\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{l49\_jgraph\_2 (TMWUtL1kHenMQZEJvKh9GKY5hcAZFjg8CW3)}
\maketitle
Let $ v4\_pre\_topc : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ 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 $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k17\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k18\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k5\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k5\_square\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k12\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_square\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k9\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k7\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 : \iota{\unicdbdjdcd}\iota .  {\unicdcdadad} X1 : \iota{\unicdbdjdcd}\iota .  (({\unicdcdadad} X2 .  v1\_xreal\_0~\\
(X1~X2)){\unicdcdcdhd}({\unicdcdadad} X2 .  v1\_xreal\_0~(X0~X2))){\unicdbdjdcd}((({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}((X1~(k5\_algstr\_0~(k15\_euclid~\\
np\_\_2)~X2~X3)=k6\_xcmplx\_0~(X1~X2)~(X1~X3)){\unicdcdcdhd}(X0~(k5\_algstr\_0~(k15\_euclid~\\
np\_\_2)~X2~X3)=k6\_xcmplx\_0~(X0~X2)~(X0~X3))))){\unicdcdcdhd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}(k5\_square\_1~(k12\_euclid~\\
(k5\_algstr\_0~(k15\_euclid~np\_\_2)~X2~X3))=k2\_xcmplx\_0~(k3\_square\_1~\\
(X1~(k5\_algstr\_0~(k15\_euclid~np\_\_2)~X2~X3)))~(k3\_square\_1~(X0~(\\
k5\_algstr\_0~(k15\_euclid~np\_\_2)~X2~X3))))))){\unicdbdjdcd}((v4\_pre\_topc~(ReplSep~\\
(toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~\\
({\uniddcgcg} X2 : \iota .  r1\_xxreal\_0~(X1~X2)~(X0~X2))~({\uniddcgcg} X2 : \iota .  X2))~(k15\_euclid~\\
np\_\_2)){\unicdcdcdhd}(m1\_subset\_1~(ReplSep~(toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~\\
X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X2 : \iota .  r1\_xxreal\_0~\\
(X1~X2)~(X0~X2))~({\uniddcgcg} X2 : \iota .  X2))~(k1\_zfmisc\_1~(u1\_struct\_0~(k15\_euclid~\\
np\_\_2))))))\end{array}
\label{hyp:dhddpfkghgchbgahigpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k1\_numbers){\unicdcdcdhd}(v1\_xreal\_0~\\
X1)){\unicdbdjdcd}(k9\_real\_1~X0~X1=k6\_xcmplx\_0~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgjdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k1\_numbers){\unicdcdcdhd}(v1\_xreal\_0~\\
X1)){\unicdbdjdcd}(k7\_real\_1~X0~X1=k2\_xcmplx\_0~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflghdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(k5\_square\_1~X0=k3\_square\_1~\\
X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfdhbhfhbgchfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{{k1\_real\_1}~{X0}}={{k4\_xcmplx\_0}~{X0}}})}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(k5\_square\_1~(k12\_euclid~(k5\_algstr\_0~(k15\_euclid~np\_\_2)~X0~X1))=\\
k7\_real\_1~(k5\_square\_1~(k1\_real\_1~(k17\_euclid~(k5\_algstr\_0~(\\
k15\_euclid~np\_\_2)~X0~X1))))~(k5\_square\_1~(k18\_euclid~(k5\_algstr\_0~\\
(k15\_euclid~np\_\_2)~X0~X1)))))\end{array}
\label{hyp:mgedidpfkghgchbgahigpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
((k1\_real\_1~(k17\_euclid~(k5\_algstr\_0~(k15\_euclid~np\_\_2)~X0~X1))=\\
k9\_real\_1~(k1\_real\_1~(k17\_euclid~X0))~(k1\_real\_1~(k17\_euclid~\\
X1))){\unicdcdcdhd}(k18\_euclid~(k5\_algstr\_0~(k15\_euclid~np\_\_2)~X0~X1)=k9\_real\_1~\\
(k18\_euclid~X0)~(k18\_euclid~X1))))\end{array}
\label{hyp:mgedhdpfkghgchbgahigpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((v1\_xcmplx\_0~(k4\_xcmplx\_0~X0)){\unicdcdcdhd}\\
(v1\_xreal\_0~(k4\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgddpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k1\_numbers){\unicdcdcdhd}(v1\_xreal\_0~\\
X1)){\unicdbdjdcd}(m1\_subset\_1~(k9\_real\_1~X0~X1)~k1\_numbers)\end{array}
\label{hyp:egehpflgjdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(m1\_subset\_1~(k5\_square\_1~\\
X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgfdpfdhbhfhbgchfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(m1\_subset\_1~(k1\_real\_1~\\
X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(m1\_subset\_1~(k18\_euclid~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdidpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(m1\_subset\_1~(k17\_euclid~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdhdpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfchfgbgmgpfbd}
\end{equation}
\begin{theorem}\label{thm:mgedjdpfkghgchbgahigpfcd}
$$\begin{array}{c}(v4\_pre\_topc~(ReplSep~(toset~({\uniddcgcg} X0 : \iota .  m1\_subset\_1~X0~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))~({\uniddcgcg} X0 : \iota .  r1\_xxreal\_0~(k1\_real\_1~(k17\_euclid~\\
X0))~(k18\_euclid~X0))~({\uniddcgcg} X0 : \iota .  X0))~(k15\_euclid~np\_\_2)){\unicdcdcdhd}(\\
m1\_subset\_1~(ReplSep~(toset~({\uniddcgcg} X0 : \iota .  m1\_subset\_1~X0~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))~({\uniddcgcg} X0 : \iota .  r1\_xxreal\_0~(k1\_real\_1~(k17\_euclid~\\
X0))~(k18\_euclid~X0))~({\uniddcgcg} X0 : \iota .  X0))~(k1\_zfmisc\_1~(u1\_struct\_0~\\
(k15\_euclid~np\_\_2))))\end{array}$$
\end{theorem}
\end{document}
