\documentclass{article}
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\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
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\newcommand\preoh[1]{\lnot #1}
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\begin{document}
\title{t45\_graphsp (TMdXThK2SPuAVb6svYAhbipas1tyRndZypG)}
\maketitle
Let $ m2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k3\_finseq\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k9\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m2\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k8\_graphsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k7\_graphsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_graphsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k5\_graphsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ m1\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ k13\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v3\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}\\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~\\
X2~X0~X1){\unicdbdjded}(m1\_subset\_1~X2~X1))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngcdpfdhfhcgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(m1\_funct\_2~\\
X2~X0~X1)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_funct\_2~X3~X0~X1~X2){\unicdbdjded}(m1\_subset\_1~X3~\\
X2))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngcdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_finseq\_2~X1~X0){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_finseq\_2~\\
X2~X0~X1){\unicdbdjded}(m1\_subset\_1~X2~X1))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngcdpfggjgogdhfgbhpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k3\_finseq\_2}~{X0}}={{k13\_finseq\_1}~{X0}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfggjgogdhfgbhpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{({\preoh{{{v1\_xboole\_0}~{k4\_ordinal1}}}})}{\unicdcdcdhd}{({{v3\_ordinal1}~{k4\_ordinal1}})}}\label{hyp:ggdggdpfpgchegjgogbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{v1\_xboole\_0}~{k1\_numbers}}}}\label{hyp:ggdgbdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {\preoh{{{v1\_xboole\_0}~{({{k13\_finseq\_1}~{X0}})}}}}}\label{hyp:ggdgbdcdpfggjgogdhfgbhpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(m1\_funct\_2~(k9\_funct\_2~\\
X0~X1)~X0~X1)\end{array}
\label{hyp:egehpflgjdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((m1\_subset\_1~X0~(k9\_funct\_2~\\
(k3\_finseq\_2~k1\_numbers)~(k3\_finseq\_2~k1\_numbers))){\unicdcdcdhd}((m1\_subset\_1~\\
X1~(k3\_finseq\_2~k1\_numbers)){\unicdcdcdhd}(v7\_ordinal1~X2))){\unicdbdjdcd}(m2\_subset\_1~\\
(k8\_graphsp~X0~X1~X2)~k1\_numbers~k5\_numbers)\end{array}
\label{hyp:egehpflgidpfhgchbgahigdhah}
\end{equation}
Assume the following.
\begin{equation}
{{{m1\_subset\_1}~{k5\_numbers}}~{({{k1\_zfmisc\_1}~{k1\_numbers}})}}\label{hyp:egehpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{m1\_finseq\_2}~{({{k3\_finseq\_2}~{X0}})}}~{X0}}}\label{hyp:egehpflgddpfggjgogdhfgbhpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m2\_funct\_2~X0~(k3\_finseq\_2~k1\_numbers)~(k3\_finseq\_2~\\
k1\_numbers)~(k9\_funct\_2~(k3\_finseq\_2~k1\_numbers)~(k3\_finseq\_2~\\
k1\_numbers))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_finseq\_2~X1~k1\_numbers~(k3\_finseq\_2~\\
k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X2 .  (v7\_ordinal1~X2){\unicdbdjdcd}(({\unicdcdaddd} X3 .  (m2\_subset\_1~\\
X3~k1\_numbers~k5\_numbers){\unicdcdcdhd}(k7\_graphsp~(k6\_graphsp~(k8\_nat\_1~\\
(k1\_funct\_2~(k3\_finseq\_2~k1\_numbers)~(k3\_finseq\_2~k1\_numbers))~\\
(k5\_graphsp~(k3\_finseq\_2~k1\_numbers)~X0)~X3)~X1)~X2=k1\_xboole\_0)){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m2\_subset\_1~X3~k1\_numbers~k5\_numbers){\unicdbdjdcd}((X3=k8\_graphsp~\\
X0~X1~X2){\unicdbdjded}((k7\_graphsp~(k6\_graphsp~(k8\_nat\_1~(k1\_funct\_2~(k3\_finseq\_2~\\
k1\_numbers)~(k3\_finseq\_2~k1\_numbers))~(k5\_graphsp~(k3\_finseq\_2~\\
k1\_numbers)~X0)~X3)~X1)~X2=k1\_xboole\_0){\unicdcdcdhd}({\unicdcdadad} X4 .  (v7\_ordinal1~\\
X4){\unicdbdjdcd}((k7\_graphsp~(k6\_graphsp~(k8\_nat\_1~(k1\_funct\_2~(k3\_finseq\_2~\\
k1\_numbers)~(k3\_finseq\_2~k1\_numbers))~(k5\_graphsp~(k3\_finseq\_2~\\
k1\_numbers)~X0)~X4)~X1)~X2=k1\_xboole\_0){\unicdbdjdcd}(r1\_xxreal\_0~X3~X4)))))))))\end{array}
\label{hyp:egedpfhgchbgahigdhah}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k4\_ordinal1}})}{\unicdbdjdcd}{({{v7\_ordinal1}~{X0}})}}}\label{hyp:dgdgidpfpgchegjgogbgmgbd}
\end{equation}
\begin{theorem}\label{thm:ehedfdpfhgchbgahigdhah}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m2\_funct\_2~X0~(k3\_finseq\_2~k1\_numbers)~(k3\_finseq\_2~\\
k1\_numbers)~(k9\_funct\_2~(k3\_finseq\_2~k1\_numbers)~(k3\_finseq\_2~\\
k1\_numbers))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_finseq\_2~X1~k1\_numbers~(k3\_finseq\_2~\\
k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~X2~k1\_numbers~k5\_numbers){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m2\_subset\_1~X3~k1\_numbers~k5\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~\\
(k8\_graphsp~X0~X1~X2)~X3){\unicdcdcdhd}(k7\_graphsp~(k6\_graphsp~(k8\_nat\_1~(k1\_funct\_2~\\
(k3\_finseq\_2~k1\_numbers)~(k3\_finseq\_2~k1\_numbers))~(k5\_graphsp~\\
(k3\_finseq\_2~k1\_numbers)~X0)~X3)~X1)~X2=k1\_xboole\_0)))))\end{array}$$
\end{theorem}
\end{document}
