\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{l23\_scmpds\_9 (TMWAfcmY6huDDa2oo496kQ4aRsmC5vBDoW1)}
\maketitle
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k18\_complex1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(r1\_xxreal\_0~k6\_numbers~(k3\_xcmplx\_0~\\
X0~X0))\end{array}
\label{hyp:ehgdddpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (\preoh \\
r1\_xxreal\_0~X1~(k4\_xcmplx\_0~X0)){\unicdcdcdhd}(r1\_xxreal\_0~(k2\_xcmplx\_0~X1~\\
X0)~k6\_numbers)))\end{array}
\label{hyp:ehgdcdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(((r1\_xxreal\_0~\\
(k4\_xcmplx\_0~X0)~X1){\unicdcdcdhd}(r1\_xxreal\_0~X1~X0)){\unicdbdjded}(r1\_xxreal\_0~(k18\_complex1~\\
X1)~X0)))\end{array}
\label{hyp:ehfdpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((\preoh (\preoh r1\_xxreal\_0~k6\_numbers~X0){\unicdcdcdhd}\\
(r1\_xxreal\_0~(k4\_xcmplx\_0~X0)~k6\_numbers)){\unicdcdcdhd}(\preoh (\preoh r1\_xxreal\_0~\\
(k4\_xcmplx\_0~X0)~k6\_numbers){\unicdcdcdhd}(r1\_xxreal\_0~k6\_numbers~X0)))\end{array}
\label{hyp:ehfdidpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}((r1\_xxreal\_0~\\
X0~X1){\unicdbdjdcd}(r1\_xxreal\_0~(k6\_xcmplx\_0~X0~X1)~k6\_numbers)))\end{array}
\label{hyp:ehedhdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (\preoh \\
r1\_xxreal\_0~X0~k6\_numbers){\unicdcdcdhd}(r1\_xxreal\_0~X1~(k6\_xcmplx\_0~X1~X0))))\end{array}
\label{hyp:ehededpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X0~k6\_numbers){\unicdcdcdhd}((r1\_xxreal\_0~\\
X1~X2){\unicdcdcdhd}(r1\_xxreal\_0~(k2\_xcmplx\_0~X0~X2)~X1)))))\end{array}
\label{hyp:ehddjdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((r1\_xxreal\_0~X0~k6\_numbers){\unicdbdjdcd}(k18\_complex1~\\
X0=k4\_xcmplx\_0~X0))\end{array}
\label{hyp:ehddadpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{{k3\_xcmplx\_0}~{X0}}~{k6\_numbers}}={k6\_numbers}})}}}\label{hyp:ehcdpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((X0=k6\_numbers){\unicdbdjded}(k18\_complex1~\\
X0=k6\_numbers))\end{array}
\label{hyp:ehcdpfbgcgdhghbgmgfhfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}(X0=\\
k2\_xcmplx\_0~(k6\_xcmplx\_0~X0~X1)~X1))\end{array}
\label{hyp:ehcdhdpfihdgngahmgihpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}((r1\_xxreal\_0~\\
(k4\_xcmplx\_0~X0)~X1){\unicdbdjdcd}(r1\_xxreal\_0~(k4\_xcmplx\_0~X1)~X0)))\end{array}
\label{hyp:ehcdgdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}(X0=\\
k6\_xcmplx\_0~(k2\_xcmplx\_0~X0~X1)~X1))\end{array}
\label{hyp:ehcdgdpfihdgngahmgihpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}((r1\_xxreal\_0~\\
X0~X1){\unicdbdjded}(r1\_xxreal\_0~(k4\_xcmplx\_0~X1)~(k4\_xcmplx\_0~X0))))\end{array}
\label{hyp:ehcdedpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}((r1\_xxreal\_0~X0~(k2\_xcmplx\_0~X1~X2)){\unicdbdjded}(r1\_xxreal\_0~\\
(k6\_xcmplx\_0~X0~X1)~X2))))\end{array}
\label{hyp:ehcdadpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(((\preoh \\
r1\_xxreal\_0~X1~(k4\_xcmplx\_0~X0)){\unicdcdcdhd}(\preoh r1\_xxreal\_0~X0~X1)){\unicdbdjded}(\preoh r1\_xxreal\_0~\\
X0~(k18\_complex1~X1))))\end{array}
\label{hyp:ehbdpfdhfgbhpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{{k2\_xcmplx\_0}~{X0}}~{k6\_numbers}}={X0}})}}}\label{hyp:ehbdpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (r1\_xxreal\_0~\\
X0~X1){\unicdcdcdhd}(r1\_xxreal\_0~(k2\_xcmplx\_0~X1~np\_\_1)~X0)))\end{array}
\label{hyp:ehbdedfdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}((k4\_xcmplx\_0~X0=k6\_numbers){\unicdbdjdcd}(\\
X0=k6\_numbers))\end{array}
\label{hyp:ehbdddedpfihdgngahmgihpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (\preoh \\
r1\_xxreal\_0~X0~k6\_numbers){\unicdcdcdhd}((\preoh r1\_xxreal\_0~k6\_numbers~X1){\unicdcdcdhd}(r1\_xxreal\_0~\\
k6\_numbers~(k3\_xcmplx\_0~X0~X1)))))\end{array}
\label{hyp:ehbdddcdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(k18\_complex1~\\
(k6\_xcmplx\_0~X0~X1)=k18\_complex1~(k6\_xcmplx\_0~X1~X0)))\end{array}
\label{hyp:ehbdbdpffhogjgggpgchngbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}(\\
k6\_xcmplx\_0~(k4\_xcmplx\_0~X0)~(k4\_xcmplx\_0~X1)=k6\_xcmplx\_0~X1~\\
X0)\end{array}
\label{hyp:dhahdgjdpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}(\\
k2\_xcmplx\_0~(k4\_xcmplx\_0~X0)~(k4\_xcmplx\_0~X1)=k4\_xcmplx\_0~(k2\_xcmplx\_0~\\
X0~X1))\end{array}
\label{hyp:dhahdgidpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}((v1\_xcmplx\_0~\\
X1){\unicdcdcdhd}(v1\_xcmplx\_0~X2))){\unicdbdjdcd}(k2\_xcmplx\_0~(k2\_xcmplx\_0~X0~X1)~X2=k2\_xcmplx\_0~\\
X0~(k2\_xcmplx\_0~X1~X2))\end{array}
\label{hyp:dhahdggdpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}(\\
k2\_xcmplx\_0~X0~(k4\_xcmplx\_0~X1)=k6\_xcmplx\_0~X0~X1)\end{array}
\label{hyp:dhahdgbdpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(((r1\_xxreal\_0~\\
X0~X1){\unicdcdcdhd}(r1\_xxreal\_0~X1~X0)){\unicdbdjdcd}(X0=X1)))\end{array}
\label{hyp:mgfdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (\preoh \\
r1\_xxreal\_0~X1~X0){\unicdcdcdhd}(r1\_xxreal\_0~k6\_numbers~(k6\_xcmplx\_0~X0~X1))))\end{array}
\label{hyp:mgfdedpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (\preoh \\
r1\_xxreal\_0~X1~X0){\unicdcdcdhd}(r1\_xxreal\_0~(k6\_xcmplx\_0~X1~X0)~k6\_numbers)))\end{array}
\label{hyp:mgedgdpfihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}((r1\_xxreal\_0~X0~X1){\unicdbdjdcd}((r1\_xxreal\_0~X2~k6\_numbers){\unicdcdcdid}\\
((\preoh r1\_xxreal\_0~(k2\_xcmplx\_0~X1~X2)~X0){\unicdcdcdhd}(\preoh r1\_xxreal\_0~X1~(k6\_xcmplx\_0~\\
X0~X2)))))))\end{array}
\label{hyp:mgbdpfchjgogggdhfhahbd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}(v1\_xreal\_0~\\
(k6\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdghdpfihchfgbgmgpfad}
\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\_xreal\_0~\\
(k2\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdgfdpfihchfgbgmgpfad}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}(\\
v1\_xcmplx\_0~(k2\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdgcdpfihdgngahmgihpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{({{k4\_xcmplx\_0}~{X0}})}})}}}\label{hyp:egehpflgedpfihdgngahmgihpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}(m1\_subset\_1~(k18\_complex1~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdidpfdgpgngahmgfgihbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}((\\
X1=k4\_xcmplx\_0~X0){\unicdbdjded}(k2\_xcmplx\_0~X0~X1=k6\_numbers)))\end{array}
\label{hyp:eggdpfihdgngahmgihpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}(\\
k2\_xcmplx\_0~X0~X1=k2\_xcmplx\_0~X1~X0)\end{array}
\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgcdpfihdgngahmgihpfad}
\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:mgcdddpfdhdgngahegdhpfjd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(\preoh (\preoh \\
r1\_xxreal\_0~X0~k6\_numbers){\unicdcdcdhd}(k2\_xcmplx\_0~(k2\_xcmplx\_0~(k18\_complex1~\\
X1)~X0)~np\_\_1=k4\_xcmplx\_0~(k2\_xcmplx\_0~(k2\_xcmplx\_0~(k2\_xcmplx\_0~\\
(k18\_complex1~X1)~X0)~X0)~X1))))\end{array}$$
\end{theorem}
\end{document}
