\documentclass{article}
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\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
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\begin{document}
\title{thm\_2Esorting\_2EFILTER\_\_EQ\_\_LENGTHS\_\_EQ (TMHGseHk5f9wtevLTScffphz1az7m9Mxxar)}
\maketitle
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
\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: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}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2Elist\_2ELENGTH : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELENGTH~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemefeoeheefie}
\end{equation}
Let $ c\_2Erich\_\_list\_2EREPLICATE : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Erich\_\_list\_2EREPLICATE~A\_27a \in \\
(((ty\_2Elist\_2Elist~A\_27a)^{{A\_27a}})^{{ty\_2Enum\_2Enum}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgdgigpfpfmgjgdhehpfcdfecffeafmejedebeeffe}
\end{equation}
Let $ c\_2Elist\_2EFILTER : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EFILTER~A\_27a \in (((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{2}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfegejemeeffecf}
\end{equation}
\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}
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} 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} V1l \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ((ap~(ap~(c\_2Elist\_2EFILTER~A\_27a)~\\
(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27a)~V0x))~V1l)=(ap~(ap~(c\_2Erich\_\_list\_2EREPLICATE~\\
A\_27a)~(ap~(c\_2Elist\_2ELENGTH~A\_27a)~(ap~(ap~(c\_2Elist\_2EFILTER~\\
A\_27a)~(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27a)~V0x))~V1l)))~V0x))))\end{array}
\label{hyp:ehigngpfcdfedhpgchehjgoghgpfcdfegejemeeffecfpfpffebfpfpfcffeaf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfedhpgchehjgoghgpfcdfegejemeeffecfpfpffebfpfpfmefeoeheefiedfpfpffebf}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1l1 \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V2l2 \in (ty\_2Elist\_2Elist~A\_27a) .  \\
(((ap~(c\_2Elist\_2ELENGTH~A\_27a)~(ap~(ap~(c\_2Elist\_2EFILTER~A\_27a)~\\
(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27a)~V0x))~V1l1))=(ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~(ap~(ap~(c\_2Elist\_2EFILTER~A\_27a)~(ap~(\mname{c\_2Emin\_2E\_3D}~A\_27a)~\\
V0x))~V2l2))){\unicdbdjdcd}((ap~(ap~(c\_2Elist\_2EFILTER~A\_27a)~(ap~(\mname{c\_2Emin\_2E\_3D}~\\
A\_27a)~V0x))~V1l1)=(ap~(ap~(c\_2Elist\_2EFILTER~A\_27a)~(ap~(\mname{c\_2Emin\_2E\_3D}~\\
A\_27a)~V0x))~V2l2))))))\end{array}$$
\end{theorem}
\end{document}
