\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}
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\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
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\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{t105\_xxreal\_3 (TMafUXBw2Z8gxwzud7av1sXN7FRWNZbYJWM)}
\maketitle
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_2 : {\iota}$ 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 $ k7\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k5\_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 $ k5\_xxreal\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{{{k4\_xxreal\_3}~{np\_\_2}}~{X0}}={{{k1\_xxreal\_3}~{X0}}~{X0}}})}}}\label{hyp:ehjdedpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{{{k4\_xxreal\_3}~{np\_\_1}}~{X0}}={X0}})}}}\label{hyp:ehidbdpfihihchfgbgmgpfdd}
\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}(k4\_xxreal\_3~X0~(k4\_xxreal\_3~X1~X2)=k4\_xxreal\_3~\\
(k4\_xxreal\_3~X0~X1)~X2)))\end{array}
\label{hyp:ehgdgdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{{k7\_xcmplx\_0}~{np\_\_1}}~{X0}}={{k5\_xcmplx\_0}~{X0}}})}}}\label{hyp:dhahdgddpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_2){\unicdcdcdhd}(m2\_subset\_1~np\_\_2~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_2~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_2~k1\_numbers))\end{array}
\label{hyp:dhahdgcdpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
{{{{k7\_xcmplx\_0}~{np\_\_2}}~{np\_\_2}}={np\_\_1}}\label{hyp:chbhcffgbgmgeejgghpfpflghdpfihdgngahmgihpfadpfpfchcdpfchcdpfchbd}
\end{equation}
Assume the following.
\begin{equation}
{{{{k7\_xcmplx\_0}~{np\_\_1}}~{({{{k7\_xcmplx\_0}~{np\_\_1}}~{np\_\_2}})}}={np\_\_2}}\label{hyp:chbhcffgbgmgeejgghpfpflghdpfihdgngahmgihpfadpfpfchbdpfchogbdegcdpfchcd}
\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}(k6\_xxreal\_3~X0~X1=k7\_xcmplx\_0~X2~X3))\end{array}
\label{hyp:jgfggdpfihihchfgbgmgpfdd}
\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}(k5\_xxreal\_3~X0=k5\_xcmplx\_0~X1))\end{array}
\label{hyp:jgfgfdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((v1\_xcmplx\_0~(k5\_xcmplx\_0~X0)){\unicdcdcdhd}\\
(v1\_xreal\_0~(k5\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgedpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((v1\_xxreal\_0~(k5\_xxreal\_3~X0)){\unicdcdcdhd}\\
(v1\_xreal\_0~(k5\_xxreal\_3~X0)))\end{array}
\label{hyp:ggdgddidpfihihchfgbgmgpfdd}
\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~(k6\_xxreal\_3~X0~X1))\end{array}
\label{hyp:egehpflggdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xxreal\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xxreal\_0}~{({{k5\_xxreal\_3}~{X0}})}})}}}\label{hyp:egehpflgfdpfihihchfgbgmgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}(k6\_xxreal\_3~\\
X0~X1=k4\_xxreal\_3~X0~(k5\_xxreal\_3~X1)))\end{array}
\label{hyp:eghdpfihihchfgbgmgpfdd}
\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\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgddpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgbdpfihdgngahmgihpfad}
\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:ehbdadfdpfihihchfgbgmgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}(k1\_xxreal\_3~(k6\_xxreal\_3~X0~np\_\_2)~\\
(k6\_xxreal\_3~X0~np\_\_2)=X0)\end{array}$$
\end{theorem}
\end{document}
