\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{t31\_xxreal\_3 (TMRwnAXUVhdAMyQ6iLdr4HEj1kBcf3mR2Nz)}
\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 $ k3\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}(k2\_xxreal\_3~\\
(k1\_xxreal\_3~X0~X1)=k1\_xxreal\_3~(k2\_xxreal\_3~X0)~(k2\_xxreal\_3~\\
X1)))\end{array}
\label{hyp:ehjdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}((\\
k1\_xxreal\_3~X0~X1=k6\_numbers){\unicdbdjdcd}(X0=k2\_xxreal\_3~X1)))\end{array}
\label{hyp:ehidpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}(k1\_xxreal\_3~X0~(k2\_xxreal\_3~X0)=\\
k6\_numbers)\end{array}
\label{hyp:ehhdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{{k2\_xxreal\_3}~{k2\_xxreal\_0}}={k1\_xxreal\_0}}\label{hyp:ehfdpfihihchfgbgmgpfdd}
\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=k2\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}(X1=k1\_xxreal\_0)){\unicdcdcdhd}((\preoh (X1=k1\_xxreal\_0){\unicdcdcdhd}\\
(X2=k2\_xxreal\_0)){\unicdcdcdhd}((\preoh (X1=k2\_xxreal\_0){\unicdcdcdhd}(X2=k1\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X2=k2\_xxreal\_0)){\unicdcdcdhd}((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}\\
(X2=k1\_xxreal\_0)){\unicdcdcdhd}(k1\_xxreal\_3~(k1\_xxreal\_3~X0~X1)~X2{\unicdcdgdad}k1\_xxreal\_3~\\
X0~(k1\_xxreal\_3~X1~X2))))))))))\end{array}
\label{hyp:ehcdjdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}((\\
k2\_xxreal\_3~(k1\_xxreal\_3~(k2\_xxreal\_3~X0)~X1)=k3\_xxreal\_3~X0~\\
X1){\unicdcdcdhd}(k2\_xxreal\_3~(k1\_xxreal\_3~(k2\_xxreal\_3~X0)~X1)=k1\_xxreal\_3~\\
X0~(k2\_xxreal\_3~X1))))\end{array}
\label{hyp:ehcdhdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}((\\
k2\_xxreal\_3~(k3\_xxreal\_3~X0~X1)=k1\_xxreal\_3~(k2\_xxreal\_3~X0)~\\
X1){\unicdcdcdhd}(k2\_xxreal\_3~(k3\_xxreal\_3~X0~X1)=k3\_xxreal\_3~X1~X0)))\end{array}
\label{hyp:ehcdgdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}(k2\_xxreal\_3~\\
(k1\_xxreal\_3~X0~X1)=k3\_xxreal\_3~(k2\_xxreal\_3~X1)~X0))\end{array}
\label{hyp:ehcdfdpfihihchfgbgmgpfdd}
\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}((\\
v1\_xreal\_0~X0){\unicdbdjdcd}((k1\_xxreal\_3~(k3\_xxreal\_3~X1~X0)~X0=X1){\unicdcdcdhd}(k3\_xxreal\_3~\\
(k1\_xxreal\_3~X1~X0)~X0=X1))))\end{array}
\label{hyp:ehcdcdpfihihchfgbgmgpfdd}
\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 \\
(\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X1=k1\_xxreal\_0)){\unicdcdcdhd}((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}\\
(X1=k2\_xxreal\_0)){\unicdcdcdhd}((k3\_xxreal\_3~X0~X1 \in k1\_numbers){\unicdcdcdhd}(\preoh (X0 \in k1\_numbers){\unicdcdcdhd}\\
(X1 \in k1\_numbers))))))\end{array}
\label{hyp:ehcdbdpfihihchfgbgmgpfdd}
\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 \\
(\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X1=k2\_xxreal\_0)){\unicdcdcdhd}((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}\\
(X1=k1\_xxreal\_0)){\unicdcdcdhd}((k1\_xxreal\_3~X0~X1 \in k1\_numbers){\unicdcdcdhd}(\preoh (X0 \in k1\_numbers){\unicdcdcdhd}\\
(X1 \in k1\_numbers))))))\end{array}
\label{hyp:ehcdadpfihihchfgbgmgpfdd}
\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=k2\_xxreal\_0){\unicdcdcdhd}((X0{\unicdcdgdad}k2\_xxreal\_0){\unicdcdcdhd}(X1{\unicdcdgdad}k1\_xxreal\_0))))\end{array}
\label{hyp:ehbdjdpfihihchfgbgmgpfdd}
\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}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{{{k3\_xxreal\_3}~{X0}}~{k6\_numbers}}={X0}})}}}\label{hyp:ehbdfdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}((X0{\unicdcdgdad}k2\_xxreal\_0){\unicdbdjdcd}((k3\_xxreal\_3~\\
k2\_xxreal\_0~X0=k2\_xxreal\_0){\unicdcdcdhd}(k3\_xxreal\_3~X0~k2\_xxreal\_0=k1\_xxreal\_0)))\end{array}
\label{hyp:ehbdedpfihihchfgbgmgpfdd}
\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}
{{{k2\_xxreal\_3}~{k1\_xxreal\_0}}={k2\_xxreal\_0}}\label{hyp:mgddidpfihihchfgbgmgpfdd}
\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 \\
(k1\_xxreal\_3~X0~X1=k2\_xxreal\_0){\unicdcdcdhd}((X0{\unicdcdgdad}k2\_xxreal\_0){\unicdcdcdhd}(X1{\unicdcdgdad}k2\_xxreal\_0))))\end{array}
\label{hyp:mgcdddpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}k2\_xxreal\_3~(k1\_xxreal\_3~k1\_xxreal\_0~k2\_xxreal\_0)=k3\_xxreal\_3~\\
(k2\_xxreal\_3~k2\_xxreal\_0)~k1\_xxreal\_0\end{array}
\label{hyp:mgbdgdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{{{k1\_xxreal\_3}~{k1\_xxreal\_0}}~{k1\_xxreal\_0}}={k1\_xxreal\_0}}\label{hyp:mgbdbdpfihihchfgbgmgpfdd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}((v1\_xxreal\_0~\\
(k3\_xxreal\_3~X0~X1)){\unicdcdcdhd}(v1\_xreal\_0~(k3\_xxreal\_3~X0~X1)))\end{array}
\label{hyp:ggdgddpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}((v1\_xxreal\_0~\\
(k1\_xxreal\_3~X0~X1)){\unicdcdcdhd}(v1\_xreal\_0~(k1\_xxreal\_3~X0~X1)))\end{array}
\label{hyp:ggdgcdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xxreal\_0}~{k2\_xxreal\_0}}\label{hyp:ggdgcdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((v1\_xxreal\_0~(k2\_xxreal\_3~X0)){\unicdcdcdhd}\\
(v1\_xreal\_0~(k2\_xxreal\_3~X0)))\end{array}
\label{hyp:ggdgbdpfihihchfgbgmgpfdd}
\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}
{{\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 .  {\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}(\\
k1\_xxreal\_3~X0~X1=k1\_xxreal\_3~X1~X0)\end{array}
\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgbdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{X0} \in {X1}})}{\unicdbdjdcd}{({\preoh{{{X1} \in {X0}}}})}}}}\label{hyp:bgogehjgdhjhngngfgehchjhpfchcdpfigjgegegfgog}
\end{equation}
\begin{theorem}\label{thm:ehddbdpfihihchfgbgmgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xxreal\_0~X2){\unicdbdjdcd}(\preoh (\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X1=k1\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}(X1=k2\_xxreal\_0)){\unicdcdcdhd}((\preoh (X1=k1\_xxreal\_0){\unicdcdcdhd}\\
(X2=k2\_xxreal\_0)){\unicdcdcdhd}((\preoh (X1=k2\_xxreal\_0){\unicdcdcdhd}(X2=k1\_xxreal\_0)){\unicdcdcdhd}\\
((\preoh (X0=k1\_xxreal\_0){\unicdcdcdhd}(X2=k1\_xxreal\_0)){\unicdcdcdhd}((\preoh (X0=k2\_xxreal\_0){\unicdcdcdhd}\\
(X2=k2\_xxreal\_0)){\unicdcdcdhd}(k3\_xxreal\_3~(k3\_xxreal\_3~X0~X1)~X2{\unicdcdgdad}k3\_xxreal\_3~\\
X0~(k1\_xxreal\_3~X1~X2))))))))))\end{array}$$
\end{theorem}
\end{document}
