\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{t20\_metric\_3 (TMbH6igiWc83W8ExNEwfLWny21zmtcVioCf)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k1\_metric\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k16\_metric\_3 : {\iota}$ be given.
Let $ k7\_metric\_1 : {\iota}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ k4\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_domain\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k2\_domain\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_domain\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ 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 $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k7\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ 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=k1\_metric\_1~k1\_numbers~k1\_numbers~k7\_metric\_1~X1~X0))\end{array}
\label{hyp:ehjdpfngfgehchjgdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  \preoh (X0{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}((X1{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}\\
(\preoh {\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k2\_zfmisc\_1~X0~X1)){\unicdbdjdcd}(X2=k4\_tarski~\\
(k1\_xtuple\_0~X2)~(k2\_xtuple\_0~X2))))\end{array}
\label{hyp:ehcdcdpfngdgbgchehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k2\_zfmisc\_1~X0~X1)))){\unicdbdjdcd}(k3\_domain\_1~X0~X1~\\
X2=k2\_xtuple\_0~X2)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfegpgngbgjgogpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k2\_zfmisc\_1~X0~X1)))){\unicdbdjdcd}(k2\_domain\_1~X0~X1~\\
X2=k1\_xtuple\_0~X2)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfegpgngbgjgogpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}\\
((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((m1\_subset\_1~X2~X0){\unicdcdcdhd}(m1\_subset\_1~X3~X1)))){\unicdbdjdcd}\\
(k1\_domain\_1~X0~X1~X2~X3=k4\_tarski~X2~X3)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdpfegpgngbgjgogpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xboole\_0}~{k1\_xboole\_0}}\label{hyp:ggdgbdpfihcgpgpgmgfgpfad}
\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 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k2\_zfmisc\_1~X0~X1)))){\unicdbdjdcd}(m1\_subset\_1~(k3\_domain\_1~\\
X0~X1~X2)~X1)\end{array}
\label{hyp:egehpflgddpfegpgngbgjgogpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k2\_zfmisc\_1~X0~X1)))){\unicdbdjdcd}(m1\_subset\_1~(k2\_domain\_1~\\
X0~X1~X2)~X0)\end{array}
\label{hyp:egehpflgcdpfegpgngbgjgogpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k16\_metric\_3){\unicdcdcdhd}((v1\_funct\_2~k16\_metric\_3~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers))~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~k16\_metric\_3~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~\\
(k2\_zfmisc\_1~k1\_numbers~k1\_numbers))~k1\_numbers))))\end{array}
\label{hyp:egehpflgbdgdpfngfgehchjgdgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers))~\\
k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers))~k1\_numbers))))){\unicdbdjdcd}((X0=k16\_metric\_3){\unicdbdjded}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~k1\_numbers){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~\\
X4~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers)){\unicdbdjdcd}(((X5=k1\_domain\_1~k1\_numbers~k1\_numbers~X1~X3){\unicdcdcdhd}\\
(X6=k1\_domain\_1~k1\_numbers~k1\_numbers~X2~X4)){\unicdbdjdcd}(k1\_metric\_1~(\\
k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~\\
X0~X5~X6=k7\_real\_1~(k1\_metric\_1~k1\_numbers~k1\_numbers~k7\_metric\_1~\\
X1~X2)~(k1\_metric\_1~k1\_numbers~k1\_numbers~k7\_metric\_1~X3~X4))))))))))\end{array}
\label{hyp:egbdgdpfngfgehchjgdgpfdd}
\end{equation}
\begin{theorem}\label{thm:ehcdadpfngfgehchjgdgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)){\unicdbdjdcd}\\
(k1\_metric\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)~k16\_metric\_3~X0~X1=k1\_metric\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~k16\_metric\_3~\\
X1~X0))\end{array}$$
\end{theorem}
\end{document}
