\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{t50\_monoid\_0 (TMXFdBuqfpipeBvJTjrxJxT3ZdsmM6Hf1GE)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_monoid\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k6\_monoid\_0 : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k6\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k11\_binop\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ l3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ 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 $ k5\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k1\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ g3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v15\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k35\_binop\_2 : {\iota}$ be given.
Let $ u2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ 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 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~X1)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~\\
X1)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
(m1\_subset\_1~X5~(u1\_struct\_0~X0)){\unicdbdjdcd}(((X2=X4){\unicdcdcdhd}(X3=X5)){\unicdbdjdcd}(k6\_algstr\_0~\\
X1~X2~X3=k6\_algstr\_0~X0~X4~X5)))))))\end{array}
\label{hyp:ehcdfdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (((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))))){\unicdcdcdhd}((m1\_subset\_1~X2~X0){\unicdcdcdhd}\\
(m1\_subset\_1~X3~X0))){\unicdbdjdcd}(k5\_binop\_1~X0~X1~X2~X3=k1\_binop\_1~X1~X2~X3)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfcgjgogpgahpfbd}
\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}(\preoh v2\_struct\_0~k6\_monoid\_0){\unicdcdcdhd}((v15\_algstr\_0~k6\_monoid\_0){\unicdcdcdhd}((\\
v1\_group\_1~k6\_monoid\_0){\unicdcdcdhd}((v3\_group\_1~k6\_monoid\_0){\unicdcdcdhd}((v5\_group\_1~\\
k6\_monoid\_0){\unicdcdcdhd}(l3\_algstr\_0~k6\_monoid\_0)))))\end{array}
\label{hyp:egehpflggdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k35\_binop\_2){\unicdcdcdhd}((v1\_funct\_2~k35\_binop\_2~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~k35\_binop\_2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~\\
k1\_numbers))))\end{array}
\label{hyp:egehpflgddfdpfcgjgogpgahpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{k6\_monoid\_0}={{{g3\_algstr\_0}~{k1\_numbers}}~{k35\_binop\_2}}}\label{hyp:egddadpfngpgogpgjgegpfad}
\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 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers)~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)~k1\_numbers))))){\unicdbdjdcd}((X0=k35\_binop\_2){\unicdbdjded}\\
({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  (v1\_xreal\_0~X2){\unicdbdjdcd}(k1\_binop\_1~\\
X0~X1~X2=k11\_binop\_2~X1~X2))))\end{array}
\label{hyp:egbdbdpfcgjgogpgahpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfihchfgbgmgpfad}
\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:ehfdadpfngpgogpgjgegpfad}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(m2\_monoid\_0~X0~k6\_monoid\_0)){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k1\_numbers){\unicdbdjdcd}(((X1=X3){\unicdcdcdhd}(X2=X4)){\unicdbdjdcd}\\
(k6\_algstr\_0~X0~X1~X2=k11\_binop\_2~X3~X4))))))\end{array}$$
\end{theorem}
\end{document}
