\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{l19\_xxreal\_3 (TMdMgomNgXDZchZEV4vWG3c49xyyb8XGZ87)}
\maketitle
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k2\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_xxreal\_0 : {\iota}$ be given.
Let $ k1\_xxreal\_0 : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Assume the following.
\begin{equation}
{{{k2\_xxreal\_3}~{k2\_xxreal\_0}}={k1\_xxreal\_0}}\label{hyp:ehfdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{X0} \in {X1}})}{\unicdbdjdcd}{({{{m1\_subset\_1}~{X0}}~{X1}})}}}}\label{hyp:ehbdpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{{k2\_xxreal\_3}~{({{k2\_xxreal\_3}~{X0}})}}={X0}})}}}\label{hyp:jgogghpgmgfhehjgghfgogfgdhdhpflgcdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xxreal\_0}~{k2\_xxreal\_0}}\label{hyp:ggdgcdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{v1\_xreal\_0}~{k1\_xxreal\_0}}}}\label{hyp:ggdgcdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xxreal\_0}~{k1\_xxreal\_0}}\label{hyp:ggdgbdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xxreal\_0}~{({{k2\_xxreal\_3}~{X0}})}})}}}\label{hyp:egehpflgcdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xxreal\_0~X2){\unicdbdjdcd}((((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}((X2=\\
k1\_xxreal\_3~X0~X1){\unicdbdjded}({\unicdcdaddd} X3 .  (v1\_xcmplx\_0~X3){\unicdcdcdhd}({\unicdcdaddd} X4 .  \\
(v1\_xcmplx\_0~X4){\unicdcdcdhd}((X0=X3){\unicdcdcdhd}((X1=X4){\unicdcdcdhd}(X2=k2\_xcmplx\_0~X3~X4))))))){\unicdcdcdhd}\\
(((((X0=k1\_xxreal\_0){\unicdcdcdhd}(X1{\unicdcdgdad}k2\_xxreal\_0)){\unicdcdcdid}((X1=k1\_xxreal\_0){\unicdcdcdhd}\\
(X0{\unicdcdgdad}k2\_xxreal\_0))){\unicdbdjdcd}((X2=k1\_xxreal\_3~X0~X1){\unicdbdjded}(X2=k1\_xxreal\_0))){\unicdcdcdhd}\\
(((((X0=k2\_xxreal\_0){\unicdcdcdhd}(X1{\unicdcdgdad}k1\_xxreal\_0)){\unicdcdcdid}((X1=k2\_xxreal\_0){\unicdcdcdhd}\\
(X0{\unicdcdgdad}k1\_xxreal\_0))){\unicdbdjdcd}((X2=k1\_xxreal\_3~X0~X1){\unicdbdjded}(X2=k2\_xxreal\_0))){\unicdcdcdhd}\\
(\preoh (\preoh (v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdcdcdhd}((\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}\\
(X1{\unicdcdgdad}k2\_xxreal\_0)){\unicdcdcdhd}((\preoh (X1=k1\_xxreal\_0){\unicdcdcdhd}(X0{\unicdcdgdad}k2\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}(X1{\unicdcdgdad}k1\_xxreal\_0)){\unicdcdcdhd}((\preoh (X1=k2\_xxreal\_0){\unicdcdcdhd}\\
(X0{\unicdcdgdad}k1\_xxreal\_0)){\unicdcdcdhd}(\preoh (X2=k1\_xxreal\_3~X0~X1){\unicdbdjded}(X2=k6\_numbers))))))))))))\end{array}
\label{hyp:egcdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{k1\_xxreal\_0}={k1\_numbers}}\label{hyp:egcdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfchfgbgmgpfbd}
\end{equation}
\begin{theorem}\label{thm:mgbdjdpfihihchfgbgmgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}((X0 \in k1\_numbers){\unicdbdjdcd}(k2\_xxreal\_3~\\
(k1\_xxreal\_3~X0~k2\_xxreal\_0)=k1\_xxreal\_3~(k2\_xxreal\_3~k2\_xxreal\_0)~\\
(k2\_xxreal\_3~X0)))\end{array}$$
\end{theorem}
\end{document}
