\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{t17\_jgraph\_6 (TMM9bS3c3z1RV47sTnQ7c7HiyanUWZZJ69o)}
\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\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_compts\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k12\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ r1\_jordan6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ 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 $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
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 .  ((\preoh v1\_xboole\_0~X4){\unicdcdcdhd}((v2\_compts\_1~X4~(k15\_euclid~\\
np\_\_2)){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2)))))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(k1\_zfmisc\_1~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2)))){\unicdbdjdcd}(((X4=ReplSep~(toset~({\uniddcgcg} X6 : \iota .  m1\_subset\_1~\\
X6~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X6 : \iota .  k12\_euclid~\\
X6=np\_\_1)~({\uniddcgcg} X6 : \iota .  X6)){\unicdcdcdhd}((r1\_jordan6~X4~X0~X1){\unicdcdcdhd}((r1\_jordan6~\\
X4~X1~X2){\unicdcdcdhd}(r1\_jordan6~X4~X2~X3)))){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}\\
((v1\_funct\_2~X6~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))){\unicdcdcdhd}(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~\\
k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))))))){\unicdbdjdcd}({\unicdcdadad} X7 .  \\
((v1\_funct\_1~X7){\unicdcdcdhd}((v1\_funct\_2~X7~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))){\unicdcdcdhd}(m1\_subset\_1~X7~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))))))){\unicdbdjdcd}\\
(\preoh (v5\_pre\_topc~X6~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~\\
X6){\unicdcdcdhd}((v5\_pre\_topc~X7~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~\\
X7){\unicdcdcdhd}((X5=ReplSep~(toset~({\uniddcgcg} X8 : \iota .  m1\_subset\_1~X8~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))~({\uniddcgcg} X8 : \iota .  r1\_xxreal\_0~(k12\_euclid~X8)~\\
np\_\_1)~({\uniddcgcg} X8 : \iota .  X8)){\unicdcdcdhd}((k1\_funct\_1~X6~k6\_numbers=X2){\unicdcdcdhd}((k1\_funct\_1~\\
X6~np\_\_1=X0){\unicdcdcdhd}((k1\_funct\_1~X7~k6\_numbers=X1){\unicdcdcdhd}((k1\_funct\_1~X7~\\
np\_\_1=X3){\unicdcdcdhd}((r1\_tarski~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X6)~X5){\unicdcdcdhd}((r1\_tarski~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X7)~X5){\unicdcdcdhd}(r1\_xboole\_0~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X6)~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X7)))))))))))))))))))))\end{array}
\label{hyp:ehbdgdpfkghgchbgahigpfgd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~(u1\_struct\_0~k5\_topmetr)~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))))))){\unicdbdjdcd}(\preoh (v5\_pre\_topc~X0~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}\\
((v2\_funct\_1~X0){\unicdcdcdhd}({\unicdcdadad} X1 .  ((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~\\
X1~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdcdcdhd}\\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~k5\_topmetr)~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2))))))){\unicdbdjdcd}(\preoh (k1\_funct\_1~X1~k6\_numbers=\\
k1\_funct\_1~X0~np\_\_1){\unicdcdcdhd}((k1\_funct\_1~X1~np\_\_1=k1\_funct\_1~X0~k6\_numbers){\unicdcdcdhd}\\
((k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1=k2\_relset\_1~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2))~X0){\unicdcdcdhd}((v5\_pre\_topc~X1~k5\_topmetr~\\
(k15\_euclid~np\_\_2)){\unicdcdcdhd}(v2\_funct\_1~X1))))))))\end{array}
\label{hyp:ehbdcdpfkghgchbgahigpffd}
\end{equation}
\begin{theorem}\label{thm:ehbdhdpfkghgchbgahigpfgd}
$$\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 .  ((\preoh v1\_xboole\_0~X4){\unicdcdcdhd}((v2\_compts\_1~X4~(k15\_euclid~\\
np\_\_2)){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2)))))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(k1\_zfmisc\_1~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2)))){\unicdbdjdcd}(((X4=ReplSep~(toset~({\uniddcgcg} X6 : \iota .  m1\_subset\_1~\\
X6~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X6 : \iota .  k12\_euclid~\\
X6=np\_\_1)~({\uniddcgcg} X6 : \iota .  X6)){\unicdcdcdhd}((r1\_jordan6~X4~X0~X1){\unicdcdcdhd}((r1\_jordan6~\\
X4~X1~X2){\unicdcdcdhd}(r1\_jordan6~X4~X2~X3)))){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}\\
((v1\_funct\_2~X6~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))){\unicdcdcdhd}(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~\\
k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))))))){\unicdbdjdcd}({\unicdcdadad} X7 .  \\
((v1\_funct\_1~X7){\unicdcdcdhd}((v1\_funct\_2~X7~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))){\unicdcdcdhd}(m1\_subset\_1~X7~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))))))){\unicdbdjdcd}\\
(\preoh (v5\_pre\_topc~X6~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~\\
X6){\unicdcdcdhd}((v5\_pre\_topc~X7~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~\\
X7){\unicdcdcdhd}((X5=ReplSep~(toset~({\uniddcgcg} X8 : \iota .  m1\_subset\_1~X8~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))~({\uniddcgcg} X8 : \iota .  r1\_xxreal\_0~(k12\_euclid~X8)~\\
np\_\_1)~({\uniddcgcg} X8 : \iota .  X8)){\unicdcdcdhd}((k1\_funct\_1~X6~k6\_numbers=X2){\unicdcdcdhd}((k1\_funct\_1~\\
X6~np\_\_1=X0){\unicdcdcdhd}((k1\_funct\_1~X7~k6\_numbers=X3){\unicdcdcdhd}((k1\_funct\_1~X7~\\
np\_\_1=X1){\unicdcdcdhd}((r1\_tarski~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X6)~X5){\unicdcdcdhd}((r1\_tarski~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X7)~X5){\unicdcdcdhd}(r1\_xboole\_0~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X6)~(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X7)))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
