\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{t6\_cayley (TMFtFLJzrsBNq9oc6hwCSTNJq4S6RHhaxSF)}
\maketitle
Let $ k1\_group\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_cayley : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_partfun1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
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 $ k2\_group\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k17\_monoid\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\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 $ v3\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ k3\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v15\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_monoid\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k6\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_cayley : {{\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}(k2\_group\_1~X0~(k1\_group\_1~X0)=k1\_group\_1~\\
X0)\end{array}
\label{hyp:ehidpfhgchpgfhahpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (k4\_binop\_1~(u1\_struct\_0~(k17\_monoid\_0~X0))~(u2\_algstr\_0~\\
(k17\_monoid\_0~X0))=k6\_partfun1~X0){\unicdcdcdhd}(k1\_group\_1~(k17\_monoid\_0~\\
X0)=k6\_partfun1~X0)\end{array}
\label{hyp:ehhdidpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k2\_cayley~\\
X0))){\unicdbdjdcd}((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~X1~X0~X0){\unicdcdcdhd}((v3\_funct\_2~X1~\\
X0~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X0))))))\end{array}
\label{hyp:ehfdpfdgbgjhmgfgjh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~X1~X0~X0){\unicdcdcdhd}\\
((v3\_funct\_2~X1~X0~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X0)))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~X0~X0){\unicdcdcdhd}\\
((v3\_funct\_2~X2~X0~X0){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X0)))))){\unicdbdjdcd}((r2\_relset\_1~X0~X0~(k1\_partfun1~X0~X0~X0~X0~X1~X2)~X2){\unicdbdjdcd}\\
(r2\_relset\_1~X0~X0~X1~(k6\_partfun1~X0))))\end{array}
\label{hyp:ehfdjdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  ((m1\_subset\_1~X2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~X0~X1)))){\unicdbdjdcd}((r2\_relset\_1~X0~X1~X2~X3){\unicdbdjded}(X2=X3))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchcdpfchfgmgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \\
(((v1\_funct\_1~X4){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X1)))){\unicdcdcdhd}((v1\_funct\_1~X5){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X2~X3))))){\unicdbdjdcd}(k1\_partfun1~X0~X1~X2~X3~X4~X5=k3\_relat\_1~X4~X5)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdpfahbgchehggfhogbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v2\_struct\_0~(k2\_cayley~X0)){\unicdcdcdhd}((v15\_algstr\_0~(k2\_cayley~\\
X0)){\unicdcdcdhd}((v2\_group\_1~(k2\_cayley~X0)){\unicdcdcdhd}((v3\_group\_1~(k2\_cayley~X0)){\unicdcdcdhd}\\
(v1\_monoid\_0~(k2\_cayley~X0)))))\end{array}
\label{hyp:ggdgedpfdgbgjhmgfgjh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_partfun1~(k6\_partfun1~X0)~X0){\unicdcdcdhd}(m1\_subset\_1~(k6\_partfun1~\\
X0)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X0)))\end{array}
\label{hyp:egehpflggdpfahbgchehggfhogbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v15\_algstr\_0~(k2\_cayley~X0)){\unicdcdcdhd}((v1\_monoid\_0~(k2\_cayley~\\
X0)){\unicdcdcdhd}(l3\_algstr\_0~(k2\_cayley~X0)))\end{array}
\label{hyp:egehpflgcdpfdgbgjhmgfgjh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l3\_algstr\_0~X0){\unicdbdjdcd}(m1\_subset\_1~(k1\_group\_1~X0)~(u1\_struct\_0~\\
X0))\end{array}
\label{hyp:egehpflgbdpfhgchpgfhahpfbd}
\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\_subset\_1~X1~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}((X2=k2\_group\_1~\\
X0~X1){\unicdbdjded}((k6\_algstr\_0~X0~X1~X2=k1\_group\_1~X0){\unicdcdcdhd}(k6\_algstr\_0~X0~X2~\\
X1=k1\_group\_1~X0)))))\end{array}
\label{hyp:egfdpfhgchpgfhahpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v15\_algstr\_0~X1){\unicdcdcdhd}((v1\_monoid\_0~X1){\unicdcdcdhd}\\
(l3\_algstr\_0~X1))){\unicdbdjdcd}((X1=k2\_cayley~X0){\unicdbdjded}((u1\_struct\_0~X1=k1\_cayley~\\
X0){\unicdcdcdhd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X1)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(u1\_struct\_0~X1)){\unicdbdjdcd}(k6\_algstr\_0~X1~X2~X3=k3\_relat\_1~\\
X2~X3)))))\end{array}
\label{hyp:egcdpfdgbgjhmgfgjh}
\end{equation}
\begin{theorem}\label{thm:ehgdpfdgbgjhmgfgjh}
${{\unicdcdadad} X0 .  {{{k1\_group\_1}~{({{k2\_cayley}~{X0}})}}={{k6\_partfun1}~{X0}}}}$.
\end{theorem}
\end{document}
