\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{t52\_xxreal\_3 (TMQNVLuZDhaovECFUZky3MXx7sQc2Q8CdYm)}
\maketitle
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_xxreal\_0 : {\iota}$ be given.
Let $ k2\_xxreal\_0 : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_numbers : {\iota}$ 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.
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{{r1\_xxreal\_0}~{k2\_xxreal\_0}}~{X0}})}}}\label{hyp:ehfdpfihihchfgbgmgpfad}
\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}((r1\_xxreal\_0~(k3\_xxreal\_3~X2~X1)~X0){\unicdbdjdcd}(((X0=\\
k2\_xxreal\_0){\unicdcdcdhd}((X1=k1\_xxreal\_0){\unicdcdcdhd}(\preoh r1\_xxreal\_0~X2~k6\_numbers))){\unicdcdcdid}\\
(((X0=k1\_xxreal\_0){\unicdcdcdhd}((X1=k2\_xxreal\_0){\unicdcdcdhd}(\preoh r1\_xxreal\_0~X2~k6\_numbers))){\unicdcdcdid}\\
(r1\_xxreal\_0~X2~(k1\_xxreal\_3~X0~X1)))))))\end{array}
\label{hyp:ehedbdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{{r1\_xxreal\_0}~{X0}}~{k1\_xxreal\_0}})}}}\label{hyp:ehddpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xxreal\_0}~{k1\_xxreal\_0}}\label{hyp:ggdgbdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(\\
v1\_xxreal\_0~(k1\_xxreal\_3~X0~X1))\end{array}
\label{hyp:egehpflgbdpfihihchfgbgmgpfdd}
\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}
\begin{theorem}\label{thm:ehfdcdpfihihchfgbgmgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xxreal\_0~X2){\unicdbdjdcd}(\preoh (\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X1=k2\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}(X1=k1\_xxreal\_0)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X2~(\\
k1\_xxreal\_3~X0~X1)){\unicdcdcdhd}(\preoh (X0{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}((X1{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}\\
((X2{\unicdcdgdad}k2\_xxreal\_0){\unicdcdcdhd}(\preoh r1\_xxreal\_0~(k3\_xxreal\_3~X2~X1)~X0)))))))))\end{array}$$
\end{theorem}
\end{document}
