\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{t82\_cat\_3 (TMWZbm1c5wwLK6qyj9R58cMjTFa5Kog6ugn)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v11\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_cat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_cat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_cat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_cat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_cat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_cat\_1 : {{\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 $ m1\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ r4\_cat\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k2\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ v2\_cat\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k5\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ k4\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}((v2\_cat\_1~\\
X0){\unicdcdcdhd}((v3\_cat\_1~X0){\unicdcdcdhd}((v4\_cat\_1~X0){\unicdcdcdhd}((v5\_cat\_1~X0){\unicdcdcdhd}((v6\_cat\_1~\\
X0){\unicdcdcdhd}(l1\_cat\_1~X0)))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(u1\_struct\_0~X0)){\unicdbdjdcd}(\preoh (k2\_cat\_1~X0~X1~X2{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}\\
((k2\_cat\_1~X0~X3~X2{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}(\preoh {\unicdcdadad} X4 .  (m1\_cat\_1~X4~X0~\\
X1~X2){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_cat\_1~X5~X0~X3~X2){\unicdbdjdcd}((r4\_cat\_3~X0~X2~X4~X5){\unicdbdjded}\\
({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(u1\_struct\_0~X0)){\unicdbdjdcd}(\preoh (k2\_cat\_1~X0~\\
X1~X6{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}((k2\_cat\_1~X0~X3~X6{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}(\preoh (k2\_cat\_1~\\
X0~X2~X6{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}({\unicdcdadad} X7 .  (m1\_cat\_1~X7~X0~X1~X6){\unicdbdjdcd}({\unicdcdadad} X8 .  \\
(m1\_cat\_1~X8~X0~X3~X6){\unicdbdjdcd}({\unicdcdaddd} X9 .  (m1\_cat\_1~X9~X0~X2~X6){\unicdcdcdhd}({\unicdcdadad} X10 .  \\
(m1\_cat\_1~X10~X0~X2~X6){\unicdbdjdcd}(((k5\_cat\_1~X0~X1~X2~X6~X4~X10=X7){\unicdcdcdhd}(k5\_cat\_1~\\
X0~X3~X2~X6~X5~X10=X8)){\unicdbdjded}(X9=X10))))))))))))))))))\end{array}
\label{hyp:ehidbdpfdgbgehpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}((v2\_cat\_1~\\
X0){\unicdcdcdhd}((v3\_cat\_1~X0){\unicdcdcdhd}((v4\_cat\_1~X0){\unicdcdcdhd}((v5\_cat\_1~X0){\unicdcdcdhd}((v6\_cat\_1~\\
X0){\unicdcdcdhd}(l1\_cat\_1~X0)))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(u1\_struct\_0~X0)){\unicdbdjdcd}(\preoh (k2\_cat\_1~X0~X1~X2{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}\\
((k2\_cat\_1~X0~X2~X3{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}(k2\_cat\_1~X0~X1~X3=k1\_xboole\_0))))))\end{array}
\label{hyp:ehcdedpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~\\
X0){\unicdcdcdhd}(l1\_cat\_1~X0))){\unicdcdcdhd}((m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(u1\_struct\_0~X0)))){\unicdbdjdcd}({\unicdcdaddd} X3 .  m1\_cat\_1~X3~X0~X1~X2)\end{array}
\label{hyp:fgihjgdhehfgogdgfgpfngbdpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}\\
((v5\_cat\_1~X0){\unicdcdcdhd}((v6\_cat\_1~X0){\unicdcdcdhd}(l1\_cat\_1~X0))))){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(u1\_struct\_0~X0))){\unicdbdjdcd}(m1\_cat\_1~(k4\_cat\_1~X0~X1)~X0~X1~X1)\end{array}
\label{hyp:egehpflgedpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}((v2\_cat\_1~\\
X0){\unicdcdcdhd}((v3\_cat\_1~X0){\unicdcdcdhd}((v4\_cat\_1~X0){\unicdcdcdhd}((v5\_cat\_1~X0){\unicdcdcdhd}((v6\_cat\_1~\\
X0){\unicdcdcdhd}(l1\_cat\_1~X0)))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_cat\_1~X3~X0~X1~X2){\unicdbdjdcd}((v2\_cat\_3~X3~X0~X1~X2){\unicdbdjded}((k2\_cat\_1~X0~X1~X2{\unicdcdgdad}\\
k1\_xboole\_0){\unicdcdcdhd}((k2\_cat\_1~X0~X2~X1{\unicdcdgdad}k1\_xboole\_0){\unicdcdcdhd}({\unicdcdaddd} X4 .  (\\
m1\_cat\_1~X4~X0~X2~X1){\unicdcdcdhd}(k5\_cat\_1~X0~X1~X2~X1~X3~X4=k4\_cat\_1~X0~X1))))))))\end{array}
\label{hyp:egjdpfdgbgehpfdd}
\end{equation}
\begin{theorem}\label{thm:ehidcdpfdgbgehpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}((v2\_cat\_1~\\
X0){\unicdcdcdhd}((v3\_cat\_1~X0){\unicdcdcdhd}((v4\_cat\_1~X0){\unicdcdcdhd}((v5\_cat\_1~X0){\unicdcdcdhd}((v6\_cat\_1~\\
X0){\unicdcdcdhd}(l1\_cat\_1~X0)))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_cat\_1~X4~X0~\\
X1~X2){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_cat\_1~X5~X0~X3~X2){\unicdbdjdcd}((r4\_cat\_3~X0~X2~X4~X5){\unicdbdjdcd}\\
((k2\_cat\_1~X0~X1~X2=k1\_xboole\_0){\unicdcdcdid}((k2\_cat\_1~X0~X3~X2=k1\_xboole\_0){\unicdcdcdid}\\
((k2\_cat\_1~X0~X1~X3=k1\_xboole\_0){\unicdcdcdid}((k2\_cat\_1~X0~X3~X1=k1\_xboole\_0){\unicdcdcdid}\\
((v2\_cat\_3~X4~X0~X1~X2){\unicdcdcdhd}(v2\_cat\_3~X5~X0~X3~X2))))))))))))\end{array}$$
\end{theorem}
\end{document}
