\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}
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\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
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\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
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\begin{document}
\title{t10\_euclid\_8 (TMR2q1QwgutRtgHiRBtHy9aTe1cNQrbX9Xj)}
\maketitle
Let $ k23\_rvsum\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_euclid\_8 : {\iota}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k12\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v7\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k15\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_square\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ np\_\_3 : {\iota}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k3\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m1\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m2\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k3\_square\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
{{{k12\_euclid}~{k2\_euclid\_8}}={np\_\_1}}\label{hyp:ehedpffgfhdgmgjgegpfid}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v7\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~\\
(k15\_euclid~X0))){\unicdbdjdcd}(k23\_rvsum\_1~X1~X1=k5\_square\_1~(k12\_euclid~\\
X1)))\end{array}
\label{hyp:ehddgdpffgfhdgmgjgegpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v7\_ordinal1~X0){\unicdbdjdcd}(u1\_struct\_0~(k15\_euclid~X0)=k1\_euclid~\\
X0)\end{array}
\label{hyp:ehcdcdpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_3){\unicdcdcdhd}(m2\_subset\_1~np\_\_3~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_3~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_3~k1\_numbers))\end{array}
\label{hyp:dhahdgddpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_1){\unicdcdcdhd}(m2\_subset\_1~np\_\_1~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_1~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_1~k1\_numbers))\end{array}
\label{hyp:dhahdgbdpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
{{{{k3\_xcmplx\_0}~{np\_\_1}}~{np\_\_1}}={np\_\_1}}\label{hyp:chbhcffgbgmgnefhmgehpfpflgddpfihdgngahmgihpfadpfpfchbdpfchbdpfchbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_finseq\_2~X1~X0){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_finseq\_2~\\
X2~X0~X1){\unicdbdjded}(m1\_subset\_1~X2~X1))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngcdpfggjgogdhfgbhpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(k5\_square\_1~X0=k3\_square\_1~\\
X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfdhbhfhbgchfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{{{m2\_finseq\_2}~{k2\_euclid\_8}}~{k1\_numbers}}~{({{k1\_euclid}~{np\_\_3}})}}\label{hyp:egehpflgcdpffgfhdgmgjgegpfid}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v7\_ordinal1}~{X0}})}{\unicdbdjdcd}{({{{m1\_finseq\_2}~{({{k1\_euclid}~{X0}})}}~{k1\_numbers}})}}}\label{hyp:egehpflgbdpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{k3\_square\_1}~{X0}}={{{k3\_xcmplx\_0}~{X0}}~{X0}}})}}}\label{hyp:egbdpfdhbhfhbgchfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k4\_ordinal1}})}{\unicdbdjdcd}{({{v7\_ordinal1}~{X0}})}}}\label{hyp:dgdgidpfpgchegjgogbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgbdpfihdgngahmgihpfad}
\end{equation}
\begin{theorem}\label{thm:ehbdadpffgfhdgmgjgegpfid}
${{{{k23\_rvsum\_1}~{k2\_euclid\_8}}~{k2\_euclid\_8}}={np\_\_1}}$.
\end{theorem}
\end{document}
