\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{t101\_jgraph\_7 (TMWR3bv6JFhnzhmK96tMd6YQ1zNVkJygcFv)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k15\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k17\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k18\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_topreal1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_jgraph\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ r1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k5\_topmetr : {\iota}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v5\_pre\_topc : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ v2\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ np\_\_1 : {\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.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~X0))){\unicdbdjdcd}({\unicdcdadad} X3 .  (\\
m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~X0))){\unicdbdjdcd}(\preoh (r1\_topreal1~\\
(k15\_euclid~X0)~X2~X3~X1){\unicdcdcdhd}({\unicdcdadad} X4 .  ((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~\\
X4~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~X0))){\unicdcdcdhd}(\\
m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~k5\_topmetr)~\\
(u1\_struct\_0~(k15\_euclid~X0))))))){\unicdbdjdcd}(\preoh (v5\_pre\_topc~X4~k5\_topmetr~\\
(k15\_euclid~X0)){\unicdcdcdhd}((v2\_funct\_1~X4){\unicdcdcdhd}((k2\_relset\_1~(u1\_struct\_0~\\
(k15\_euclid~X0))~X4=X1){\unicdcdcdhd}((k1\_funct\_1~X4~k6\_numbers=X2){\unicdcdcdhd}(k1\_funct\_1~\\
X4~np\_\_1=X3))))))))))\end{array}
\label{hyp:ehcdpfkghgchbgahigpfhd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  (v1\_xreal\_0~X4){\unicdbdjdcd}({\unicdcdadad} X5 .  (v1\_xreal\_0~X5){\unicdbdjdcd}({\unicdcdadad} X6 .  \\
(v1\_xreal\_0~X6){\unicdbdjdcd}({\unicdcdadad} X7 .  (v1\_xreal\_0~X7){\unicdbdjdcd}({\unicdcdadad} X8 .  ((v1\_funct\_1~\\
X8){\unicdcdcdhd}((v1\_funct\_2~X8~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))){\unicdcdcdhd}(m1\_subset\_1~X8~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~\\
k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))))))){\unicdbdjdcd}({\unicdcdadad} X9 .  \\
((v1\_funct\_1~X9){\unicdcdcdhd}((v1\_funct\_2~X9~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))){\unicdcdcdhd}(m1\_subset\_1~X9~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))))))){\unicdbdjdcd}\\
(\preoh (\preoh r1\_xxreal\_0~X5~X4){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X7~X6){\unicdcdcdhd}((k17\_euclid~X0=\\
X4){\unicdcdcdhd}((k17\_euclid~X1=X5){\unicdcdcdhd}((k17\_euclid~X2=X5){\unicdcdcdhd}((k17\_euclid~X3=\\
X5){\unicdcdcdhd}((r1\_xxreal\_0~X6~(k18\_euclid~X0)){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~\\
X0)~X7){\unicdcdcdhd}((r1\_xxreal\_0~X6~(k18\_euclid~X3)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~(k18\_euclid~\\
X2)~(k18\_euclid~X3)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~(k18\_euclid~X1)~(k18\_euclid~\\
X2)){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X1)~X7){\unicdcdcdhd}((k1\_funct\_1~X8~k6\_numbers=\\
X0){\unicdcdcdhd}((k1\_funct\_1~X8~np\_\_1=X2){\unicdcdcdhd}((k1\_funct\_1~X9~k6\_numbers=X1){\unicdcdcdhd}\\
((k1\_funct\_1~X9~np\_\_1=X3){\unicdcdcdhd}((v5\_pre\_topc~X8~k5\_topmetr~(k15\_euclid~\\
np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~X8){\unicdcdcdhd}((v5\_pre\_topc~X9~k5\_topmetr~(k15\_euclid~\\
np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~X9){\unicdcdcdhd}((r1\_tarski~(k2\_relset\_1~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))~X8)~(k2\_jgraph\_6~X4~X5~X6~X7)){\unicdcdcdhd}((r1\_tarski~\\
(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X9)~(k2\_jgraph\_6~\\
X4~X5~X6~X7)){\unicdcdcdhd}(r1\_xboole\_0~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X8)~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X9)))))))))))))))))))))))))))))))))\end{array}
\label{hyp:ehbdadadpfkghgchbgahigpfhd}
\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}
\begin{theorem}\label{thm:ehbdadbdpfkghgchbgahigpfhd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  (v1\_xreal\_0~X4){\unicdbdjdcd}({\unicdcdadad} X5 .  (v1\_xreal\_0~X5){\unicdbdjdcd}({\unicdcdadad} X6 .  \\
(v1\_xreal\_0~X6){\unicdbdjdcd}({\unicdcdadad} X7 .  (v1\_xreal\_0~X7){\unicdbdjdcd}({\unicdcdadad} X8 .  (m1\_subset\_1~\\
X8~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}({\unicdcdadad} X9 .  \\
(m1\_subset\_1~X9~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}\\
(\preoh (\preoh r1\_xxreal\_0~X5~X4){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X7~X6){\unicdcdcdhd}((k17\_euclid~X0=\\
X4){\unicdcdcdhd}((k17\_euclid~X1=X5){\unicdcdcdhd}((k17\_euclid~X2=X5){\unicdcdcdhd}((k17\_euclid~X3=\\
X5){\unicdcdcdhd}((r1\_xxreal\_0~X6~(k18\_euclid~X0)){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~\\
X0)~X7){\unicdcdcdhd}((r1\_xxreal\_0~X6~(k18\_euclid~X3)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~(k18\_euclid~\\
X2)~(k18\_euclid~X3)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~(k18\_euclid~X1)~(k18\_euclid~\\
X2)){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X1)~X7){\unicdcdcdhd}((r1\_topreal1~(k15\_euclid~\\
np\_\_2)~X0~X2~X8){\unicdcdcdhd}((r1\_topreal1~(k15\_euclid~np\_\_2)~X1~X3~X9){\unicdcdcdhd}((r1\_tarski~\\
X8~(k2\_jgraph\_6~X4~X5~X6~X7)){\unicdcdcdhd}((r1\_tarski~X9~(k2\_jgraph\_6~X4~X5~X6~\\
X7)){\unicdcdcdhd}(r1\_xboole\_0~X8~X9))))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
