\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{t37\_limfunc4 (TMS3URTuVd3awFaNYfWjKdLaf8GQvt5Rgca)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_numbers : {\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 $ r3\_limfunc3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v7\_limfunc1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ r2\_limfunc3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_seq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r6\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r5\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r3\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r2\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)))){\unicdbdjdcd}((v7\_limfunc1~X0){\unicdbdjded}(({\unicdcdadad} X1 .  (\\
m1\_subset\_1~X1~k1\_numbers){\unicdbdjdcd}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~k1\_numbers){\unicdcdcdhd}\\
((\preoh r1\_xxreal\_0~X1~X2){\unicdcdcdhd}(X2 \in k1\_relset\_1~k1\_numbers~X0)))){\unicdcdcdhd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~k1\_numbers){\unicdbdjdcd}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~k1\_numbers){\unicdcdcdhd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X2~X3){\unicdcdcdhd}\\
((X3 \in k1\_relset\_1~k1\_numbers~X0){\unicdcdcdhd}(r1\_xxreal\_0~(k1\_seq\_1~X0~X3)~\\
X1))))))))\end{array}
\label{hyp:ehedidpfmgjgngggfhogdgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)))){\unicdbdjdcd}(((r6\_limfunc2~X1~X0){\unicdcdcdhd}((v7\_limfunc1~\\
X2){\unicdcdcdhd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~\\
X3~X0){\unicdcdcdhd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~\\
X3~X4){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X4~X0){\unicdcdcdhd}(X4 \in k1\_relset\_1~k1\_numbers~(k1\_partfun1~\\
k1\_numbers~k1\_numbers~k1\_numbers~k1\_numbers~X1~X2))))))))){\unicdbdjdcd}(\\
r5\_limfunc2~(k1\_partfun1~k1\_numbers~k1\_numbers~k1\_numbers~k1\_numbers~\\
X1~X2)~X0))))\end{array}
\label{hyp:ehbdhdpfmgjgngggfhogdged}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)))){\unicdbdjdcd}(((r3\_limfunc2~X1~X0){\unicdcdcdhd}((v7\_limfunc1~\\
X2){\unicdcdcdhd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~\\
X0~X3){\unicdcdcdhd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~\\
X4~X3){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X0~X4){\unicdcdcdhd}(X4 \in k1\_relset\_1~k1\_numbers~(k1\_partfun1~\\
k1\_numbers~k1\_numbers~k1\_numbers~k1\_numbers~X1~X2))))))))){\unicdbdjdcd}(\\
r2\_limfunc2~(k1\_partfun1~k1\_numbers~k1\_numbers~k1\_numbers~k1\_numbers~\\
X1~X2)~X0))))\end{array}
\label{hyp:ehbdddpfmgjgngggfhogdged}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
((r3\_limfunc3~X1~X0){\unicdbdjded}((r3\_limfunc2~X1~X0){\unicdcdcdhd}(r6\_limfunc2~X1~X0))))\end{array}
\label{hyp:ehbdddpfmgjgngggfhogdgdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
((r2\_limfunc3~X1~X0){\unicdbdjded}((r2\_limfunc2~X1~X0){\unicdcdcdhd}(r5\_limfunc2~X1~X0))))\end{array}
\label{hyp:ehbdcdpfmgjgngggfhogdgdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
((r6\_limfunc2~X1~X0){\unicdbdjded}(({\unicdcdadad} X2 .  (m1\_subset\_1~X2~k1\_numbers){\unicdbdjdcd}\\
(\preoh (\preoh r1\_xxreal\_0~X2~X0){\unicdcdcdhd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}\\
(\preoh (\preoh r1\_xxreal\_0~X2~X3){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X3~X0){\unicdcdcdhd}(X3 \in k1\_relset\_1~\\
k1\_numbers~X1)))))){\unicdcdcdhd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~k1\_numbers){\unicdbdjdcd}\\
({\unicdcdaddd} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X3~X0){\unicdcdcdhd}\\
({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X3~X4){\unicdcdcdhd}\\
((\preoh r1\_xxreal\_0~X4~X0){\unicdcdcdhd}((X4 \in k1\_relset\_1~k1\_numbers~X1){\unicdcdcdhd}(r1\_xxreal\_0~\\
X2~(k1\_seq\_1~X1~X4))))))))))))\end{array}
\label{hyp:ehbdcdpfmgjgngggfhogdgcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \\
(((v1\_funct\_1~X4){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X1)))){\unicdcdcdhd}((v1\_funct\_1~X5){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X2~X3))))){\unicdbdjdcd}((v1\_funct\_1~(k1\_partfun1~X0~X1~X2~X3~X4~X5)){\unicdcdcdhd}(m1\_subset\_1~\\
(k1\_partfun1~X0~X1~X2~X3~X4~X5)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X3))))\end{array}
\label{hyp:egehpflgbdpfahbgchehggfhogbd}
\end{equation}
\begin{theorem}\label{thm:ehddhdpfmgjgngggfhogdged}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)))){\unicdbdjdcd}(((r3\_limfunc3~X1~X0){\unicdcdcdhd}((v7\_limfunc1~\\
X2){\unicdcdcdhd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~\\
X4~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X0~X3){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X4~X0){\unicdcdcdhd}\\
({\unicdcdadad} X5 .  (m1\_subset\_1~X5~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X6 .  (m1\_subset\_1~\\
X6~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X5~X3){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X0~X5){\unicdcdcdhd}\\
((X5 \in k1\_relset\_1~k1\_numbers~(k1\_partfun1~k1\_numbers~k1\_numbers~\\
k1\_numbers~k1\_numbers~X1~X2)){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X4~X6){\unicdcdcdhd}((\preoh r1\_xxreal\_0~\\
X6~X0){\unicdcdcdhd}(X6 \in k1\_relset\_1~k1\_numbers~(k1\_partfun1~k1\_numbers~k1\_numbers~\\
k1\_numbers~k1\_numbers~X1~X2))))))))))))))){\unicdbdjdcd}(r2\_limfunc3~(k1\_partfun1~\\
k1\_numbers~k1\_numbers~k1\_numbers~k1\_numbers~X1~X2)~X0))))\end{array}$$
\end{theorem}
\end{document}
