\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{t81\_xxreal\_3 (TMP31C8fYf9hgvJ6GuX767CVLHJCNsFKger)}
\maketitle
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k4\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k1\_xxreal\_0 : {\iota}$ be given.
Let $ k2\_xxreal\_0 : {\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 $ k5\_numbers : {\iota}$ be given.
Let $ v3\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{{k3\_xcmplx\_0}~{np\_\_1}}~{X0}}={X0}})}}}\label{hyp:ehddpfbgchjgehigng}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}(\preoh (\preoh X0 \in k1\_numbers){\unicdcdcdhd}((X0{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}\\
(X0{\unicdcdgdad}k2\_xxreal\_0)))\end{array}
\label{hyp:ehbdedpfihihchfgbgmgpfad}
\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}
{{v3\_xxreal\_0}~{k2\_xxreal\_0}}\label{hyp:ggdgedpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{v2\_xxreal\_0}~{k1\_xxreal\_0}}\label{hyp:ggdgddpfihihchfgbgmgpfad}
\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=\\
k4\_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=k3\_xcmplx\_0~X3~X4))))))){\unicdcdcdhd}\\
((\preoh (\preoh (v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdcdcdhd}((((v2\_xxreal\_0~X0){\unicdcdcdhd}\\
(v2\_xxreal\_0~X1)){\unicdcdcdid}((v3\_xxreal\_0~X0){\unicdcdcdhd}(v3\_xxreal\_0~X1))){\unicdcdcdhd}(\preoh (\\
X2=k4\_xxreal\_3~X0~X1){\unicdbdjded}(X2=k1\_xxreal\_0)))){\unicdcdcdhd}((\preoh (\preoh (v1\_xreal\_0~\\
X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdcdcdhd}((((v2\_xxreal\_0~X0){\unicdcdcdhd}(v3\_xxreal\_0~X1)){\unicdcdcdid}\\
((v3\_xxreal\_0~X0){\unicdcdcdhd}(v2\_xxreal\_0~X1))){\unicdcdcdhd}(\preoh (X2=k4\_xxreal\_3~X0~X1){\unicdbdjded}\\
(X2=k2\_xxreal\_0)))){\unicdcdcdhd}(\preoh (\preoh (v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdcdcdhd}\\
((\preoh (\preoh (v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdcdcdhd}(((v2\_xxreal\_0~X0){\unicdcdcdhd}(\\
v2\_xxreal\_0~X1)){\unicdcdcdid}((v3\_xxreal\_0~X0){\unicdcdcdhd}(v3\_xxreal\_0~X1)))){\unicdcdcdhd}((\preoh \\
(\preoh (v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdcdcdhd}(((v2\_xxreal\_0~X0){\unicdcdcdhd}(v3\_xxreal\_0~\\
X1)){\unicdcdcdid}((v3\_xxreal\_0~X0){\unicdcdcdhd}(v2\_xxreal\_0~X1)))){\unicdcdcdhd}(\preoh (X2=k4\_xxreal\_3~\\
X0~X1){\unicdbdjded}(X2=k6\_numbers))))))))))\end{array}
\label{hyp:egfdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{k1\_xxreal\_0}={k1\_numbers}}\label{hyp:egcdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xxreal\_0}~{X0}})}}}\label{hyp:dgdgedpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgddpfihchfgbgmgpfad}
\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:ehidbdpfihihchfgbgmgpfdd}
${{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{{{k4\_xxreal\_3}~{np\_\_1}}~{X0}}={X0}})}}}$.
\end{theorem}
\end{document}
