\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{t25\_sgraph1 (TMaAU5Xu8hGn1DTLNDSrF1gdWwrMRb2F3md)}
\maketitle
Let $ r2\_sgraph1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k3\_sgraph1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_sgraph1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ g1\_sgraph1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_finset\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_tarski : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ k1\_subset\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_sgraph1~X1))){\unicdbdjdcd}({\unicdcdadad} X3 .  {\unicdcdadad} X4 .  \\
\preoh (X3 \in X1){\unicdcdcdhd}((X4 \in X1){\unicdcdcdhd}((X3{\unicdcdgdad}X4){\unicdcdcdhd}((g1\_sgraph1~X1~X2 \in k3\_sgraph1~X0){\unicdcdcdhd}\\
({\unicdcdadad} X5 .  ((v1\_finset\_1~X5){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(\\
k2\_sgraph1~X1)))){\unicdbdjdcd}(\preoh (X5=k2\_xboole\_0~X2~(k1\_tarski~(k2\_tarski~\\
X3~X4))){\unicdcdcdhd}(g1\_sgraph1~X1~X5 \in k3\_sgraph1~X0))))))))\end{array}
\label{hyp:ehcdedpfdhhgchbgahigbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_sgraph1~X1))){\unicdbdjdcd}({\unicdcdadad} X3 .  {\unicdcdadad} X4 .  \\
((v1\_finset\_1~X4){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_sgraph1~(\\
k2\_xboole\_0~X1~(k1\_tarski~X3)))))){\unicdbdjdcd}(((g1\_sgraph1~X1~X2 \in k3\_sgraph1~\\
X0){\unicdcdcdhd}(X3 \in X0)){\unicdbdjdcd}(g1\_sgraph1~(k2\_xboole\_0~X1~(k1\_tarski~X3))~X4 \in \\
k3\_sgraph1~X0))))\end{array}
\label{hyp:ehcdddpfdhhgchbgahigbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  g1\_sgraph1~k1\_xboole\_0~(k1\_subset\_1~(k2\_sgraph1~k1\_xboole\_0)) \in \\
k3\_sgraph1~X0\end{array}
\label{hyp:ehbdgdpfdhhgchbgahigbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (r2\_sgraph1~X0~X1){\unicdbdjded}((g1\_sgraph1~k1\_xboole\_0~\\
(k1\_subset\_1~(k2\_sgraph1~k1\_xboole\_0)) \in X1){\unicdcdcdhd}(({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(k1\_zfmisc\_1~\\
(k2\_sgraph1~X2))){\unicdbdjdcd}({\unicdcdadad} X4 .  {\unicdcdadad} X5 .  ((v1\_finset\_1~X5){\unicdcdcdhd}\\
(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_sgraph1~(k2\_xboole\_0~X2~(k1\_tarski~\\
X4)))))){\unicdbdjdcd}(((g1\_sgraph1~X2~X3 \in X1){\unicdcdcdhd}(X4 \in X0)){\unicdbdjdcd}((X4 \in X2){\unicdcdcdid}(g1\_sgraph1~\\
(k2\_xboole\_0~X2~(k1\_tarski~X4))~X5 \in X1)))))){\unicdcdcdhd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(k1\_zfmisc\_1~\\
(k2\_sgraph1~X2))){\unicdbdjdcd}({\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \preoh (g1\_sgraph1~X2~X3 \in \\
X1){\unicdcdcdhd}((X4 \in X2){\unicdcdcdhd}((X5 \in X2){\unicdcdcdhd}((X4{\unicdcdgdad}X5){\unicdcdcdhd}((\preoh k2\_tarski~X4~X5 \in X3){\unicdcdcdhd}({\unicdcdadad} X6 .  \\
((v1\_finset\_1~X6){\unicdcdcdhd}(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_sgraph1~X2)))){\unicdbdjdcd}\\
(\preoh (X6=k2\_xboole\_0~X3~(k1\_tarski~(k2\_tarski~X4~X5))){\unicdcdcdhd}(g1\_sgraph1~\\
X2~X6 \in X1))))))))))))\end{array}
\label{hyp:eggdpfdhhgchbgahigbd}
\end{equation}
\begin{theorem}\label{thm:ehcdfdpfdhhgchbgahigbd}
${{\unicdcdadad} X0 .  {{{r2\_sgraph1}~{X0}}~{({{k3\_sgraph1}~{X0}})}}}$.
\end{theorem}
\end{document}
