\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{t8\_autgroup (TMTkW3qxeFWsDaXpnAagCxT8cugm675beRM)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v15\_algstr\_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 $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_autgroup : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k1\_autgroup : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ g3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_autgroup : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ u2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~X1~(k2\_zfmisc\_1~\\
X0~X0)~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X0)~X0))))){\unicdbdjdcd}({\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (g3\_algstr\_0~X0~X1=g3\_algstr\_0~\\
X2~X3){\unicdbdjdcd}((X0=X2){\unicdcdcdhd}(X1=X3)))\end{array}
\label{hyp:ggchfgfgpfhgddpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v15\_algstr\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0))))){\unicdbdjdcd}((\preoh v2\_struct\_0~(\\
k3\_autgroup~X0)){\unicdcdcdhd}((v15\_algstr\_0~(k3\_autgroup~X0)){\unicdcdcdhd}((v2\_group\_1~\\
(k3\_autgroup~X0)){\unicdcdcdhd}((v3\_group\_1~(k3\_autgroup~X0)){\unicdcdcdhd}(l3\_algstr\_0~\\
(k3\_autgroup~X0))))))\end{array}
\label{hyp:egehpflgddpfbgfhehhgchpgfhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v15\_algstr\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0))))){\unicdbdjdcd}((v1\_funct\_1~(k2\_autgroup~\\
X0)){\unicdcdcdhd}((v1\_funct\_2~(k2\_autgroup~X0)~(k2\_zfmisc\_1~(k1\_autgroup~\\
X0)~(k1\_autgroup~X0))~(k1\_autgroup~X0)){\unicdcdcdhd}(m1\_subset\_1~(k2\_autgroup~\\
X0)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~(k1\_autgroup~X0)~(\\
k1\_autgroup~X0))~(k1\_autgroup~X0))))))\end{array}
\label{hyp:egehpflgcdpfbgfhehhgchpgfhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v15\_algstr\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0))))){\unicdbdjdcd}(k3\_autgroup~X0=\\
g3\_algstr\_0~(k1\_autgroup~X0)~(k2\_autgroup~X0))\end{array}
\label{hyp:egddpfbgfhehhgchpgfhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v15\_algstr\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}((v1\_funct\_2~X1~(k2\_zfmisc\_1~(k1\_autgroup~X0)~(k1\_autgroup~\\
X0))~(k1\_autgroup~X0)){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k1\_autgroup~X0)~(k1\_autgroup~X0))~(k1\_autgroup~\\
X0)))))){\unicdbdjdcd}((X1=k2\_autgroup~X0){\unicdbdjded}({\unicdcdadad} X2 .  (m2\_funct\_2~X2~(u1\_struct\_0~\\
X0)~(u1\_struct\_0~X0)~(k1\_autgroup~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_funct\_2~\\
X3~(u1\_struct\_0~X0)~(u1\_struct\_0~X0)~(k1\_autgroup~X0)){\unicdbdjdcd}(k5\_binop\_1~\\
(k1\_autgroup~X0)~X1~X2~X3=k1\_partfun1~(u1\_struct\_0~X0)~(u1\_struct\_0~\\
X0)~(u1\_struct\_0~X0)~(u1\_struct\_0~X0)~X3~X2)))))\end{array}
\label{hyp:egcdpfbgfhehhgchpgfhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l3\_algstr\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}(k6\_algstr\_0~\\
X0~X1~X2=k5\_binop\_1~(u1\_struct\_0~X0)~(u2\_algstr\_0~X0)~X1~X2)))\end{array}
\label{hyp:egbdidpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l3\_algstr\_0~X0){\unicdbdjdcd}((v15\_algstr\_0~X0){\unicdbdjdcd}(X0=g3\_algstr\_0~\\
(u1\_struct\_0~X0)~(u2\_algstr\_0~X0)))\end{array}
\label{hyp:bgcgdhehchbgdgehogfgdhdhpfghbdfdpfbgmghgdhehchpfad}
\end{equation}
\begin{theorem}\label{thm:ehidpfbgfhehhgchpgfhah}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v15\_algstr\_0~X0){\unicdcdcdhd}((v2\_group\_1~\\
X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~(u1\_struct\_0~(k3\_autgroup~X0))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~(k3\_autgroup~X0))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_funct\_2~X3~\\
(u1\_struct\_0~X0)~(u1\_struct\_0~X0)~(k1\_autgroup~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
(m2\_funct\_2~X4~(u1\_struct\_0~X0)~(u1\_struct\_0~X0)~(k1\_autgroup~\\
X0)){\unicdbdjdcd}(((X1=X3){\unicdcdcdhd}(X2=X4)){\unicdbdjdcd}(k6\_algstr\_0~(k3\_autgroup~X0)~X1~X2=\\
k1\_partfun1~(u1\_struct\_0~X0)~(u1\_struct\_0~X0)~(u1\_struct\_0~X0)~\\
(u1\_struct\_0~X0)~X4~X3))))))\end{array}$$
\end{theorem}
\end{document}
