\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{t55\_xxreal\_3 (TMYiaeFkCBVauusCGvJzJmquvt58fd4o4tv)}
\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 $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ v3\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ 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}(\preoh (\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X1=k1\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}(X1=k2\_xxreal\_0)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~(k3\_xxreal\_3~\\
X0~X1)~X2){\unicdcdcdhd}(\preoh (X2{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}((X1{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}((X0{\unicdcdgdad}k2\_xxreal\_0){\unicdcdcdhd}\\
(\preoh r1\_xxreal\_0~X0~(k1\_xxreal\_3~X2~X1))))))))))\end{array}
\label{hyp:ehfdddpfihihchfgbgmgpfdd}
\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){\unicdbdjdcd}((r1\_xxreal\_0~(k1\_xxreal\_3~\\
X1~X0)~X2){\unicdbdjded}(r1\_xxreal\_0~X1~(k3\_xxreal\_3~X2~X0))))))\end{array}
\label{hyp:ehedfdpfihihchfgbgmgpfdd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}(((X0=k1\_xxreal\_0){\unicdbdjdcd}(k2\_xxreal\_3~\\
X0=k2\_xxreal\_0)){\unicdcdcdhd}(((k2\_xxreal\_3~X0=k2\_xxreal\_0){\unicdbdjdcd}(X0=k1\_xxreal\_0)){\unicdcdcdhd}\\
(((X0=k2\_xxreal\_0){\unicdbdjdcd}(k2\_xxreal\_3~X0=k1\_xxreal\_0)){\unicdcdcdhd}((k2\_xxreal\_3~\\
X0=k1\_xxreal\_0){\unicdbdjdcd}(X0=k2\_xxreal\_0)))))\end{array}
\label{hyp:ehcdddpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}((\\
(r1\_xxreal\_0~X0~X1){\unicdcdcdhd}(r1\_xxreal\_0~X1~X0)){\unicdbdjdcd}(X0=X1)))\end{array}
\label{hyp:ehbdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}(\preoh \\
(k3\_xxreal\_3~X0~X1=k1\_xxreal\_0){\unicdcdcdhd}((X0{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}(X1{\unicdcdgdad}k2\_xxreal\_0))))\end{array}
\label{hyp:ehbdidpfihihchfgbgmgpfdd}
\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}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}((X0{\unicdcdgdad}k1\_xxreal\_0){\unicdbdjdcd}((k3\_xxreal\_3~\\
k1\_xxreal\_0~X0=k1\_xxreal\_0){\unicdcdcdhd}(k3\_xxreal\_3~X0~k1\_xxreal\_0=k2\_xxreal\_0)))\end{array}
\label{hyp:ehbdddpfihihchfgbgmgpfdd}
\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}
{{v1\_xxreal\_0}~{k2\_xxreal\_0}}\label{hyp:ggdgcdpfihihchfgbgmgpfad}
\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~(k3\_xxreal\_3~X0~X1))\end{array}
\label{hyp:egehpflgddpfihihchfgbgmgpfdd}
\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}(k3\_xxreal\_3~\\
X0~X1=k1\_xxreal\_3~X0~(k2\_xxreal\_3~X1)))\end{array}
\label{hyp:egedpfihihchfgbgmgpfdd}
\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}})}{\unicdbdjded}{({{X0} \in {k1\_numbers}})}}}\label{hyp:egbdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(\\
(r1\_xxreal\_0~X0~X1){\unicdcdcdid}(r1\_xxreal\_0~X1~X0))\end{array}
\label{hyp:dgpgogogfgdgehfgegogfgdhdhpfchbdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(\\
k1\_xxreal\_3~X0~X1=k1\_xxreal\_3~X1~X0)\end{array}
\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgbdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v2\_xxreal\_0~X0)){\unicdbdjdcd}((\preoh v1\_xboole\_0~\\
X0){\unicdcdcdhd}((v1\_xxreal\_0~X0){\unicdcdcdhd}(\preoh v3\_xxreal\_0~X0)))\end{array}
\label{hyp:dgdgddpfihihchfgbgmgpfad}
\end{equation}
\begin{theorem}\label{thm:ehfdfdpfihihchfgbgmgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xxreal\_0~X2){\unicdbdjdcd}(\preoh (\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X1=k2\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}(X1=k1\_xxreal\_0)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~(k1\_xxreal\_3~\\
X0~X1)~X2){\unicdcdcdhd}(\preoh (X2{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}((X1{\unicdcdgdad}k2\_xxreal\_0){\unicdcdcdhd}((X0{\unicdcdgdad}k2\_xxreal\_0){\unicdcdcdhd}\\
(\preoh r1\_xxreal\_0~X0~(k3\_xxreal\_3~X2~X1))))))))))\end{array}$$
\end{theorem}
\end{document}
