\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{t69\_group\_5 (TMLjPGiqHtZcnVa3t26Nz6g5WJ5XgSpAQLc)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_group\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m1\_group\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r1\_group\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k8\_group\_5 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v15\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k8\_group\_4 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_group\_1~X0){\unicdcdcdhd}((v3\_group\_1~\\
X0){\unicdcdcdhd}(l3\_algstr\_0~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_group\_2~X1~X0){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((v15\_algstr\_0~X2){\unicdcdcdhd}(m1\_group\_2~X2~X0)){\unicdbdjdcd}((m1\_group\_2~X1~X2){\unicdbdjded}(\\
r1\_group\_2~X0~(k8\_group\_4~X0~X1~X2)~X2))))\end{array}
\label{hyp:ehgdbdpfhgchpgfhahpfed}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_group\_1~X0){\unicdcdcdhd}((v3\_group\_1~\\
X0){\unicdcdcdhd}(l3\_algstr\_0~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_group\_2~X1~X0){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_group\_2~X2~X0){\unicdbdjdcd}(r1\_group\_2~X0~(k8\_group\_4~X0~X1~X2)~(k8\_group\_4~\\
X0~X2~X1))))\end{array}
\label{hyp:ehfdgdpfhgchpgfhahpfed}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0)))){\unicdcdcdhd}(((v15\_algstr\_0~X1){\unicdcdcdhd}\\
(m1\_group\_2~X1~X0)){\unicdcdcdhd}((v15\_algstr\_0~X2){\unicdcdcdhd}(m1\_group\_2~X2~X0)))){\unicdbdjdcd}\\
((r1\_group\_2~X0~X1~X2){\unicdbdjded}(X1=X2))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchbdpfhgchpgfhahpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_group\_1~X0){\unicdcdcdhd}((v3\_group\_1~\\
X0){\unicdcdcdhd}(l3\_algstr\_0~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_group\_3~X1~X0){\unicdcdcdhd}(m1\_group\_2~\\
X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_group\_3~X2~X0){\unicdcdcdhd}(m1\_group\_2~X2~X0)){\unicdbdjdcd}(\\
m1\_group\_2~(k8\_group\_5~X0~X1~X2)~(k8\_group\_5~X0~X2~X1))))\end{array}
\label{hyp:mghdidpfhgchpgfhahpffd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0)))){\unicdcdcdhd}((m1\_group\_2~X1~X0){\unicdcdcdhd}\\
(m1\_group\_2~X2~X0))){\unicdbdjdcd}((v15\_algstr\_0~(k8\_group\_5~X0~X1~X2)){\unicdcdcdhd}(m1\_group\_2~\\
(k8\_group\_5~X0~X1~X2)~X0))\end{array}
\label{hyp:egehpflgidpfhgchpgfhahpffd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0)))){\unicdcdcdhd}((m1\_group\_2~X1~X0){\unicdcdcdhd}\\
(m1\_group\_2~X2~X0))){\unicdbdjdcd}((v15\_algstr\_0~(k8\_group\_4~X0~X1~X2)){\unicdcdcdhd}(m1\_group\_2~\\
(k8\_group\_4~X0~X1~X2)~X0))\end{array}
\label{hyp:egehpflgidpfhgchpgfhahpfed}
\end{equation}
\begin{theorem}\label{thm:ehgdjdpfhgchpgfhahpffd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_group\_1~X0){\unicdcdcdhd}((v3\_group\_1~\\
X0){\unicdcdcdhd}(l3\_algstr\_0~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_group\_3~X1~X0){\unicdcdcdhd}(m1\_group\_2~\\
X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_group\_3~X2~X0){\unicdcdcdhd}(m1\_group\_2~X2~X0)){\unicdbdjdcd}(\\
r1\_group\_2~X0~(k8\_group\_5~X0~X1~X2)~(k8\_group\_5~X0~X2~X1))))\end{array}$$
\end{theorem}
\end{document}
