\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{t19\_extreal2 (TMU5vzLQVjWdBSduAW7rWEqkRmpsJVkRp2U)}
\maketitle
Let $ k3\_extreal1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_supinf\_1 : {\iota}$ be given.
Let $ k2\_supinf\_1 : {\iota}$ be given.
Let $ k2\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_xxreal\_0 : {\iota}$ be given.
Let $ k2\_xxreal\_0 : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k7\_numbers : {\iota}$ be given.
Let $ k2\_supinf\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v3\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
{{{k2\_xxreal\_3}~{k1\_xxreal\_0}}={k2\_xxreal\_0}}\label{hyp:ehgdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k7\_numbers){\unicdbdjdcd}((k3\_extreal1~X0=X0){\unicdcdcdid}\\
(k3\_extreal1~X0=k2\_supinf\_2~X0))\end{array}
\label{hyp:ehcdpffgihehchfgbgmgcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k7\_numbers){\unicdbdjdcd}(k3\_extreal1~X0=k3\_extreal1~\\
(k2\_supinf\_2~X0))\end{array}
\label{hyp:ehbdidpffgihehchfgbgmgcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k7\_numbers){\unicdbdjdcd}(k2\_supinf\_2~X0=k2\_xxreal\_3~\\
X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfdhfhahjgogggpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{k2\_supinf\_1}={k2\_xxreal\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfdhfhahjgogggpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k1\_supinf\_1}={k1\_xxreal\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdpfdhfhahjgogggpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{v3\_xxreal\_0}~{k2\_xxreal\_0}}\label{hyp:ggdgedpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k7\_numbers){\unicdbdjdcd}(\preoh v3\_xxreal\_0~(k3\_extreal1~\\
X0))\end{array}
\label{hyp:ggdgbdpffgihehchfgbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
{{{m1\_subset\_1}~{k1\_supinf\_1}}~{k7\_numbers}}\label{hyp:egehpflgbdpfdhfhahjgogggpfbd}
\end{equation}
\begin{theorem}\label{thm:ehbdjdpffgihehchfgbgmgcd}
$$\begin{array}{c}(k3\_extreal1~k1\_supinf\_1=k1\_supinf\_1){\unicdcdcdhd}(k3\_extreal1~k2\_supinf\_1=\\
k1\_supinf\_1)\end{array}$$
\end{theorem}
\end{document}
