool\_2EF})~(ap~c\_2EDeepSyntax\_2ENegn~V4f)))){\unicdcdcdhd}(p~(ap~V3P~V17a)))){\unicdbdjded}\\
({\unicdcdaddd} V18a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V18a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\mname{c\_2Ebool\_2ET})~\\
V4f))){\unicdcdcdhd}(p~(ap~V3P~V18a))))){\unicdcdcdhd}((({\unicdcdaddd} V19a \in ty\_2Einteger\_2Eint .  \\
((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V19a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
V0pos)~(ap~c\_2EDeepSyntax\_2EUnrelatedBool~V5a0)))){\unicdcdcdhd}(p~(ap~V3P~\\
V19a)))){\unicdbdjded}False){\unicdcdcdhd}((({\unicdcdaddd} V20a \in ty\_2Einteger\_2Eint .  ((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V20a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
\mname{c\_2Ebool\_2ET})~(ap~c\_2EDeepSyntax\_2ExLT~V6i)))){\unicdcdcdhd}(p~(ap~V3P~V20a)))){\unicdbdjded}\\
(p~(ap~V3P~V6i))){\unicdcdcdhd}((({\unicdcdaddd} V21a \in ty\_2Einteger\_2Eint .  ((p~(ap~\\
(ap~(\mname{c\_2Ebool\_2EIN}~ty\_2Einteger\_2Eint)~V21a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\\
\mname{c\_2Ebool\_2EF})~(ap~c\_2EDeepSyntax\_2ExLT~V6i)))){\unicdcdcdhd}(p~(ap~V3P~V21a)))){\unicdbdjded}\\
False){\unicdcdcdhd}((({\unicdcdaddd} V22a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V22a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\mname{c\_2Ebool\_2ET})~\\
(ap~c\_2EDeepSyntax\_2ELTx~V6i)))){\unicdcdcdhd}(p~(ap~V3P~V22a)))){\unicdbdjded}False){\unicdcdcdhd}\\
((({\unicdcdaddd} V23a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V23a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\mname{c\_2Ebool\_2EF})~\\
(ap~c\_2EDeepSyntax\_2ELTx~V6i)))){\unicdcdcdhd}(p~(ap~V3P~V23a)))){\unicdbdjded}(p~(ap~V3P~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V6i)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))))){\unicdcdcdhd}\\
((({\unicdcdaddd} V24a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V24a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\mname{c\_2Ebool\_2ET})~\\
(ap~c\_2EDeepSyntax\_2ExEQ~V6i)))){\unicdcdcdhd}(p~(ap~V3P~V24a)))){\unicdbdjded}(p~(ap~V3P~\\
(ap~(ap~\mname{c\_2Einteger\_2Eint\_\_add}~V6i)~(ap~c\_2Einteger\_2Eint\_\_of\_\_num~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))))){\unicdcdcdhd}\\
((({\unicdcdaddd} V25a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V25a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~\mname{c\_2Ebool\_2EF})~\\
(ap~c\_2EDeepSyntax\_2ExEQ~V6i)))){\unicdcdcdhd}(p~(ap~V3P~V25a)))){\unicdbdjded}(p~(ap~V3P~\\
V6i))){\unicdcdcdhd}(({\unicdcdaddd} V26a \in ty\_2Einteger\_2Eint .  ((p~(ap~(ap~(\mname{c\_2Ebool\_2EIN}~\\
ty\_2Einteger\_2Eint)~V26a)~(ap~(ap~c\_2EDeepSyntax\_2EAset~V0pos)~\\
(ap~(ap~c\_2EDeepSyntax\_2ExDivided~V7i1)~V8i2)))){\unicdcdcdhd}(p~(ap~V3P~V26a)))){\unicdbdjded}\\
False)))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
