\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{l3\_algstr\_2 (TMdtNThc5NZ1NwgENJZtkL4MLEcA5HMP9BV)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_vectsp\_1 : {\iota}$ be given.
Let $ k4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_group\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k3\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k4\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k5\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}((k3\_rlvect\_1~k2\_vectsp\_1~X0~X1=\\
k3\_rlvect\_1~k2\_vectsp\_1~X1~X0){\unicdcdcdhd}((k3\_rlvect\_1~k2\_vectsp\_1~(k3\_rlvect\_1~\\
k2\_vectsp\_1~X0~X1)~X2=k3\_rlvect\_1~k2\_vectsp\_1~X0~(k3\_rlvect\_1~\\
k2\_vectsp\_1~X1~X2)){\unicdcdcdhd}((k3\_rlvect\_1~k2\_vectsp\_1~X0~(k4\_struct\_0~\\
k2\_vectsp\_1)=X0){\unicdcdcdhd}((k3\_rlvect\_1~k2\_vectsp\_1~X0~(k4\_algstr\_0~\\
k2\_vectsp\_1~X0)=k4\_struct\_0~k2\_vectsp\_1){\unicdcdcdhd}((k8\_group\_1~k2\_vectsp\_1~\\
X0~X1=k8\_group\_1~k2\_vectsp\_1~X1~X0){\unicdcdcdhd}((k8\_group\_1~k2\_vectsp\_1~\\
(k8\_group\_1~k2\_vectsp\_1~X0~X1)~X2=k8\_group\_1~k2\_vectsp\_1~X0~(k8\_group\_1~\\
k2\_vectsp\_1~X1~X2)){\unicdcdcdhd}((k8\_group\_1~k2\_vectsp\_1~(k5\_struct\_0~k2\_vectsp\_1)~\\
X0=X0){\unicdcdcdhd}((\preoh (X0{\unicdcdgdad}k4\_struct\_0~k2\_vectsp\_1){\unicdcdcdhd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}(k8\_group\_1~k2\_vectsp\_1~X3~X0{\unicdcdgdad}\\
k5\_struct\_0~k2\_vectsp\_1))){\unicdcdcdhd}((k8\_group\_1~k2\_vectsp\_1~X0~(k3\_rlvect\_1~\\
k2\_vectsp\_1~X1~X2)=k3\_rlvect\_1~k2\_vectsp\_1~(k8\_group\_1~k2\_vectsp\_1~\\
X0~X1)~(k8\_group\_1~k2\_vectsp\_1~X0~X2)){\unicdcdcdhd}(k8\_group\_1~k2\_vectsp\_1~\\
(k3\_rlvect\_1~k2\_vectsp\_1~X1~X2)~X0=k3\_rlvect\_1~k2\_vectsp\_1~(k8\_group\_1~\\
k2\_vectsp\_1~X1~X0)~(k8\_group\_1~k2\_vectsp\_1~X2~X0)))))))))))))\end{array}
\label{hyp:ehddpfghfgdgehdhahpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}(\preoh (X0{\unicdcdgdad}k4\_struct\_0~\\
k2\_vectsp\_1){\unicdcdcdhd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}\\
(k8\_group\_1~k2\_vectsp\_1~X0~X2{\unicdcdgdad}X1))))\end{array}
\label{hyp:mgcdpfbgmghgdhehchpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{k6\_numbers}={{k4\_struct\_0}~{k2\_vectsp\_1}}}\label{hyp:mgbdpfbgmghgdhehchpfcd}
\end{equation}
\begin{theorem}\label{thm:mgddpfbgmghgdhehchpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}(\preoh (X0{\unicdcdgdad}k4\_struct\_0~\\
k2\_vectsp\_1){\unicdcdcdhd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~k2\_vectsp\_1)){\unicdbdjdcd}\\
(k8\_group\_1~k2\_vectsp\_1~X2~X0{\unicdcdgdad}X1))))\end{array}$$
\end{theorem}
\end{document}
