\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\_2Enumeral\_\_bit\_2Enumeral\_\_imod\_\_2exp (TMMAJYNQv28GJzZReeNGMb4vY5g9E59GN5W)}
\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:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Earithmetic\_2E\_2D : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2D \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdee}
\end{equation}
Let $ c\_2Earithmetic\_2EDIV : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EDIV \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfeeejegf}
\end{equation}
Let $ c\_2Earithmetic\_2E\_2A : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2A \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdbe}
\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:dgpfcdfebgchjgehigngfgehjgdgpfcdfeoeffnefecfbeme}
We define $\mname{c\_2Earithmetic\_2ENUMERAL}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {V0x}}$.
\end{definition}
Let $ c\_2Earithmetic\_2EEXP : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EEXP \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfefeifaf}
\end{equation}
Let $ c\_2Earithmetic\_2EMOD : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2EMOD \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfenepeee}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfecepeffoeeefeee}
We define $\mname{c\_2Ebool\_2EBOUNDED}$ to be ${({{\uniddcgcg} V0v \in {2} .  {\mname{c\_2Ebool\_2ET}}})}$.
\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\_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:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\end{definition}
Let $ c\_2Earithmetic\_2E\_2B : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Earithmetic\_2E\_2B \in ((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfebgchjgehigngfgehjgdgpfcdfepfcdce}
\end{equation}
\begin{definition}\label{def:dgpfcdfebgchjgehigngfgehjgdgpfcdfecejeefcd}
We define $\mname{c\_2Earithmetic\_2EBIT2}$ to be ${{\uniddcgcg} V0n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0n}})}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{\mname{c\_2Enum\_2E0}}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def: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:dgpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpe}
We define $\mname{c\_2Earithmetic\_2EZERO}$ to be ${\mname{c\_2Enum\_2E0}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfcdfejgeeffce}
We define $\mname{c\_2Enumeral\_2EiDUB}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2E\_2B}}~{V0x}})}}~{V0x}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgehpfcdfenepeeepfpfcdfeifaf}
We define $\mname{c\_2Ebit\_2EMOD\_\_2EXP}$ to be ${{\uniddcgcg} V0x \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EMOD}}~{V1n}})}}~{({{{ap}~{({{{ap}~{c\_2Earithmetic\_2EEXP}}~{({{{ap}~{\mname{c\_2Earithmetic\_2ENUMERAL}}}~{({{{ap}~{\mname{c\_2Earithmetic\_2EBIT2}}}~{\mname{c\_2Earithmetic\_2EZERO}}})}})}})}}~{V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfpfcgjgehpfcdfejgnepeeepfpfcdfeifaf}
We define $\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}$ to be ${\mname{c\_2Ebit\_2EMOD\_\_2EXP}}$.
\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: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:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\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}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~\\
\mname{c\_2Enum\_2E0})~V0n)=V0n)){\unicdcdcdhd}({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\mname{c\_2Enum\_2ESUC}~\\
V1m))~V2n)=(ap~\mname{c\_2Enum\_2ESUC}~(ap~(ap~c\_2Earithmetic\_2E\_2B~V1m)~\\
V2n))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~\\
V0m)~\mname{c\_2Enum\_2E0})=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))){\unicdcdcdhd}({\unicdcdadad} V1m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2n \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EEXP~V1m)~(ap~\mname{c\_2Enum\_2ESUC}~\\
V2n))=(ap~(ap~c\_2Earithmetic\_2E\_2A~V1m)~(ap~(ap~c\_2Earithmetic\_2EEXP~\\
V1m)~V2n))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfefeifaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))=\\
(ap~\mname{c\_2Enum\_2ESUC}~\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfepeoefe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO}))=\\
(ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO}))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeefhfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V1n)~V2p))=(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~(ap~c\_2Earithmetic\_2E\_2B~V0m)~V1n))~V2p)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeepfpfbedfdfpede}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((V0m=\mname{c\_2Enum\_2E0}){\unicdcdcdid}({\unicdcdaddd} V1n \in \\
ty\_2Enum\_2Enum .  (V0m=(ap~\mname{c\_2Enum\_2ESUC}~V1n)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeogfhngpfpfdebedffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Enum\_2ESUC}~V0m)=(ap~(ap~\\
c\_2Earithmetic\_2E\_2B~V0m)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfebeeeeebd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2D~(\\
ap~\mname{c\_2Enum\_2ESUC}~V0m))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})))=V0m))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfedfffdepfpfdfffcebd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~\\
\mname{c\_2Enum\_2E0})=\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeneffmeefpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=\\
V0m))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeneffmeefpfpfcfjeheieefpfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
(ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~V1n)=(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V1n)~V0m))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeneffmeefpfpfdepenene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2E\_2A~(\\
ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~\\
V0n)=(ap~(ap~c\_2Earithmetic\_2E\_2B~V0n)~V0n)))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeefjenefedfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~(ap~(ap~c\_2Earithmetic\_2EEXP~\\
(ap~\mname{c\_2Enum\_2ESUC}~V1n))~V0m)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpepfpfmefedfdfpfpffeifaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0m \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2p \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(\\
ap~(ap~c\_2Earithmetic\_2E\_2A~V0m)~V1n))~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V0m)~V2p))){\unicdbdjded}((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~V0m)){\unicdcdcdhd}\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1n)~V2p)))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemeefpfpfneffmeefpfpfmedebeoedefeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V0n)){\unicdbdjdcd}({\unicdcdadad} V1k \in ty\_2Enum\_2Enum .  ((V1k=(ap~(\\
ap~c\_2Earithmetic\_2E\_2B~(ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~(ap~c\_2Earithmetic\_2EDIV~\\
V1k)~V0n))~V0n))~(ap~(ap~c\_2Earithmetic\_2EMOD~V1k)~V0n))){\unicdcdcdhd}(p~(ap~\\
(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~(ap~c\_2Earithmetic\_2EMOD~V1k)~V0n))~\\
V0n))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeeejegfjedfjepeoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0k \in ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EMOD~V0k)~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))=\\
\mname{c\_2Enum\_2E0}))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfenepeeepfpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1k \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1k)~V0n)){\unicdbdjdcd}((ap~(ap~c\_2Earithmetic\_2EMOD~\\
V1k)~V0n)=V1k))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfemefedfdfpfpfnepeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V0n)){\unicdbdjdcd}((ap~(ap~c\_2Earithmetic\_2EMOD~\mname{c\_2Enum\_2E0})~\\
V0n)=\mname{c\_2Enum\_2E0})))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfekffecfpepfpfnepeee}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1r \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V1r)~V0n)){\unicdbdjdcd}({\unicdcdadad} V2q \in ty\_2Enum\_2Enum .  \\
((ap~(ap~c\_2Earithmetic\_2EMOD~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\\
(ap~c\_2Earithmetic\_2E\_2A~V2q)~V0n))~V1r))~V0n)=V1r)))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfenepeeepfpfneffmeef}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V0n)){\unicdbdjdcd}({\unicdcdadad} V1j \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2k \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~c\_2Earithmetic\_2EMOD~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
(ap~(ap~c\_2Earithmetic\_2EMOD~V1j)~V0n))~(ap~(ap~c\_2Earithmetic\_2EMOD~\\
V2k)~V0n)))~V0n)=(ap~(ap~c\_2Earithmetic\_2EMOD~(ap~(ap~c\_2Earithmetic\_2E\_2B~\\
V1j)~V2k))~V0n))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfenepeeepfpfafmeffdf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1p \in ty\_2Enum\_2Enum .  (\\
{\unicdcdadad} V2q \in ty\_2Enum\_2Enum .  (((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
\mname{c\_2Enum\_2E0})~V0n)){\unicdcdcdhd}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
V2q))){\unicdbdjdcd}((ap~(ap~c\_2Earithmetic\_2E\_2A~V0n)~(ap~(ap~c\_2Earithmetic\_2EMOD~\\
V1p)~V2q))=(ap~(ap~c\_2Earithmetic\_2EMOD~(ap~(ap~c\_2Earithmetic\_2E\_2A~\\
V0n)~V1p))~(ap~(ap~c\_2Earithmetic\_2E\_2A~V0n)~V2q)))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfenepeeepfpfdepenenepeoepfpfgebedeefpecf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0b \in ty\_2Enum\_2Enum .  ((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO})))~\\
V0b)){\unicdbdjdcd}({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2m \in ty\_2Enum\_2Enum .  \\
((p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~(ap~c\_2Earithmetic\_2EEXP~V0b)~\\
V2m))~(ap~(ap~c\_2Earithmetic\_2EEXP~V0b)~V1n))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
V2m)~V1n)))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfefeifafpfpfcebedffepfpfmeefpfpfnepeoepe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0p \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V1q \in ty\_2Enum\_2Enum .  (\\
(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~(ap~(ap~c\_2Earithmetic\_2E\_2B~(ap~\\
(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
\mname{c\_2Earithmetic\_2EZERO})))~V0p))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~(ap~\\
c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
\mname{c\_2Earithmetic\_2EZERO})))~V1q))){\unicdbdjded}(p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT2}~\mname{c\_2Earithmetic\_2EZERO})))~V0p))~(ap~(ap~\\
c\_2Earithmetic\_2E\_2A~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
\mname{c\_2Earithmetic\_2EZERO})))~V1q))))))\end{array}
\label{hyp:ehigngpfcdfebgchjgehigngfgehjgdgpfcdfeeepeffcemefepfpfmeef}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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 .  ({\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~V0A){\unicdcdcdid}(\\
(p~V1B){\unicdcdcdhd}(p~V2C))){\unicdbdjded}(((p~V0A){\unicdcdcdid}(p~V1B)){\unicdcdcdhd}((p~V0A){\unicdcdcdid}(p~V2C)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfemefegeefpfpfpecfpfpfpegffecfpfpfbeoeee}
\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}
{({{\unicdcdadad} V0v \in {2} .  {({{({{p}~{({{{ap}~{\mname{c\_2Ebool\_2EBOUNDED}}}~{V0v}})}})}{\unicdbdjded}{True}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecepeffoeeefeeepfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(((ap~\mname{c\_2Enum\_2ESUC}~\mname{c\_2Earithmetic\_2EZERO})=(ap~\mname{c\_2Earithmetic\_2EBIT1}~\\
\mname{c\_2Earithmetic\_2EZERO})){\unicdcdcdhd}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  ((ap~\\
\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V0n))=(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V0n))){\unicdcdcdhd}({\unicdcdadad} V1n \in ty\_2Enum\_2Enum .  ((ap~\mname{c\_2Enum\_2ESUC}~(ap~\mname{c\_2Earithmetic\_2EBIT2}~\\
V1n))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~\mname{c\_2Enum\_2ESUC}~V1n))))))\end{array}
\label{hyp:ehigngpfcdfeogfhngfgchbgmgpfcdfeogfhngfgchbgmgpfpfdhfhdg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~\mname{c\_2Enum\_2E0})~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  (p~(ap~(ap~\mname{c\_2Eprim\_\_rec\_2E\_3C}~V0n)~\\
(ap~\mname{c\_2Enum\_2ESUC}~V0n))))\end{array}
\label{hyp:ehigngpfcdfeahchjgngpfpfchfgdgpfcdfemefedfdfpfpfdfffdepfpfcffegeme}
\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:ehigngpfcdfeogfhngfgchbgmgpfpfcgjgehpfcdfeogfhngfgchbgmgpfpfjgngpgegpfpfcdfgihah}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~\mname{c\_2Enum\_2E0})~V0n)=\mname{c\_2Earithmetic\_2EZERO})){\unicdcdcdhd}\\
(({\unicdcdadad} V1x \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V2n \in A\_27a .  ((ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~\\
V1x)~\mname{c\_2Earithmetic\_2EZERO})=\mname{c\_2Earithmetic\_2EZERO}))){\unicdcdcdhd}(({\unicdcdadad} V3x \in \\
ty\_2Enum\_2Enum .  ({\unicdcdadad} V4n \in ty\_2Enum\_2Enum .  ((ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V3x)))~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~V4n))=(ap~\mname{c\_2Earithmetic\_2EBIT1}~(\\
ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~(ap~(ap~c\_2Earithmetic\_2E\_2D~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V3x)))~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~\\
V4n))))){\unicdcdcdhd}(({\unicdcdadad} V5x \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V6n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V5x)))~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V6n))=\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~(ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V5x)))~\\
V6n))))){\unicdcdcdhd}(({\unicdcdadad} V7x \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V8n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT1}~V7x)))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V8n))=\\
(ap~\mname{c\_2Enumeral\_2EiDUB}~(ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~\\
(ap~(ap~c\_2Earithmetic\_2E\_2D~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\\
\mname{c\_2Earithmetic\_2EBIT1}~V7x)))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(\\
ap~\mname{c\_2Earithmetic\_2EBIT1}~\mname{c\_2Earithmetic\_2EZERO}))))~(ap~\mname{c\_2Enum\_2ESUC}~\\
V8n)))))){\unicdcdcdhd}({\unicdcdadad} V9x \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V10n \in ty\_2Enum\_2Enum .  \\
((ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~\mname{c\_2Earithmetic\_2EBIT2}~V9x)))~(ap~\mname{c\_2Earithmetic\_2EBIT2}~V10n))=\\
(ap~\mname{c\_2Enumeral\_2EiDUB}~(ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiMOD\_\_2EXP}~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~\mname{c\_2Earithmetic\_2EBIT1}~V9x)))~\\
(ap~\mname{c\_2Enum\_2ESUC}~V10n)))))))))))\end{array}$$
\end{theorem}
\end{document}
