\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{t35\_closure1 (TMWRoGqUtTX6Qqb54RkVr7fHyP4S7YpnSWd)}
\maketitle
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v4\_mssubfam : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ m3\_pboole : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_mboolean : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m1\_pboole : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k5\_mssubfam : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m2\_pboole : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ r6\_pboole : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k2\_closure1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ v2\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_pboole : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r2\_pboole : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k4\_mssubfam : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v5\_mssubfam : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v5\_mssubfam~\\
X2~X0~X1){\unicdcdcdhd}(m3\_pboole~X2~X0~(k1\_mboolean~X0~X1))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_pboole~\\
X3~X0~(k5\_mssubfam~X0~X1)){\unicdbdjdcd}({\unicdcdaddd} X4 .  ((v2\_relat\_1~X4){\unicdcdcdhd}(m3\_pboole~\\
X4~X0~(k1\_mboolean~X0~X1))){\unicdcdcdhd}({\unicdcdadad} X5 .  ((v1\_relat\_1~X5){\unicdcdcdhd}((v4\_relat\_1~\\
X5~X0){\unicdcdcdhd}((v1\_funct\_1~X5){\unicdcdcdhd}(v1\_partfun1~X5~X0)))){\unicdbdjdcd}((r1\_pboole~X0~\\
X5~X4){\unicdbdjded}((r1\_pboole~X0~X5~X2){\unicdcdcdhd}(r2\_pboole~X0~X3~X5)))))))\end{array}
\label{hyp:ehddbdpfdgmgpgdhfhchfgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 : \iota{\unicdbdjdcd}o .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (((v1\_relat\_1~\\
X3){\unicdcdcdhd}((v2\_relat\_1~X3){\unicdcdcdhd}((v4\_relat\_1~X3~X2){\unicdcdcdhd}((v1\_funct\_1~X3){\unicdcdcdhd}(\\
v1\_partfun1~X3~X2))))){\unicdcdcdhd}((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X2){\unicdcdcdhd}\\
((v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X2))))){\unicdbdjdcd}(({\unicdcdadad} X4 .  ((v1\_relat\_1~\\
X4){\unicdcdcdhd}((v4\_relat\_1~X4~X2){\unicdcdcdhd}((v1\_funct\_1~X4){\unicdcdcdhd}(v1\_partfun1~X4~X2)))){\unicdbdjdcd}\\
((r1\_pboole~X2~X4~X3){\unicdbdjded}((r1\_pboole~X2~X4~X1){\unicdcdcdhd}(X0~X4)))){\unicdbdjdcd}(r2\_pboole~\\
X2~X3~X1))\end{array}
\label{hyp:dhbdpfdgmgpgdhfhchfgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~\\
X1~X0){\unicdcdcdhd}((v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdcdcdhd}((v1\_relat\_1~\\
X2){\unicdcdcdhd}((v4\_relat\_1~X2~X0){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}(v1\_partfun1~X2~X0))))){\unicdbdjdcd}\\
((r6\_pboole~X0~X1~X2){\unicdbdjded}(X1=X2))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchgdpfahcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}(k5\_mssubfam~X0~X1=k1\_mboolean~\\
X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfngdhdhfhcgggbgng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m3\_pboole~\\
X2~X0~X1){\unicdbdjdcd}((v1\_relat\_1~X2){\unicdcdcdhd}((v4\_relat\_1~X2~X0){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}\\
(v1\_partfun1~X2~X0)))))\end{array}
\label{hyp:egehpfngddpfahcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_pboole~\\
X2~X0~X1){\unicdbdjdcd}((v1\_relat\_1~X2){\unicdcdcdhd}((v4\_relat\_1~X2~X0){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}\\
(v1\_partfun1~X2~X0)))))\end{array}
\label{hyp:egehpfngbdpfahcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~\\
X1~X0){\unicdcdcdhd}((v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdcdcdhd}(m3\_pboole~X2~\\
X0~(k1\_mboolean~X0~X1))){\unicdbdjdcd}(m3\_pboole~(k4\_mssubfam~X0~X1~X2)~X0~X1)\end{array}
\label{hyp:egehpflgedpfngdhdhfhcgggbgng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (((v1\_relat\_1~X1){\unicdcdcdhd}\\
((v4\_relat\_1~X1~X0){\unicdcdcdhd}((v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdcdcdhd}\\
((m2\_pboole~X2~X0~(k5\_mssubfam~X0~X1)~(k5\_mssubfam~X0~X1)){\unicdcdcdhd}(m1\_pboole~\\
X3~X0~(k5\_mssubfam~X0~X1)))){\unicdbdjdcd}(m1\_pboole~(k2\_closure1~X0~X1~X2~X3)~\\
X0~(k5\_mssubfam~X0~X1))\end{array}
\label{hyp:egehpflgcdpfdgmgpgdhfhchfgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}((v1\_relat\_1~(k1\_mboolean~\\
X0~X1)){\unicdcdcdhd}((v4\_relat\_1~(k1\_mboolean~X0~X1)~X0){\unicdcdcdhd}((v1\_funct\_1~(k1\_mboolean~\\
X0~X1)){\unicdcdcdhd}(v1\_partfun1~(k1\_mboolean~X0~X1)~X0))))\end{array}
\label{hyp:egehpflgbdpfngcgpgpgmgfgbgog}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m3\_pboole~\\
X2~X0~(k1\_mboolean~X0~X1)){\unicdbdjdcd}((v4\_mssubfam~X2~X0~X1){\unicdbdjded}({\unicdcdadad} X3 .  \\
(m3\_pboole~X3~X0~(k1\_mboolean~X0~X1)){\unicdbdjdcd}((r2\_pboole~X0~X3~X2){\unicdbdjdcd}(r1\_pboole~\\
X0~(k4\_mssubfam~X0~X1~X3)~X2)))))\end{array}
\label{hyp:egfdpfngdhdhfhcgggbgng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m3\_pboole~\\
X2~X0~(k1\_mboolean~X0~X1)){\unicdbdjdcd}((v4\_mssubfam~X2~X0~X1){\unicdbdjdcd}(v5\_mssubfam~\\
X2~X0~X1)))\end{array}
\label{hyp:dgdgfdpfngdhdhfhcgggbgng}
\end{equation}
\begin{theorem}\label{thm:ehddfdpfdgmgpgdhfhchfgbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}((v4\_relat\_1~X1~X0){\unicdcdcdhd}(\\
(v1\_funct\_1~X1){\unicdcdcdhd}(v1\_partfun1~X1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v4\_mssubfam~\\
X2~X0~X1){\unicdcdcdhd}(m3\_pboole~X2~X0~(k1\_mboolean~X0~X1))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_pboole~\\
X3~X0~(k5\_mssubfam~X0~X1)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_pboole~X4~X0~(k5\_mssubfam~\\
X0~X1)~(k5\_mssubfam~X0~X1)){\unicdbdjdcd}(((r6\_pboole~X0~(k2\_closure1~X0~X1~X4~\\
X3)~X3){\unicdcdcdhd}({\unicdcdadad} X5 .  (m1\_pboole~X5~X0~(k5\_mssubfam~X0~X1)){\unicdbdjdcd}({\unicdcdadad} X6 .  \\
((v2\_relat\_1~X6){\unicdcdcdhd}(m3\_pboole~X6~X0~(k1\_mboolean~X0~X1))){\unicdbdjdcd}(({\unicdcdadad} X7 .  \\
((v1\_relat\_1~X7){\unicdcdcdhd}((v4\_relat\_1~X7~X0){\unicdcdcdhd}((v1\_funct\_1~X7){\unicdcdcdhd}(v1\_partfun1~\\
X7~X0)))){\unicdbdjdcd}((r1\_pboole~X0~X7~X6){\unicdbdjded}((r1\_pboole~X0~X7~X2){\unicdcdcdhd}(r2\_pboole~\\
X0~X5~X7)))){\unicdbdjdcd}(r6\_pboole~X0~(k2\_closure1~X0~X1~X4~X5)~(k4\_mssubfam~\\
X0~X1~X6)))))){\unicdbdjdcd}(r1\_pboole~X0~X3~X2)))))\end{array}$$
\end{theorem}
\end{document}
