\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{t13\_cayley (TMdQ5QFD1225cQ38ffWDY1v5Nk4tF7LtPCU)}
\maketitle
Let $ r2\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r2\_group\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_cayley : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k10\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k9\_xtuple\_0 : {{\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 $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r2\_wellord2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v3\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ v2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k3\_cayley : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v2\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_group\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
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 $ r1\_group\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v5\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_monoid\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xboole\_0}~{X0}})}{\unicdbdjdcd}{({{X0}={k1\_xboole\_0}})}}}\label{hyp:ehgdpfcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}(v1\_funct\_1~X1)){\unicdbdjdcd}((r1\_tarski~\\
(k10\_xtuple\_0~X1)~X0){\unicdbdjdcd}((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~X1~(k9\_xtuple\_0~\\
X1)~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k9\_xtuple\_0~\\
X1)~X0))))))\end{array}
\label{hyp:ehcdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{r2\_wellord2}~{X0}}~{k1\_xboole\_0}})}{\unicdbdjdcd}{({{X0}={k1\_xboole\_0}})}}}\label{hyp:ehcdgdpfdgbgchegpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))))){\unicdbdjdcd}((v3\_funct\_2~X2~X0~X1){\unicdbdjdcd}\\
(v2\_funct\_2~(k3\_cayley~X0~X1~X2)~(u1\_struct\_0~(k2\_cayley~X1))))))\end{array}
\label{hyp:ehbdcdpfdgbgjhmgfgjh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))))){\unicdbdjdcd}((v3\_funct\_2~X2~X0~X1){\unicdbdjdcd}\\
(v2\_funct\_1~(k3\_cayley~X0~X1~X2)))))\end{array}
\label{hyp:ehbdbdpfdgbgjhmgfgjh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))))){\unicdbdjdcd}((v3\_funct\_2~X2~X0~X1){\unicdbdjdcd}\\
(v1\_group\_6~(k3\_cayley~X0~X1~X2)~(k2\_cayley~X0)~(k2\_cayley~X1)))))\end{array}
\label{hyp:ehbdadpfdgbgjhmgfgjh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{{r2\_wellord2}~{X0}}~{X1}})}{\unicdbdjdcd}{({{{r2\_wellord2}~{X1}}~{X0}})}}}}\label{hyp:dhjhngngfgehchjhpfchcdpfhhfgmgmgpgchegcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\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))))){\unicdcdcdhd}((\preoh \\
v2\_struct\_0~X1){\unicdcdcdhd}((v15\_algstr\_0~X1){\unicdcdcdhd}((v2\_group\_1~X1){\unicdcdcdhd}((v3\_group\_1~\\
X1){\unicdcdcdhd}(l3\_algstr\_0~X1)))))){\unicdbdjdcd}(r2\_group\_6~X0~X0)\end{array}
\label{hyp:chfgggmgfgihjgghjgehjhpfchcdpfhgchpgfhahpfgd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{{r1\_tarski}~{X0}}~{X0}}}}\label{hyp:chfgggmgfgihjgghjgehjhpfchbdpfehbgchdhlgjg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{{r2\_wellord2}~{X0}}~{X1}})}{\unicdbdjded}{({{{r2\_tarski}~{X0}}~{X1}})}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpfchcdpfhhfgmgmgpgchegcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\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))))){\unicdcdcdhd}((\preoh \\
v2\_struct\_0~X1){\unicdcdcdhd}((v15\_algstr\_0~X1){\unicdcdcdhd}((v2\_group\_1~X1){\unicdcdcdhd}((v3\_group\_1~\\
X1){\unicdcdcdhd}(l3\_algstr\_0~X1)))))){\unicdbdjdcd}((r2\_group\_6~X0~X1){\unicdbdjded}(r1\_group\_6~X0~\\
X1))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchcdpfhgchpgfhahpfgd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}(v5\_relat\_1~X1~X0)){\unicdbdjdcd}(\\
k2\_relset\_1~X0~X1=k10\_xtuple\_0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfchfgmgdhfgehpfbd}
\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 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))))){\unicdbdjdcd}\\
((v1\_funct\_1~(k3\_cayley~X0~X1~X2)){\unicdcdcdhd}((v1\_funct\_2~(k3\_cayley~X0~\\
X1~X2)~(u1\_struct\_0~(k2\_cayley~X0))~(u1\_struct\_0~(k2\_cayley~X1))){\unicdcdcdhd}\\
(m1\_subset\_1~(k3\_cayley~X0~X1~X2)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~\\
(k2\_cayley~X0))~(u1\_struct\_0~(k2\_cayley~X1)))))))\end{array}
\label{hyp:egehpflgddpfdgbgjhmgfgjh}
\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 .  {\unicdcdadad} X1 .  (r2\_wellord2~X0~X1){\unicdbdjded}({\unicdcdaddd} X2 .  ((v1\_relat\_1~\\
X2){\unicdcdcdhd}(v1\_funct\_1~X2)){\unicdcdcdhd}((v2\_funct\_1~X2){\unicdcdcdhd}((k9\_xtuple\_0~X2=X0){\unicdcdcdhd}\\
(k10\_xtuple\_0~X2=X1))))\end{array}
\label{hyp:egedpfhhfgmgmgpgchegcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}(v5\_relat\_1~X1~X0)){\unicdbdjdcd}(\\
(v2\_funct\_2~X1~X0){\unicdbdjded}(k2\_relset\_1~X0~X1=X0))\end{array}
\label{hyp:egddpfggfhogdgehpfcd}
\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 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v2\_group\_1~\\
X1){\unicdcdcdhd}((v3\_group\_1~X1){\unicdcdcdhd}(l3\_algstr\_0~X1)))){\unicdbdjdcd}((r1\_group\_6~X0~X1){\unicdbdjded}\\
({\unicdcdaddd} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~(u1\_struct\_0~X0)~\\
(u1\_struct\_0~X1)){\unicdcdcdhd}((v1\_group\_6~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~X0)~(u1\_struct\_0~X1))))))){\unicdcdcdhd}(v3\_funct\_2~\\
X2~(u1\_struct\_0~X0)~(u1\_struct\_0~X1)))))\end{array}
\label{hyp:egbdbdpfhgchpgfhahpfgd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~X0~X1))){\unicdbdjdcd}(((v1\_funct\_1~X2){\unicdcdcdhd}((v2\_funct\_1~X2){\unicdcdcdhd}(v2\_funct\_2~\\
X2~X1))){\unicdbdjdcd}((v1\_funct\_1~X2){\unicdcdcdhd}(v3\_funct\_2~X2~X0~X1)))\end{array}
\label{hyp:dgdggdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~X0~X1))){\unicdbdjdcd}((v4\_relat\_1~X2~X0){\unicdcdcdhd}(v5\_relat\_1~X2~X1))\end{array}
\label{hyp:dgdgcdpfchfgmgdhfgehpfbd}
\end{equation}
\begin{theorem}\label{thm:ehbdddpfdgbgjhmgfgjh}
$$\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (r2\_tarski~X0~X1){\unicdbdjdcd}(r2\_group\_6~(k2\_cayley~\\
X0)~(k2\_cayley~X1))\end{array}$$
\end{theorem}
\end{document}
