\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{t91\_xxreal\_3 (TMH64UVKQBiXghzDPnhH1Nf5cmz4TEfYtqN)}
\maketitle
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k4\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_xxreal\_3 : {{\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 $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\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 $ k1\_numbers : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_xxreal\_0 : {\iota}$ be given.
Let $ k1\_xxreal\_0 : {\iota}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}(k3\_xcmplx\_0~X0~(k4\_xcmplx\_0~np\_\_1)=\\
k4\_xcmplx\_0~X0)\end{array}
\label{hyp:dhahdgcdpfbgchjgehigng}
\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}
{{{k4\_xcmplx\_0}~{({{k4\_xcmplx\_0}~{np\_\_1}})}}={np\_\_1}}\label{hyp:chbhcffgbgmgoefghgpfpflgedpfihdgngahmgihpfadpfpfchngbdpfchbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{k4\_xcmplx\_0}~{({{k4\_xcmplx\_0}~{X0}})}}={X0}})}}}\label{hyp:jgogghpgmgfhehjgghfgogfgdhdhpflgedpfihdgngahmgihpfad}
\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 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}\\
((v1\_xreal\_0~X1){\unicdcdcdhd}((v1\_xcmplx\_0~X2){\unicdcdcdhd}(v1\_xcmplx\_0~X3)))){\unicdbdjdcd}(((\\
X0=X2){\unicdcdcdhd}(X1=X3)){\unicdbdjdcd}(k4\_xxreal\_3~X0~X1=k3\_xcmplx\_0~X2~X3))\end{array}
\label{hyp:jgfgedpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}((\\
X0=X1){\unicdbdjdcd}(k2\_xxreal\_3~X0=k4\_xcmplx\_0~X1))\end{array}
\label{hyp:jgfgcdpfihihchfgbgmgpfdd}
\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\_xreal\_0~X0){\unicdbdjdcd}((v1\_xcmplx\_0~(k4\_xcmplx\_0~X0)){\unicdcdcdhd}\\
(v1\_xreal\_0~(k4\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgddpfihchfgbgmgpfad}
\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}
{\preoh{{{v1\_xreal\_0}~{k2\_xxreal\_0}}}}\label{hyp:ggdgbdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v3\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X0)){\unicdbdjdcd}((v1\_xcmplx\_0~\\
(k4\_xcmplx\_0~X0)){\unicdcdcdhd}(\preoh v2\_xxreal\_0~(k4\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgbdgdpfihchfgbgmgpfad}
\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=\\
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}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}((\\
(v1\_xreal\_0~X0){\unicdbdjdcd}((X1=k2\_xxreal\_3~X0){\unicdbdjded}({\unicdcdaddd} X2 .  (v1\_xcmplx\_0~\\
X2){\unicdcdcdhd}((X0=X2){\unicdcdcdhd}(X1=k4\_xcmplx\_0~X2))))){\unicdcdcdhd}(((X0=k1\_xxreal\_0){\unicdbdjdcd}\\
((X1=k2\_xxreal\_3~X0){\unicdbdjded}(X1=k2\_xxreal\_0))){\unicdcdcdhd}(\preoh (\preoh v1\_xreal\_0~X0){\unicdcdcdhd}\\
((X0{\unicdcdgdad}k1\_xxreal\_0){\unicdcdcdhd}(\preoh (X1=k2\_xxreal\_3~X0){\unicdbdjded}(X1=k1\_xxreal\_0)))))))\end{array}
\label{hyp:egddpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{k1\_xxreal\_0}={k1\_numbers}}\label{hyp:egcdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(\\
k4\_xxreal\_3~X0~X1=k4\_xxreal\_3~X1~X0)\end{array}
\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgedpfihihchfgbgmgpfdd}
\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:ehjdbdpfihihchfgbgmgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}(k4\_xxreal\_3~(k2\_xxreal\_3~np\_\_1)~\\
X0=k2\_xxreal\_3~X0)\end{array}$$
\end{theorem}
\end{document}
