\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{t8\_topmetr3 (TMb2PupSahapgp9WrRBa4Jec3beMUKMPPvJ)}
\maketitle
Let $ v1\_xreal\_0 : {{\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\_numbers : {\iota}$ be given.
Let $ k1\_numbers : {\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 $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_topmetr : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v2\_comseq\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_tbsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_seq\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_tbsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_metric\_1 : {\iota}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v8\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v9\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_topmetr : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~k5\_numbers~(u1\_struct\_0~(k2\_topmetr~\\
X0~X1))){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~\\
(u1\_struct\_0~(k2\_topmetr~X0~X1))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~\\
X3){\unicdcdcdhd}((v1\_funct\_2~X3~k5\_numbers~(u1\_struct\_0~k8\_metric\_1)){\unicdcdcdhd}(\\
m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~(u1\_struct\_0~\\
k8\_metric\_1)))))){\unicdbdjdcd}(((r1\_funct\_2~k5\_numbers~(u1\_struct\_0~k8\_metric\_1)~\\
k5\_numbers~(u1\_struct\_0~(k2\_topmetr~X0~X1))~X3~X2){\unicdcdcdhd}(r1\_xxreal\_0~\\
X0~X1)){\unicdbdjdcd}(((v2\_tbsp\_1~X3~k8\_metric\_1){\unicdbdjdcd}(v2\_tbsp\_1~X2~(k2\_topmetr~\\
X0~X1))){\unicdcdcdhd}(((v2\_tbsp\_1~X2~(k2\_topmetr~X0~X1)){\unicdbdjdcd}(v2\_tbsp\_1~X3~k8\_metric\_1)){\unicdcdcdhd}\\
((v2\_tbsp\_1~X3~k8\_metric\_1){\unicdbdjdcd}(k1\_tbsp\_1~k8\_metric\_1~X3=k1\_tbsp\_1~\\
(k2\_topmetr~X0~X1)~X2))))))))\end{array}
\label{hyp:ehhdpfehpgahngfgehchdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~k5\_numbers~(u1\_struct\_0~(k2\_topmetr~\\
X0~X1))){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~\\
(u1\_struct\_0~(k2\_topmetr~X0~X1))))))){\unicdbdjdcd}((r1\_xxreal\_0~X0~X1){\unicdbdjdcd}(\\
(v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~k5\_numbers~(u1\_struct\_0~k8\_metric\_1)){\unicdcdcdhd}\\
(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~(u1\_struct\_0~\\
k8\_metric\_1)))))))))\end{array}
\label{hyp:ehgdpfehpgahngfgehchdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k1\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  ((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~X1~k5\_numbers~(u1\_struct\_0~\\
k8\_metric\_1)){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~\\
(u1\_struct\_0~k8\_metric\_1)))))){\unicdbdjdcd}((r1\_funct\_2~k5\_numbers~k1\_numbers~\\
k5\_numbers~(u1\_struct\_0~k8\_metric\_1)~X0~X1){\unicdbdjdcd}(((v2\_comseq\_2~X0){\unicdbdjdcd}\\
(v2\_tbsp\_1~X1~k8\_metric\_1)){\unicdcdcdhd}(((v2\_tbsp\_1~X1~k8\_metric\_1){\unicdbdjdcd}(v2\_comseq\_2~\\
X0)){\unicdcdcdhd}((v2\_comseq\_2~X0){\unicdbdjdcd}(k2\_seq\_2~X0=k1\_tbsp\_1~k8\_metric\_1~X1))))))\end{array}
\label{hyp:ehedpfehpgahngfgehchdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \\
((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((\preoh v1\_xboole\_0~X3){\unicdcdcdhd}(((v1\_funct\_1~X4){\unicdcdcdhd}((\\
v1\_funct\_2~X4~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X1))))){\unicdcdcdhd}((v1\_funct\_1~X5){\unicdcdcdhd}((v1\_funct\_2~X5~X2~X3){\unicdcdcdhd}(m1\_subset\_1~\\
X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X2~X3)))))))){\unicdbdjdcd}((r1\_funct\_2~X0~X1~\\
X2~X3~X4~X5){\unicdbdjded}(X4=X5))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchbdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_metric\_1~k8\_metric\_1){\unicdcdcdhd}((v6\_metric\_1~k8\_metric\_1){\unicdcdcdhd}((v7\_metric\_1~\\
k8\_metric\_1){\unicdcdcdhd}((v8\_metric\_1~k8\_metric\_1){\unicdcdcdhd}(v9\_metric\_1~k8\_metric\_1))))\end{array}
\label{hyp:ggdggdpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{({\preoh{{{v2\_struct\_0}~{k8\_metric\_1}}}})}{\unicdcdcdhd}{({{v1\_metric\_1}~{k8\_metric\_1}})}}\label{hyp:ggdgfdpfngfgehchjgdgpfbd}
\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}
{\preoh{{{v1\_xboole\_0}~{k1\_numbers}}}}\label{hyp:ggdgbdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v6\_metric\_1~X0){\unicdcdcdhd}((v7\_metric\_1~X0){\unicdcdcdhd}((v8\_metric\_1~\\
X0){\unicdcdcdhd}((v9\_metric\_1~X0){\unicdcdcdhd}(l1\_metric\_1~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_topmetr~\\
X1~X0){\unicdbdjdcd}((v6\_metric\_1~X1){\unicdcdcdhd}((v7\_metric\_1~X1){\unicdcdcdhd}((v8\_metric\_1~X1){\unicdcdcdhd}\\
((v9\_metric\_1~X1){\unicdcdcdhd}(l1\_metric\_1~X1))))))\end{array}
\label{hyp:egehpfngbdpfehpgahngfgehch}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_metric\_1}~{X0}})}{\unicdbdjdcd}{({{l1\_struct\_0}~{X0}})}}}\label{hyp:egehpfmgbdpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{({{v1\_metric\_1}~{k8\_metric\_1}})}{\unicdcdcdhd}{({{l1\_metric\_1}~{k8\_metric\_1}})}}\label{hyp:egehpflgidpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}((\preoh \\
v2\_struct\_0~(k2\_topmetr~X0~X1)){\unicdcdcdhd}((v1\_metric\_1~(k2\_topmetr~X0~\\
X1)){\unicdcdcdhd}(m1\_topmetr~(k2\_topmetr~X0~X1)~k8\_metric\_1)))\end{array}
\label{hyp:egehpflgcdpfehpgahngfgehch}
\end{equation}
\begin{theorem}\label{thm:ehidpfehpgahngfgehchdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~k5\_numbers~(u1\_struct\_0~(k2\_topmetr~\\
X0~X1))){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~\\
(u1\_struct\_0~(k2\_topmetr~X0~X1))))))){\unicdbdjdcd}(((r1\_funct\_2~k5\_numbers~\\
(u1\_struct\_0~(k2\_topmetr~X0~X1))~k5\_numbers~k1\_numbers~X3~X2){\unicdcdcdhd}\\
((r1\_xxreal\_0~X0~X1){\unicdcdcdhd}(v2\_comseq\_2~X2))){\unicdbdjdcd}((v2\_tbsp\_1~X3~(k2\_topmetr~\\
X0~X1)){\unicdcdcdhd}(k2\_seq\_2~X2=k1\_tbsp\_1~(k2\_topmetr~X0~X1)~X3))))))\end{array}$$
\end{theorem}
\end{document}
