\documentclass{article}
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\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
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\begin{document}
\title{thm\_2Enumeral\_\_bit\_2ENUMERAL\_\_BITWISE (TMRYCNTCbpynvyJ3uTjG2De7qA8ZWJvXwZH)}
\maketitle
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: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}
\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}
Let $ c\_2Ebit\_2EBITWISE : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}c\_2Ebit\_2EBITWISE \in ((((ty\_2Enum\_2Enum^{{ty\_2Enum\_2Enum}})^{{ty\_2Enum\_2Enum}})^{{({{({{2}^{{2}}})}^{{2}}})}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgehpfcdfecejeefhfjedffe}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngfgchbgmgpfpfcgjgehpfcdfejgcejeefhfjedffe}
We define $\mname{c\_2Enumeral\_\_bit\_2EiBITWISE}$ to be ${c\_2Ebit\_2EBITWISE}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} 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} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeogfhngfgchbgmgpfpfcgjgehpfcdfeoeffnefecfbemepfpfcejeefhfjedffe}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0x \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V1f \in ((2^{{2}})^{{2}}) .  ({\unicdcdadad} V2a \in A\_27a .  ((ap~(ap~(ap~(ap~c\_2Ebit\_2EBITWISE~\\
V0x)~V1f)~\mname{c\_2Enum\_2E0})~\mname{c\_2Enum\_2E0})=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~(ap~(ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiBITWISE}~V0x)~V1f)~\mname{c\_2Enum\_2E0})~\\
\mname{c\_2Enum\_2E0})))))){\unicdcdcdhd}(({\unicdcdadad} V3x \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V4f \in \\
((2^{{2}})^{{2}}) .  ({\unicdcdadad} V5a \in ty\_2Enum\_2Enum .  ((ap~(ap~(ap~(ap~c\_2Ebit\_2EBITWISE~\\
V3x)~V4f)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V5a))~\mname{c\_2Enum\_2E0})=(ap~\\
\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~(ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiBITWISE}~\\
V3x)~V4f)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V5a))~\mname{c\_2Enum\_2E0})))))){\unicdcdcdhd}\\
(({\unicdcdadad} V6x \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V7f \in ((2^{{2}})^{{2}}) .  ({\unicdcdadad} V8b \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~(ap~(ap~c\_2Ebit\_2EBITWISE~V6x)~V7f)~\mname{c\_2Enum\_2E0})~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V8b))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
(ap~(ap~(ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiBITWISE}~V6x)~V7f)~\mname{c\_2Enum\_2E0})~\\
(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V8b))))))){\unicdcdcdhd}({\unicdcdadad} V9x \in ty\_2Enum\_2Enum .  \\
({\unicdcdadad} V10f \in ((2^{{2}})^{{2}}) .  ({\unicdcdadad} V11a \in ty\_2Enum\_2Enum .  ({\unicdcdadad} V12b \in \\
ty\_2Enum\_2Enum .  ((ap~(ap~(ap~(ap~c\_2Ebit\_2EBITWISE~V9x)~V10f)~(\\
ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V11a))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V12b))=(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~(ap~(ap~(ap~(ap~\mname{c\_2Enumeral\_\_bit\_2EiBITWISE}~\\
V9x)~V10f)~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~V11a))~(ap~\mname{c\_2Earithmetic\_2ENUMERAL}~\\
V12b)))))))))))\end{array}$$
\end{theorem}
\end{document}
