\documentclass{article}
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\newtheorem{lemma}{Lemma}
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\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
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\newcommand\unicdbdjded{\Leftrightarrow}
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\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
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\begin{document}
\title{t29\_metric\_1 (TMFcf553cYVWbwazNKkkVBbBfv8oBdLmdbi)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_metric\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k12\_metric\_1 : {\iota}$ be given.
Let $ k7\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k7\_metric\_1 : {\iota}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ v1\_funct\_1 : {{\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 $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k1\_numbers){\unicdbdjdcd}((k1\_metric\_1~k1\_numbers~k1\_numbers~k7\_metric\_1~\\
X0~X1=k6\_numbers){\unicdbdjded}(X0=X1)))\end{array}
\label{hyp:ehidpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~np\_\_2){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~\\
np\_\_2){\unicdbdjdcd}(k1\_metric\_1~np\_\_2~np\_\_2~k12\_metric\_1~X0~X1=k6\_numbers))\end{array}
\label{hyp:ehcdhdpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~k1\_numbers){\unicdbdjdcd}(r1\_xxreal\_0~\\
(k1\_metric\_1~k1\_numbers~k1\_numbers~k7\_metric\_1~X0~X1)~(k7\_real\_1~\\
(k1\_metric\_1~k1\_numbers~k1\_numbers~k7\_metric\_1~X0~X2)~(k1\_metric\_1~\\
k1\_numbers~k1\_numbers~k7\_metric\_1~X2~X1)))))\end{array}
\label{hyp:ehbdadpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k6\_numbers}={k1\_xboole\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflggdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  (((v1\_funct\_1~\\
X2){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~X0~\\
X1)~k1\_numbers)))){\unicdcdcdhd}((m1\_subset\_1~X3~X0){\unicdcdcdhd}(m1\_subset\_1~X4~X1))){\unicdbdjdcd}\\
(m1\_subset\_1~(k1\_metric\_1~X0~X1~X2~X3~X4)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k12\_metric\_1){\unicdcdcdhd}((v1\_funct\_2~k12\_metric\_1~(k2\_zfmisc\_1~\\
np\_\_2~np\_\_2)~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~k12\_metric\_1~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k2\_zfmisc\_1~np\_\_2~np\_\_2)~k1\_numbers))))\end{array}
\label{hyp:egehpflgbdcdpfngfgehchjgdgpfbd}
\end{equation}
\begin{theorem}\label{thm:ehcdjdpfngfgehchjgdgpfbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~np\_\_2){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~\\
np\_\_2){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~np\_\_2){\unicdbdjdcd}(r1\_xxreal\_0~(k1\_metric\_1~\\
np\_\_2~np\_\_2~k12\_metric\_1~X0~X2)~(k7\_real\_1~(k1\_metric\_1~np\_\_2~np\_\_2~\\
k12\_metric\_1~X0~X1)~(k1\_metric\_1~np\_\_2~np\_\_2~k12\_metric\_1~X1~X2)))))\end{array}$$
\end{theorem}
\end{document}
