\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{t41\_monoid\_0 (TMb39FhXBaUk7hEFJLFgRUY76psLTDwrZBQ)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v15\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_monoid\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_monoid\_0 : {\iota}$ be given.
Let $ k3\_monoid\_0 : {\iota}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_numbers : {\iota}$ be given.
Let $ l3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ g3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_gr\_cy\_1 : {\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 $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k44\_binop\_2 : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l3\_algstr\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((\preoh v2\_struct\_0~X1){\unicdcdcdhd}(m2\_monoid\_0~X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~\\
X2){\unicdcdcdhd}(m2\_monoid\_0~X2~X0)){\unicdbdjdcd}((u1\_struct\_0~X1=u1\_struct\_0~X2){\unicdbdjdcd}(\\
g3\_algstr\_0~(u1\_struct\_0~X1)~(u2\_algstr\_0~X1)=g3\_algstr\_0~(u1\_struct\_0~\\
X2)~(u2\_algstr\_0~X2)))))\end{array}
\label{hyp:ehcdgdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
{{k3\_monoid\_0}={k2\_gr\_cy\_1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfngpgogpgjgegpfad}
\end{equation}
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 .  (l3\_algstr\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_monoid\_0~X1~X0){\unicdbdjdcd}\\
(l3\_algstr\_0~X1))\end{array}
\label{hyp:egehpfngcdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k44\_binop\_2){\unicdcdcdhd}((v1\_funct\_2~k44\_binop\_2~(k2\_zfmisc\_1~\\
k4\_numbers~k4\_numbers)~k4\_numbers){\unicdcdcdhd}(m1\_subset\_1~k44\_binop\_2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~k4\_numbers~k4\_numbers)~\\
k4\_numbers))))\end{array}
\label{hyp:egehpflgededpfcgjgogpgahpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\preoh v2\_struct\_0~k3\_monoid\_0){\unicdcdcdhd}((v15\_algstr\_0~k3\_monoid\_0){\unicdcdcdhd}(m2\_monoid\_0~\\
k3\_monoid\_0~k2\_monoid\_0))\end{array}
\label{hyp:egehpflgddpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
{{({\preoh{{{v2\_struct\_0}~{k2\_monoid\_0}}}})}{\unicdcdcdhd}{({{l3\_algstr\_0}~{k2\_monoid\_0}})}}\label{hyp:egehpflgcdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\preoh v2\_struct\_0~k2\_gr\_cy\_1){\unicdcdcdhd}((v15\_algstr\_0~k2\_gr\_cy\_1){\unicdcdcdhd}(l3\_algstr\_0~\\
k2\_gr\_cy\_1))\end{array}
\label{hyp:egehpflgcdpfhgchpfdgjhpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k2\_gr\_cy\_1}={{{g3\_algstr\_0}~{k4\_numbers}}~{k44\_binop\_2}}}\label{hyp:egddpfhgchpfdgjhpfbd}
\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:ehedbdpfngpgogpgjgegpfad}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v15\_algstr\_0~X0){\unicdcdcdhd}(m2\_monoid\_0~\\
X0~k2\_monoid\_0))){\unicdbdjdcd}((X0=k3\_monoid\_0){\unicdbdjded}(u1\_struct\_0~X0=k4\_numbers))\end{array}$$
\end{theorem}
\end{document}
