\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{l48\_isocat\_2 (TMcSTvCbGihaX6DrpzLAWcTMf95YdFeK6bu)}
\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 $ m2\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r1\_nattra\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ m1\_nattra\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\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 $ k3\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k14\_funct\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k2\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k8\_cat\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k8\_isocat\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k4\_nattra\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ k9\_cat\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k10\_cat\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ m1\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ 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 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\\
\preoh v11\_struct\_0~X1){\unicdcdcdhd}((v2\_cat\_1~X1){\unicdcdcdhd}((v3\_cat\_1~X1){\unicdcdcdhd}((v4\_cat\_1~\\
X1){\unicdcdcdhd}((v5\_cat\_1~X1){\unicdcdcdhd}((v6\_cat\_1~X1){\unicdcdcdhd}(l1\_cat\_1~X1)))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m2\_cat\_1~X2~X0~X1){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_cat\_1~X3~X0~X1){\unicdbdjdcd}((r1\_nattra\_1~\\
X0~X1~X2~X3){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_nattra\_1~X4~X0~X1~X2~X3){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
(m1\_subset\_1~X5~(u1\_struct\_0~X0)){\unicdbdjdcd}(k4\_nattra\_1~X0~X1~X2~X3~X4~X5 \in \\
k2\_cat\_1~X1~(k8\_cat\_1~X0~X1~X2~X5)~(k8\_cat\_1~X0~X1~X3~X5))))))))\end{array}
\label{hyp:ehedpfjgdhpgdgbgehpfbd}
\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 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\\
\preoh v11\_struct\_0~X1){\unicdcdcdhd}((v2\_cat\_1~X1){\unicdcdcdhd}((v3\_cat\_1~X1){\unicdcdcdhd}((v4\_cat\_1~\\
X1){\unicdcdcdhd}((v5\_cat\_1~X1){\unicdcdcdhd}((v6\_cat\_1~X1){\unicdcdcdhd}(l1\_cat\_1~X1)))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((\preoh v11\_struct\_0~X2){\unicdcdcdhd}((v2\_cat\_1~X2){\unicdcdcdhd}((v3\_cat\_1~\\
X2){\unicdcdcdhd}((v4\_cat\_1~X2){\unicdcdcdhd}((v5\_cat\_1~X2){\unicdcdcdhd}((v6\_cat\_1~X2){\unicdcdcdhd}(l1\_cat\_1~\\
X2)))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_cat\_1~X3~X0~X1){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_cat\_1~\\
X4~X0~X2){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X0)){\unicdbdjdcd}(k8\_cat\_1~\\
X0~(k8\_cat\_2~X1~X2)~(k8\_isocat\_2~X0~X1~X2~X3~X4)~X5=k9\_cat\_2~X1~X2~(\\
k8\_cat\_1~X0~X1~X3~X5)~(k8\_cat\_1~X0~X2~X4~X5)))))))\end{array}
\label{hyp:ehddfdpfjgdhpgdgbgehpfcd}
\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 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\\
\preoh v11\_struct\_0~X1){\unicdcdcdhd}((v2\_cat\_1~X1){\unicdcdcdhd}((v3\_cat\_1~X1){\unicdcdcdhd}((v4\_cat\_1~\\
X1){\unicdcdcdhd}((v5\_cat\_1~X1){\unicdcdcdhd}((v6\_cat\_1~X1){\unicdcdcdhd}(l1\_cat\_1~X1)))))))){\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\_subset\_1~X4~(u1\_struct\_0~X1)){\unicdbdjdcd}\\
({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X1)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m1\_subset\_1~\\
X6~(u4\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~X7~(u4\_struct\_0~\\
X1)){\unicdbdjdcd}(((X6 \in k2\_cat\_1~X0~X2~X3){\unicdcdcdhd}(X7 \in k2\_cat\_1~X1~X4~X5)){\unicdbdjdcd}(k10\_cat\_2~\\
X0~X1~X6~X7 \in k2\_cat\_1~(k8\_cat\_2~X0~X1)~(k9\_cat\_2~X0~X1~X2~X4)~(k9\_cat\_2~\\
X0~X1~X3~X5))))))))))\end{array}
\label{hyp:ehbdedpfjgdhpgdgbgehpfcd}
\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 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\\
\preoh v11\_struct\_0~X1){\unicdcdcdhd}((v2\_cat\_1~X1){\unicdcdcdhd}((v3\_cat\_1~X1){\unicdcdcdhd}((v4\_cat\_1~\\
X1){\unicdcdcdhd}((v5\_cat\_1~X1){\unicdcdcdhd}((v6\_cat\_1~X1){\unicdcdcdhd}(l1\_cat\_1~X1)))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((\preoh v11\_struct\_0~X2){\unicdcdcdhd}((v2\_cat\_1~X2){\unicdcdcdhd}((v3\_cat\_1~\\
X2){\unicdcdcdhd}((v4\_cat\_1~X2){\unicdcdcdhd}((v5\_cat\_1~X2){\unicdcdcdhd}((v6\_cat\_1~X2){\unicdcdcdhd}(l1\_cat\_1~\\
X2)))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_cat\_1~X3~X0~X1){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_cat\_1~\\
X4~X0~X1){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_cat\_1~X5~X0~X2){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_cat\_1~\\
X6~X0~X2){\unicdbdjdcd}(((r1\_nattra\_1~X0~X1~X3~X4){\unicdcdcdhd}(r1\_nattra\_1~X0~X2~X5~X6)){\unicdbdjdcd}\\
({\unicdcdadad} X7 .  (m1\_nattra\_1~X7~X0~X1~X3~X4){\unicdbdjdcd}({\unicdcdadad} X8 .  (m1\_nattra\_1~\\
X8~X0~X2~X5~X6){\unicdbdjdcd}({\unicdcdadad} X9 .  (m1\_subset\_1~X9~(u1\_struct\_0~X0)){\unicdbdjdcd}(\\
k3\_funct\_2~(u1\_struct\_0~X0)~(k2\_zfmisc\_1~(u4\_struct\_0~X1)~(u4\_struct\_0~\\
X2))~(k14\_funct\_3~(u1\_struct\_0~X0)~(u4\_struct\_0~X1)~(u4\_struct\_0~\\
X2)~X7~X8)~X9=k10\_cat\_2~X1~X2~(k4\_nattra\_1~X0~X1~X3~X4~X7~X9)~(k4\_nattra\_1~\\
X0~X2~X5~X6~X8~X9))))))))))))\end{array}
\label{hyp:mgedfdpfjgdhpgdgbgehpfcd}
\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}({\unicdcdadad} X3 .  (m1\_cat\_1~X3~X0~X1~X2){\unicdbdjdcd}(m1\_subset\_1~\\
X3~(u4\_struct\_0~X0)))\end{array}
\label{hyp:egehpfngbdpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (((\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)))))))){\unicdcdcdhd}(((\\
\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\preoh v11\_struct\_0~X1){\unicdcdcdhd}((v2\_cat\_1~X1){\unicdcdcdhd}((v3\_cat\_1~\\
X1){\unicdcdcdhd}((v4\_cat\_1~X1){\unicdcdcdhd}((v5\_cat\_1~X1){\unicdcdcdhd}((v6\_cat\_1~X1){\unicdcdcdhd}(l1\_cat\_1~\\
X1)))))))){\unicdcdcdhd}((m2\_cat\_1~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X3~(u1\_struct\_0~\\
X0))))){\unicdbdjdcd}(m1\_subset\_1~(k8\_cat\_1~X0~X1~X2~X3)~(u1\_struct\_0~X1))\end{array}
\label{hyp:egehpflgidpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \\
(((\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)))))))){\unicdcdcdhd}(((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\preoh v11\_struct\_0~X1){\unicdcdcdhd}((v2\_cat\_1~\\
X1){\unicdcdcdhd}((v3\_cat\_1~X1){\unicdcdcdhd}((v4\_cat\_1~X1){\unicdcdcdhd}((v5\_cat\_1~X1){\unicdcdcdhd}((v6\_cat\_1~\\
X1){\unicdcdcdhd}(l1\_cat\_1~X1)))))))){\unicdcdcdhd}((m2\_cat\_1~X2~X0~X1){\unicdcdcdhd}((m2\_cat\_1~X3~X0~\\
X1){\unicdcdcdhd}((m1\_nattra\_1~X4~X0~X1~X2~X3){\unicdcdcdhd}(m1\_subset\_1~X5~(u1\_struct\_0~\\
X0))))))){\unicdbdjdcd}(m1\_cat\_1~(k4\_nattra\_1~X0~X1~X2~X3~X4~X5)~X1~(k8\_cat\_1~\\
X0~X1~X2~X5)~(k8\_cat\_1~X0~X1~X3~X5))\end{array}
\label{hyp:egehpflgedpfogbgehehchbgpfbd}
\end{equation}
\begin{theorem}\label{thm:mgedidpfjgdhpgdgbgehpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}((v2\_cat\_1~\\
X0){\unicdcdcdhd}((v3\_cat\_1~X0){\unicdcdcdhd}((v4\_cat\_1~X0){\unicdcdcdhd}((v5\_cat\_1~X0){\unicdcdcdhd}((v6\_cat\_1~\\
X0){\unicdcdcdhd}(l1\_cat\_1~X0)))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\\
\preoh v11\_struct\_0~X1){\unicdcdcdhd}((v2\_cat\_1~X1){\unicdcdcdhd}((v3\_cat\_1~X1){\unicdcdcdhd}((v4\_cat\_1~\\
X1){\unicdcdcdhd}((v5\_cat\_1~X1){\unicdcdcdhd}((v6\_cat\_1~X1){\unicdcdcdhd}(l1\_cat\_1~X1)))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((\preoh v11\_struct\_0~X2){\unicdcdcdhd}((v2\_cat\_1~X2){\unicdcdcdhd}((v3\_cat\_1~\\
X2){\unicdcdcdhd}((v4\_cat\_1~X2){\unicdcdcdhd}((v5\_cat\_1~X2){\unicdcdcdhd}((v6\_cat\_1~X2){\unicdcdcdhd}(l1\_cat\_1~\\
X2)))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_cat\_1~X3~X0~X1){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_cat\_1~\\
X4~X0~X1){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_cat\_1~X5~X0~X2){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_cat\_1~\\
X6~X0~X2){\unicdbdjdcd}(((r1\_nattra\_1~X0~X1~X3~X4){\unicdcdcdhd}(r1\_nattra\_1~X0~X2~X5~X6)){\unicdbdjdcd}\\
({\unicdcdadad} X7 .  (m1\_nattra\_1~X7~X0~X1~X3~X4){\unicdbdjdcd}({\unicdcdadad} X8 .  (m1\_nattra\_1~\\
X8~X0~X2~X5~X6){\unicdbdjdcd}({\unicdcdadad} X9 .  (m1\_subset\_1~X9~(u1\_struct\_0~X0)){\unicdbdjdcd}(\\
k3\_funct\_2~(u1\_struct\_0~X0)~(k2\_zfmisc\_1~(u4\_struct\_0~X1)~(u4\_struct\_0~\\
X2))~(k14\_funct\_3~(u1\_struct\_0~X0)~(u4\_struct\_0~X1)~(u4\_struct\_0~\\
X2)~X7~X8)~X9 \in k2\_cat\_1~(k8\_cat\_2~X1~X2)~(k8\_cat\_1~X0~(k8\_cat\_2~X1~\\
X2)~(k8\_isocat\_2~X0~X1~X2~X3~X5)~X9)~(k8\_cat\_1~X0~(k8\_cat\_2~X1~X2)~(\\
k8\_isocat\_2~X0~X1~X2~X4~X6)~X9))))))))))))\end{array}$$
\end{theorem}
\end{document}
