\documentclass{article}
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\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\newcommand\unicdbdjded{\Leftrightarrow}
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\begin{document}
\title{t40\_rfunct\_2 (TMRMaeMV2NzCQxaPnu19rfcCykwg2uCBi19)}
\maketitle
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\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 $ k1\_numbers : {\iota}$ be given.
Let $ v8\_valued\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ v3\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_valued\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k3\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v5\_valued\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_valued\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_valued\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
((v1\_funct\_1~X3){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers)))){\unicdbdjdcd}((((v5\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~\\
X2~X0)){\unicdcdcdhd}(v5\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~X3~X1))){\unicdbdjdcd}\\
(v5\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~(k3\_valued\_1~\\
k1\_numbers~k1\_numbers~k1\_numbers~X2~X3)~(k3\_xboole\_0~X0~X1)))){\unicdcdcdhd}\\
((((v6\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~X2~X0)){\unicdcdcdhd}(\\
v6\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~X3~X1))){\unicdbdjdcd}(v6\_valued\_0~\\
(k2\_partfun1~k1\_numbers~k1\_numbers~(k3\_valued\_1~k1\_numbers~k1\_numbers~\\
k1\_numbers~X2~X3)~(k3\_xboole\_0~X0~X1)))){\unicdcdcdhd}((((v7\_valued\_0~(k2\_partfun1~\\
k1\_numbers~k1\_numbers~X2~X0)){\unicdcdcdhd}(v7\_valued\_0~(k2\_partfun1~k1\_numbers~\\
k1\_numbers~X3~X1))){\unicdbdjdcd}(v7\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~\\
(k3\_valued\_1~k1\_numbers~k1\_numbers~k1\_numbers~X2~X3)~(k3\_xboole\_0~\\
X0~X1)))){\unicdcdcdhd}(((v8\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~\\
X2~X0)){\unicdcdcdhd}(v8\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~X3~X1))){\unicdbdjdcd}\\
(v8\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~(k3\_valued\_1~\\
k1\_numbers~k1\_numbers~k1\_numbers~X2~X3)~(k3\_xboole\_0~X0~X1))))))))\end{array}
\label{hyp:ehddhdpfchggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  ((v1\_funct\_1~X2){\unicdcdcdhd}\\
(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1)))){\unicdbdjdcd}((v1\_funct\_1~\\
(k2\_partfun1~X0~X1~X2~X3)){\unicdcdcdhd}(m1\_subset\_1~(k2\_partfun1~X0~X1~X2~X3)~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))))\end{array}
\label{hyp:egehpflgcdpfahbgchehggfhogbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers))){\unicdbdjdcd}(((v1\_funct\_1~X0){\unicdcdcdhd}(v3\_funct\_1~X0)){\unicdbdjdcd}((v1\_funct\_1~\\
X0){\unicdcdcdhd}((v7\_valued\_0~X0){\unicdcdcdhd}(v8\_valued\_0~X0))))\end{array}
\label{hyp:dgdgfdpfchggfhogdgehpfcd}
\end{equation}
\begin{theorem}\label{thm:ehedadpfchggfhogdgehpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
((v1\_funct\_1~X3){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers)))){\unicdbdjdcd}(((v8\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~\\
X2~X0)){\unicdcdcdhd}(v3\_funct\_1~(k2\_partfun1~k1\_numbers~k1\_numbers~X3~X1))){\unicdbdjdcd}\\
(v8\_valued\_0~(k2\_partfun1~k1\_numbers~k1\_numbers~(k3\_valued\_1~\\
k1\_numbers~k1\_numbers~k1\_numbers~X2~X3)~(k3\_xboole\_0~X0~X1)))))\end{array}$$
\end{theorem}
\end{document}
