\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{l45\_euclidlp (TMWWYUzeMh3kLSH9yAJ22zLkwJTHn1ojc9b)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ m2\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k1\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r2\_euclidlp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r3\_euclidlp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k6\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k9\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v7\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k5\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k7\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r1\_euclidlp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ m1\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ v6\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_finseq\_2~X2~k1\_numbers~(k1\_euclid~\\
X1)){\unicdbdjdcd}((k6\_euclid~X1~(k9\_euclid~X1~X2~X0)=k9\_euclid~X1~X2~(k1\_real\_1~\\
X0)){\unicdcdcdhd}(k6\_euclid~X1~(k9\_euclid~X1~X2~X0)=k9\_euclid~X1~(k6\_euclid~\\
X1~X2)~X0))))\end{array}
\label{hyp:ehddpffgfhdgmgjgegmgah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v7\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_finseq\_2~X1~k1\_numbers~\\
(k1\_euclid~X0)){\unicdbdjdcd}((k9\_euclid~X0~X1~np\_\_1=X1){\unicdcdcdhd}(k9\_euclid~X0~X1~k6\_numbers=\\
k5\_euclid~X0)))\end{array}
\label{hyp:ehddpffgfhdgmgjgegpfed}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_finseq\_2~\\
X1~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}((k8\_euclid~X0~X1~X1=k5\_euclid~X0){\unicdcdcdhd}\\
(k7\_euclid~X0~X1~(k6\_euclid~X0~X1)=k5\_euclid~X0)))\end{array}
\label{hyp:ehcdpffgfhdgmgjgegmgah}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((m1\_subset\_1~X0~k5\_numbers){\unicdcdcdhd}\\
((m1\_subset\_1~X1~(k1\_euclid~X0)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_euclid~X0)))){\unicdbdjdcd}\\
((r2\_euclidlp~X0~X1~X2){\unicdbdjded}(r1\_euclidlp~X0~X1~X2))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchcdpffgfhdgmgjgegmgah}
\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}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{v6\_membered}~{k4\_ordinal1}}\label{hyp:ggdggdpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(m1\_subset\_1~(k1\_real\_1~\\
X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdpfchfgbgmgpfbd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_finseq\_2~\\
X1~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_finseq\_2~X2~k1\_numbers~\\
(k1\_euclid~X0)){\unicdbdjdcd}((r3\_euclidlp~X0~X1~X2){\unicdbdjded}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k1\_numbers){\unicdbdjdcd}((k7\_euclid~\\
X0~(k9\_euclid~X0~X1~X3)~(k9\_euclid~X0~X2~X4)=k5\_euclid~X0){\unicdbdjdcd}((X3=\\
k6\_numbers){\unicdcdcdhd}(X4=k6\_numbers))))))))\end{array}
\label{hyp:egcdpffgfhdgmgjgegmgah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_finseq\_2~\\
X1~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_finseq\_2~X2~k1\_numbers~\\
(k1\_euclid~X0)){\unicdbdjdcd}((r1\_euclidlp~X0~X1~X2){\unicdbdjded}((X1{\unicdcdgdad}k5\_euclid~X0){\unicdcdcdhd}(\\
(X2{\unicdcdgdad}k5\_euclid~X0){\unicdcdcdhd}({\unicdcdaddd} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdcdcdhd}(\\
X1=k9\_euclid~X0~X2~X3)))))))\end{array}
\label{hyp:egbdpffgfhdgmgjgegmgah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v6\_membered~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
(v7\_ordinal1~X1))\end{array}
\label{hyp:dgdgbdhdpfngfgngcgfgchfgeg}
\end{equation}
\begin{theorem}\label{thm:mgedfdpffgfhdgmgjgegmgah}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_finseq\_2~\\
X1~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_finseq\_2~X2~k1\_numbers~\\
(k1\_euclid~X0)){\unicdbdjdcd}(\preoh (r2\_euclidlp~X0~X1~X2){\unicdcdcdhd}(r3\_euclidlp~X0~X1~X2))))\end{array}$$
\end{theorem}
\end{document}
