\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{t15\_jgraph\_5 (TML66t9UzTkU3tGRea3Bw68bH4yETrZE4vd)}
\maketitle
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_topmetr : {\iota}$ be given.
Let $ k15\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_2 : {\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\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ np\_\_1 : {\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 $ k12\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k18\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k17\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\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.
Let $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k17\_borsuk\_1 : {\iota}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
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}({\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}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  (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}({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(k1\_zfmisc\_1~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~\\
X7~(u1\_struct\_0~k5\_topmetr)){\unicdbdjdcd}({\unicdcdadad} X8 .  (m1\_subset\_1~X8~(u1\_struct\_0~\\
k5\_topmetr)){\unicdbdjdcd}(\preoh (X7=k6\_numbers){\unicdcdcdhd}((X8=np\_\_1){\unicdcdcdhd}((v5\_pre\_topc~\\
X0~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~X0){\unicdcdcdhd}((v5\_pre\_topc~\\
X1~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~X1){\unicdcdcdhd}((X2=ReplSep~\\
(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~\\
({\uniddcgcg} X9 : \iota .  r1\_xxreal\_0~np\_\_1~(k12\_euclid~X9))~({\uniddcgcg} X9 : \iota .  \\
X9)){\unicdcdcdhd}((X3=ReplSep~(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~X9~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))~({\uniddcgcg} X9 : \iota .  (k12\_euclid~X9=np\_\_1){\unicdcdcdhd}((\\
r1\_xxreal\_0~(k18\_euclid~X9)~(k17\_euclid~X9)){\unicdcdcdhd}(r1\_xxreal\_0~(k1\_real\_1~\\
(k17\_euclid~X9))~(k18\_euclid~X9))))~({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}((X4=ReplSep~\\
(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~\\
({\uniddcgcg} X9 : \iota .  (k12\_euclid~X9=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~\\
X9)~(k18\_euclid~X9)){\unicdcdcdhd}(r1\_xxreal\_0~(k18\_euclid~X9)~(k1\_real\_1~\\
(k17\_euclid~X9)))))~({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}((X5=ReplSep~(toset~({\uniddcgcg} X9 : \\
\iota .  m1\_subset\_1~X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X9 : \\
\iota .  (k12\_euclid~X9=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~X9)~(k18\_euclid~\\
X9)){\unicdcdcdhd}(r1\_xxreal\_0~(k1\_real\_1~(k17\_euclid~X9))~(k18\_euclid~X9))))~\\
({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}((X6=ReplSep~(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~\\
X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X9 : \iota .  (k12\_euclid~\\
X9=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X9)~(k17\_euclid~X9)){\unicdcdcdhd}(r1\_xxreal\_0~\\
(k18\_euclid~X9)~(k1\_real\_1~(k17\_euclid~X9)))))~({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}\\
((k3\_funct\_2~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X0~X7 \in X4){\unicdcdcdhd}((k3\_funct\_2~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))~X0~X8 \in X3){\unicdcdcdhd}((k3\_funct\_2~(u1\_struct\_0~k5\_topmetr)~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1~X7 \in X6){\unicdcdcdhd}((k3\_funct\_2~(u1\_struct\_0~\\
k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1~X8 \in X5){\unicdcdcdhd}((r1\_tarski~\\
(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X0)~X2){\unicdcdcdhd}((r1\_tarski~\\
(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1)~X2){\unicdcdcdhd}(r1\_xboole\_0~\\
(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X0)~(k2\_relset\_1~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1)))))))))))))))))))))))))))\end{array}
\label{hyp:ehbdedpfkghgchbgahigpffd}
\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}
Assume the following.
\begin{equation}
{{k5\_topmetr}={k17\_borsuk\_1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfehpgahngfgehch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}\\
(((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~X0~X1))))){\unicdcdcdhd}(m1\_subset\_1~X3~X0))){\unicdbdjdcd}(k3\_funct\_2~X0~\\
X1~X2~X3=k1\_funct\_1~X2~X3)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\preoh v2\_struct\_0~k17\_borsuk\_1){\unicdcdcdhd}((v1\_pre\_topc~k17\_borsuk\_1){\unicdcdcdhd}(\\
v2\_pre\_topc~k17\_borsuk\_1))\end{array}
\label{hyp:ggdgjdpfcgpgchdhfhlgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l1\_struct\_0~X0)){\unicdbdjdcd}(\preoh v1\_xboole\_0~\\
(u1\_struct\_0~X0))\end{array}
\label{hyp:ggdgcdpfdhehchfhdgehpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_pre\_topc}~{X0}})}{\unicdbdjdcd}{({{l1\_struct\_0}~{X0}})}}}\label{hyp:egehpfmgbdpfahchfgpfehpgahdg}
\end{equation}
Assume the following.
\begin{equation}
{{l1\_pre\_topc}~{k17\_borsuk\_1}}\label{hyp:egehpflgbdhdpfcgpgchdhfhlgpfbd}
\end{equation}
\begin{theorem}\label{thm:ehbdfdpfkghgchbgahigpffd}
$$\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}({\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}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  (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}({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(k1\_zfmisc\_1~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2)))){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~\\
X7~(u1\_struct\_0~k5\_topmetr)){\unicdbdjdcd}({\unicdcdadad} X8 .  (m1\_subset\_1~X8~(u1\_struct\_0~\\
k5\_topmetr)){\unicdbdjdcd}(\preoh (X7=k6\_numbers){\unicdcdcdhd}((X8=np\_\_1){\unicdcdcdhd}((v5\_pre\_topc~\\
X0~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~X0){\unicdcdcdhd}((v5\_pre\_topc~\\
X1~k5\_topmetr~(k15\_euclid~np\_\_2)){\unicdcdcdhd}((v2\_funct\_1~X1){\unicdcdcdhd}((X2=ReplSep~\\
(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~\\
({\uniddcgcg} X9 : \iota .  r1\_xxreal\_0~np\_\_1~(k12\_euclid~X9))~({\uniddcgcg} X9 : \iota .  \\
X9)){\unicdcdcdhd}((X3=ReplSep~(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~X9~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))~({\uniddcgcg} X9 : \iota .  (k12\_euclid~X9=np\_\_1){\unicdcdcdhd}((\\
r1\_xxreal\_0~(k18\_euclid~X9)~(k17\_euclid~X9)){\unicdcdcdhd}(r1\_xxreal\_0~(k1\_real\_1~\\
(k17\_euclid~X9))~(k18\_euclid~X9))))~({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}((X4=ReplSep~\\
(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~\\
({\uniddcgcg} X9 : \iota .  (k12\_euclid~X9=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~\\
X9)~(k18\_euclid~X9)){\unicdcdcdhd}(r1\_xxreal\_0~(k18\_euclid~X9)~(k1\_real\_1~\\
(k17\_euclid~X9)))))~({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}((X5=ReplSep~(toset~({\uniddcgcg} X9 : \\
\iota .  m1\_subset\_1~X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X9 : \\
\iota .  (k12\_euclid~X9=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~X9)~(k18\_euclid~\\
X9)){\unicdcdcdhd}(r1\_xxreal\_0~(k1\_real\_1~(k17\_euclid~X9))~(k18\_euclid~X9))))~\\
({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}((X6=ReplSep~(toset~({\uniddcgcg} X9 : \iota .  m1\_subset\_1~\\
X9~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X9 : \iota .  (k12\_euclid~\\
X9=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X9)~(k17\_euclid~X9)){\unicdcdcdhd}(r1\_xxreal\_0~\\
(k18\_euclid~X9)~(k1\_real\_1~(k17\_euclid~X9)))))~({\uniddcgcg} X9 : \iota .  X9)){\unicdcdcdhd}\\
((k3\_funct\_2~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))~X0~X7 \in X4){\unicdcdcdhd}((k3\_funct\_2~(u1\_struct\_0~k5\_topmetr)~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))~X0~X8 \in X3){\unicdcdcdhd}((k3\_funct\_2~(u1\_struct\_0~k5\_topmetr)~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1~X7 \in X5){\unicdcdcdhd}((k3\_funct\_2~(u1\_struct\_0~\\
k5\_topmetr)~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1~X8 \in X6){\unicdcdcdhd}((r1\_tarski~\\
(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X0)~X2){\unicdcdcdhd}((r1\_tarski~\\
(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1)~X2){\unicdcdcdhd}(r1\_xboole\_0~\\
(k2\_relset\_1~(u1\_struct\_0~(k15\_euclid~np\_\_2))~X0)~(k2\_relset\_1~\\
(u1\_struct\_0~(k15\_euclid~np\_\_2))~X1)))))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
