\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{t25\_jgraph\_3 (TMZEcopW3RCoo9Y1uwdXPGoytqHDCgWJn38)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k15\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k17\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k18\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_topreal2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_compts\_1 : {{\iota}{\unicdbdjdcd}{{\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 $ k1\_pre\_topc : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v3\_tops\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ v6\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_pre\_topc : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ l1\_rltopsp1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_pre\_topc : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_rltopsp1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~(u1\_struct\_0~(k1\_pre\_topc~(k15\_euclid~np\_\_2)~X1))~(u1\_struct\_0~\\
(k1\_pre\_topc~(k15\_euclid~np\_\_2)~X0))){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~(k1\_pre\_topc~(k15\_euclid~np\_\_2)~X1))~\\
(u1\_struct\_0~(k1\_pre\_topc~(k15\_euclid~np\_\_2)~X0))))))){\unicdbdjdcd}(((X1=\\
ReplSep~(toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))))~({\uniddcgcg} X3 : \iota .  \preoh (\preoh (k1\_real\_1~np\_\_1=k17\_euclid~X3){\unicdcdcdhd}(\\
(r1\_xxreal\_0~(k1\_real\_1~np\_\_1)~(k18\_euclid~X3)){\unicdcdcdhd}(r1\_xxreal\_0~\\
(k18\_euclid~X3)~np\_\_1))){\unicdcdcdhd}((\preoh (k17\_euclid~X3=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~\\
(k1\_real\_1~np\_\_1)~(k18\_euclid~X3)){\unicdcdcdhd}(r1\_xxreal\_0~(k18\_euclid~\\
X3)~np\_\_1))){\unicdcdcdhd}((\preoh (k1\_real\_1~np\_\_1=k18\_euclid~X3){\unicdcdcdhd}((r1\_xxreal\_0~\\
(k1\_real\_1~np\_\_1)~(k17\_euclid~X3)){\unicdcdcdhd}(r1\_xxreal\_0~(k17\_euclid~\\
X3)~np\_\_1))){\unicdcdcdhd}(\preoh (np\_\_1=k18\_euclid~X3){\unicdcdcdhd}((r1\_xxreal\_0~(k1\_real\_1~\\
np\_\_1)~(k17\_euclid~X3)){\unicdcdcdhd}(r1\_xxreal\_0~(k17\_euclid~X3)~np\_\_1))))))~\\
({\uniddcgcg} X3 : \iota .  X3)){\unicdcdcdhd}(v3\_tops\_2~X2~(k1\_pre\_topc~(k15\_euclid~np\_\_2)~\\
X1)~(k1\_pre\_topc~(k15\_euclid~np\_\_2)~X0))){\unicdbdjdcd}(v1\_topreal2~X0))))\end{array}
\label{hyp:ehcdedpfkghgchbgahigpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_2){\unicdcdcdhd}(m2\_subset\_1~np\_\_2~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_2~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_2~k1\_numbers))\end{array}
\label{hyp:dhahdgcdpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{v6\_membered}~{k4\_ordinal1}}\label{hyp:ggdggdpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_pre\_topc~X0){\unicdbdjdcd}((v1\_funct\_1~(k3\_struct\_0~X0)){\unicdcdcdhd}\\
((v1\_funct\_2~(k3\_struct\_0~X0)~(u1\_struct\_0~X0)~(u1\_struct\_0~X0)){\unicdcdcdhd}\\
(v3\_tops\_2~(k3\_struct\_0~X0)~X0~X0)))\end{array}
\label{hyp:ggdgbdpfcgpgchdhfhlgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_pre\_topc~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_pre\_topc~X1~X0){\unicdbdjdcd}\\
(l1\_pre\_topc~X1))\end{array}
\label{hyp:egehpfngbdpfahchfgpfehpgahdg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_rltopsp1}~{X0}})}{\unicdbdjdcd}{({{({{l1\_rlvect\_1}~{X0}})}{\unicdcdcdhd}{({{l1\_pre\_topc}~{X0}})}})}}}\label{hyp:egehpfmgbdpfchmgehpgahdhahbd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_struct\_0~X0){\unicdbdjdcd}((v1\_funct\_1~(k3\_struct\_0~X0)){\unicdcdcdhd}\\
((v1\_funct\_2~(k3\_struct\_0~X0)~(u1\_struct\_0~X0)~(u1\_struct\_0~X0)){\unicdcdcdhd}\\
(m1\_subset\_1~(k3\_struct\_0~X0)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~\\
X0)~(u1\_struct\_0~X0))))))\end{array}
\label{hyp:egehpflgddpfdhehchfhdgehpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((l1\_pre\_topc~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}((v1\_pre\_topc~(k1\_pre\_topc~X0~X1)){\unicdcdcdhd}(m1\_pre\_topc~\\
(k1\_pre\_topc~X0~X1)~X0))\end{array}
\label{hyp:egehpflgbdpfahchfgpfehpgahdg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v7\_ordinal1~X0){\unicdbdjdcd}((v5\_rltopsp1~(k15\_euclid~X0)){\unicdcdcdhd}\\
(l1\_rltopsp1~(k15\_euclid~X0)))\end{array}
\label{hyp:egehpflgbdfdpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2)))){\unicdbdjdcd}((v1\_topreal2~X0){\unicdbdjdcd}((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v2\_compts\_1~\\
X0~(k15\_euclid~np\_\_2))))\end{array}
\label{hyp:dgdgbdpfehpgahchfgbgmgcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v6\_membered~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
(v7\_ordinal1~X1))\end{array}
\label{hyp:dgdgbdhdpfngfgngcgfgchfgeg}
\end{equation}
\begin{theorem}\label{thm:ehcdfdpfkghgchbgahigpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2)))){\unicdbdjdcd}((X0=ReplSep~(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))~({\uniddcgcg} X1 : \iota .  \preoh (\preoh (k1\_real\_1~np\_\_1=k17\_euclid~\\
X1){\unicdcdcdhd}((r1\_xxreal\_0~(k1\_real\_1~np\_\_1)~(k18\_euclid~X1)){\unicdcdcdhd}(r1\_xxreal\_0~\\
(k18\_euclid~X1)~np\_\_1))){\unicdcdcdhd}((\preoh (k17\_euclid~X1=np\_\_1){\unicdcdcdhd}((r1\_xxreal\_0~\\
(k1\_real\_1~np\_\_1)~(k18\_euclid~X1)){\unicdcdcdhd}(r1\_xxreal\_0~(k18\_euclid~\\
X1)~np\_\_1))){\unicdcdcdhd}((\preoh (k1\_real\_1~np\_\_1=k18\_euclid~X1){\unicdcdcdhd}((r1\_xxreal\_0~\\
(k1\_real\_1~np\_\_1)~(k17\_euclid~X1)){\unicdcdcdhd}(r1\_xxreal\_0~(k17\_euclid~\\
X1)~np\_\_1))){\unicdcdcdhd}(\preoh (np\_\_1=k18\_euclid~X1){\unicdcdcdhd}((r1\_xxreal\_0~(k1\_real\_1~\\
np\_\_1)~(k17\_euclid~X1)){\unicdcdcdhd}(r1\_xxreal\_0~(k17\_euclid~X1)~np\_\_1))))))~\\
({\uniddcgcg} X1 : \iota .  X1)){\unicdbdjdcd}((v1\_topreal2~X0){\unicdcdcdhd}(v2\_compts\_1~X0~(k15\_euclid~\\
np\_\_2))))\end{array}$$
\end{theorem}
\end{document}
