\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{t19\_topreala (TMJZvYHspKtfYsko99N1SvhyAXzRaeeW1MW)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_topmetr : {\iota}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ m1\_rcomp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m1\_connsp\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_pre\_topc : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_rcomp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k6\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v3\_topmetr : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_pre\_topc~X0){\unicdcdcdhd}(l1\_pre\_topc~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}(((v3\_pre\_topc~\\
X1~X0){\unicdcdcdhd}(X2 \in X1)){\unicdbdjdcd}(m1\_connsp\_2~X1~X0~X2))))\end{array}
\label{hyp:ehddpfdgpgogogdhahpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(k1\_zfmisc\_1~(u1\_struct\_0~k3\_topmetr))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  (v1\_xreal\_0~X2){\unicdbdjdcd}((X0=\\
k2\_rcomp\_1~X1~X2){\unicdbdjdcd}(v3\_pre\_topc~X0~k3\_topmetr))))\end{array}
\label{hyp:ehddfdpfkgpgchegbgoggd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}((\preoh r1\_xxreal\_0~\\
X1~k6\_numbers){\unicdbdjdcd}(X0 \in k2\_rcomp\_1~(k6\_xcmplx\_0~X0~X1)~(k2\_xcmplx\_0~\\
X0~X1))))\end{array}
\label{hyp:ehbdfdpfehpgahchfgbgmggd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}(v1\_xreal\_0~\\
(k6\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdghdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}(v1\_xreal\_0~\\
(k2\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdgfdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{({{v2\_pre\_topc}~{k3\_topmetr}})}{\unicdcdcdhd}{({{v3\_topmetr}~{k3\_topmetr}})}}\label{hyp:ggdgfdpfehpgahngfgehch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v3\_topmetr~X0){\unicdcdcdhd}(l1\_struct\_0~X0)){\unicdbdjdcd}(v3\_membered~\\
(u1\_struct\_0~X0))\end{array}
\label{hyp:ggdgedpfehpgahngfgehch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\preoh v2\_struct\_0~k3\_topmetr){\unicdcdcdhd}((v1\_pre\_topc~k3\_topmetr){\unicdcdcdhd}(v2\_pre\_topc~\\
k3\_topmetr))\end{array}
\label{hyp:ggdgcdpfehpgahngfgehch}
\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}
{{({{v2\_pre\_topc}~{k3\_topmetr}})}{\unicdcdcdhd}{({{l1\_pre\_topc}~{k3\_topmetr}})}}\label{hyp:egehpflgddpfehpgahngfgehch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~\\
k1\_numbers)){\unicdbdjdcd}((m1\_rcomp\_1~X1~X0){\unicdbdjded}({\unicdcdaddd} X2 .  (v1\_xreal\_0~X2){\unicdcdcdhd}\\
((\preoh r1\_xxreal\_0~X2~k6\_numbers){\unicdcdcdhd}(X1=k2\_rcomp\_1~(k6\_xcmplx\_0~X0~\\
X2)~(k2\_xcmplx\_0~X0~X2))))))\end{array}
\label{hyp:eggdpfchdgpgngahpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v3\_membered~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
(v1\_xreal\_0~X1))\end{array}
\label{hyp:dgdgbdedpfngfgngcgfgchfgeg}
\end{equation}
\begin{theorem}\label{thm:ehbdjdpfehpgahchfgbgmgbg}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~k3\_topmetr)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(u1\_struct\_0~k3\_topmetr))){\unicdbdjdcd}(((X2=X1){\unicdcdcdhd}(m1\_rcomp\_1~\\
X1~X0)){\unicdbdjdcd}(m1\_connsp\_2~X2~k3\_topmetr~X0))))\end{array}$$
\end{theorem}
\end{document}
