\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{t18\_limfunc3 (TMPmM2gvJyg14Ye8HyFX8x31EBy3jx2f8L4)}
\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 $ r2\_limfunc3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k26\_valued\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ r3\_limfunc3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r5\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r6\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r2\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r3\_limfunc2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v3\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xreal\_0 : {{\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}({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}(((r5\_limfunc2~\\
X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~X1~k6\_numbers){\unicdcdcdid}(r5\_limfunc2~(k26\_valued\_1~\\
k1\_numbers~k1\_numbers~X2~X1)~X0))){\unicdcdcdhd}(((r5\_limfunc2~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~\\
k6\_numbers~X1){\unicdcdcdid}(r6\_limfunc2~(k26\_valued\_1~k1\_numbers~k1\_numbers~\\
X2~X1)~X0))){\unicdcdcdhd}(((r6\_limfunc2~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~X1~k6\_numbers){\unicdcdcdid}\\
(r6\_limfunc2~(k26\_valued\_1~k1\_numbers~k1\_numbers~X2~X1)~X0))){\unicdcdcdhd}\\
((r6\_limfunc2~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~k6\_numbers~X1){\unicdcdcdid}(r5\_limfunc2~\\
(k26\_valued\_1~k1\_numbers~k1\_numbers~X2~X1)~X0))))))))\end{array}
\label{hyp:ehcdcdpfmgjgngggfhogdgcd}
\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 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}(((r2\_limfunc2~\\
X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~X1~k6\_numbers){\unicdcdcdid}(r2\_limfunc2~(k26\_valued\_1~\\
k1\_numbers~k1\_numbers~X2~X1)~X0))){\unicdcdcdhd}(((r2\_limfunc2~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~\\
k6\_numbers~X1){\unicdcdcdid}(r3\_limfunc2~(k26\_valued\_1~k1\_numbers~k1\_numbers~\\
X2~X1)~X0))){\unicdcdcdhd}(((r3\_limfunc2~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~X1~k6\_numbers){\unicdcdcdid}\\
(r3\_limfunc2~(k26\_valued\_1~k1\_numbers~k1\_numbers~X2~X1)~X0))){\unicdcdcdhd}\\
((r3\_limfunc2~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~k6\_numbers~X1){\unicdcdcdid}(r2\_limfunc2~\\
(k26\_valued\_1~k1\_numbers~k1\_numbers~X2~X1)~X0))))))))\end{array}
\label{hyp:ehcdbdpfmgjgngggfhogdgcd}
\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}
{{v3\_membered}~{k1\_numbers}}\label{hyp:ggdgddpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  ((v3\_membered~X1){\unicdcdcdhd}\\
(((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X1)))){\unicdcdcdhd}(v1\_xreal\_0~X3))){\unicdbdjdcd}((v1\_funct\_1~(k26\_valued\_1~X0~X1~\\
X2~X3)){\unicdcdcdhd}(m1\_subset\_1~(k26\_valued\_1~X0~X1~X2~X3)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~k1\_numbers))))\end{array}
\label{hyp:egehpflgcdgdpfghbgmgfhfgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfihchfgbgmgpfad}
\end{equation}
\begin{theorem}\label{thm:ehbdidpfmgjgngggfhogdgdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}(((r2\_limfunc3~\\
X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~X1~k6\_numbers){\unicdcdcdid}(r2\_limfunc3~(k26\_valued\_1~\\
k1\_numbers~k1\_numbers~X2~X1)~X0))){\unicdcdcdhd}(((r2\_limfunc3~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~\\
k6\_numbers~X1){\unicdcdcdid}(r3\_limfunc3~(k26\_valued\_1~k1\_numbers~k1\_numbers~\\
X2~X1)~X0))){\unicdcdcdhd}(((r3\_limfunc3~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~X1~k6\_numbers){\unicdcdcdid}\\
(r3\_limfunc3~(k26\_valued\_1~k1\_numbers~k1\_numbers~X2~X1)~X0))){\unicdcdcdhd}\\
((r3\_limfunc3~X2~X0){\unicdbdjdcd}((r1\_xxreal\_0~k6\_numbers~X1){\unicdcdcdid}(r2\_limfunc3~\\
(k26\_valued\_1~k1\_numbers~k1\_numbers~X2~X1)~X0))))))))\end{array}$$
\end{theorem}
\end{document}
