\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{t25\_monoid\_1 (TMb5PP8wtMC1U6MCDAf5WtTvQobt51kDyns)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_monoid\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k4\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m3\_monoid\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k9\_monoid\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k11\_monoid\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k5\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_monoid\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k9\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_monoid\_1 : {{\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 $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_funcop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v22\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_monoid\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l4\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{\preoh{{({{X0} \in {X1}})}}}{\unicdcdcdhd}{({{v1\_xboole\_0}~{X1}})}}}}\label{hyp:ehhdpfcgpgpgmgfg}
\end{equation}
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}(((v1\_group\_1~X0){\unicdcdcdhd}(\\
k4\_binop\_1~(u1\_struct\_0~X0)~(u2\_algstr\_0~X0) \in u1\_struct\_0~X1)){\unicdbdjdcd}\\
((v1\_group\_1~X1){\unicdcdcdhd}(k4\_binop\_1~(u1\_struct\_0~X0)~(u2\_algstr\_0~X0)=\\
k4\_binop\_1~(u1\_struct\_0~X1)~(u2\_algstr\_0~X1)))))\end{array}
\label{hyp:ehddadpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v1\_group\_1~X1){\unicdcdcdhd}(\\
l3\_algstr\_0~X1))){\unicdbdjdcd}(k5\_struct\_0~(k11\_monoid\_1~X1~X0)=k5\_monoid\_1~\\
(u1\_struct\_0~X1)~X0~(k4\_binop\_1~(u1\_struct\_0~X1)~(u2\_algstr\_0~\\
X1)))\end{array}
\label{hyp:ehcdcdpfngpgogpgjgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{X0} \in {X1}})}{\unicdbdjdcd}{({{{m1\_subset\_1}~{X0}}~{X1}})}}}}\label{hyp:ehbdpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}(l3\_algstr\_0~X1)){\unicdbdjdcd}\\
((u1\_struct\_0~(k9\_monoid\_1~X1~X0)=k9\_funct\_2~X0~(u1\_struct\_0~\\
X1)){\unicdcdcdhd}(r1\_funct\_2~(k2\_zfmisc\_1~(u1\_struct\_0~(k9\_monoid\_1~X1~X0))~\\
(u1\_struct\_0~(k9\_monoid\_1~X1~X0)))~(u1\_struct\_0~(k9\_monoid\_1~\\
X1~X0))~(k2\_zfmisc\_1~(k9\_funct\_2~X0~(u1\_struct\_0~X1))~(k9\_funct\_2~\\
X0~(u1\_struct\_0~X1)))~(k9\_funct\_2~X0~(u1\_struct\_0~X1))~(u2\_algstr\_0~\\
(k9\_monoid\_1~X1~X0))~(k8\_monoid\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X1)~(u2\_algstr\_0~X1)~X0)))\end{array}
\label{hyp:ehbdhdpfngpgogpgjgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  (\preoh v1\_xboole\_0~\\
X2){\unicdbdjdcd}({\unicdcdadad} X3 .  (\preoh v1\_xboole\_0~X3){\unicdbdjdcd}({\unicdcdadad} X4 .  (\preoh v1\_xboole\_0~\\
X4){\unicdbdjdcd}({\unicdcdadad} X5 .  (\preoh v1\_xboole\_0~X5){\unicdbdjdcd}({\unicdcdadad} X6 .  (\preoh v1\_xboole\_0~\\
X6){\unicdbdjdcd}({\unicdcdadad} X7 .  ((v1\_funct\_1~X7){\unicdcdcdhd}((v1\_funct\_2~X7~(k2\_zfmisc\_1~\\
X1~X2)~X3){\unicdcdcdhd}(m1\_subset\_1~X7~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
X1~X2)~X3))))){\unicdbdjdcd}({\unicdcdadad} X8 .  ((v1\_funct\_1~X8){\unicdcdcdhd}((v1\_funct\_2~X8~(\\
k2\_zfmisc\_1~X4~X5)~X6){\unicdcdcdhd}(m1\_subset\_1~X8~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~X4~X5)~X6))))){\unicdbdjdcd}((r1\_relset\_1~(k2\_zfmisc\_1~X1~X2)~\\
X3~X7~X8){\unicdbdjdcd}(r1\_relset\_1~(k2\_zfmisc\_1~(k9\_funct\_2~X0~X1)~(k9\_funct\_2~\\
X0~X2))~(k9\_funct\_2~X0~X3)~(k8\_monoid\_1~X1~X2~X3~X7~X0)~(k8\_monoid\_1~\\
X4~X5~X6~X8~X0))))))))))\end{array}
\label{hyp:ehbdgdpfngpgogpgjgegpfbd}
\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))){\unicdbdjdcd}((r1\_relset\_1~X0~X1~X2~X3){\unicdbdjded}(\\
r1\_tarski~X2~X3))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchbdpfchfgmgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \\
((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((\preoh v1\_xboole\_0~X3){\unicdcdcdhd}(((v1\_funct\_1~X4){\unicdcdcdhd}((\\
v1\_funct\_2~X4~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X1))))){\unicdcdcdhd}((v1\_funct\_1~X5){\unicdcdcdhd}((v1\_funct\_2~X5~X2~X3){\unicdcdcdhd}(m1\_subset\_1~\\
X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X2~X3)))))))){\unicdbdjdcd}((r1\_funct\_2~X0~X1~\\
X2~X3~X4~X5){\unicdbdjded}(X4=X5))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchbdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(k9\_funct\_2~X0~X1=k1\_funct\_2~\\
X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgjdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(m1\_subset\_1~\\
X2~X0)){\unicdbdjdcd}(k5\_monoid\_1~X0~X1~X2=k2\_funcop\_1~X1~X2)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfngpgogpgjgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v1\_group\_1~X0){\unicdcdcdhd}(\\
l3\_algstr\_0~X0))){\unicdbdjdcd}(k11\_monoid\_1~X0~X1=k9\_monoid\_1~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdbdpfngpgogpgjgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l1\_struct\_0~X0)){\unicdbdjdcd}(\preoh v1\_xboole\_0~\\
(u1\_struct\_0~X0))\end{array}
\label{hyp:ggdgcdpfdhehchfhdgehpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(\preoh v1\_xboole\_0~(k1\_funct\_2~\\
X0~X1))\end{array}
\label{hyp:ggdgbdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l3\_algstr\_0~X0){\unicdbdjdcd}((v1\_funct\_1~(u2\_algstr\_0~X0)){\unicdcdcdhd}\\
((v1\_funct\_2~(u2\_algstr\_0~X0)~(k2\_zfmisc\_1~(u1\_struct\_0~X0)~(\\
u1\_struct\_0~X0))~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~(u2\_algstr\_0~\\
X0)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X0)~(\\
u1\_struct\_0~X0))~(u1\_struct\_0~X0))))))\end{array}
\label{hyp:egehpffhcdpfbgmghgdhehchpfad}
\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}
{{\unicdcdadad} X0 .  {{({{l3\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{l1\_struct\_0}~{X0}})}}}\label{hyp:egehpfmgddpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l3\_algstr\_0~X0)){\unicdbdjdcd}\\
(l3\_algstr\_0~(k9\_monoid\_1~X0~X1))\end{array}
\label{hyp:egehpflgjdpfngpgogpgjgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  ((\preoh v1\_xboole\_0~\\
X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((\preoh v1\_xboole\_0~X2){\unicdcdcdhd}((v1\_funct\_1~X3){\unicdcdcdhd}\\
((v1\_funct\_2~X3~(k2\_zfmisc\_1~X0~X1)~X2){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1)~X2)))))))){\unicdbdjdcd}((v1\_funct\_1~(k8\_monoid\_1~\\
X0~X1~X2~X3~X4)){\unicdcdcdhd}((v1\_funct\_2~(k8\_monoid\_1~X0~X1~X2~X3~X4)~(k2\_zfmisc\_1~\\
(k9\_funct\_2~X4~X0)~(k9\_funct\_2~X4~X1))~(k9\_funct\_2~X4~X2)){\unicdcdcdhd}(m1\_subset\_1~\\
(k8\_monoid\_1~X0~X1~X2~X3~X4)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k9\_funct\_2~X4~X0)~(k9\_funct\_2~X4~X1))~(k9\_funct\_2~X4~X2))))))\end{array}
\label{hyp:egehpflgidpfngpgogpgjgegpfbd}
\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}(m1\_subset\_1~(k4\_binop\_1~X0~X1)~X0)\end{array}
\label{hyp:egehpflgedpfcgjgogpgahpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v1\_group\_1~X0){\unicdcdcdhd}(\\
l3\_algstr\_0~X0))){\unicdbdjdcd}((\preoh v2\_struct\_0~(k11\_monoid\_1~X0~X1)){\unicdcdcdhd}((v22\_algstr\_0~\\
(k11\_monoid\_1~X0~X1)){\unicdcdcdhd}((v4\_vectsp\_1~(k11\_monoid\_1~X0~X1)){\unicdcdcdhd}((\\
v1\_monoid\_0~(k11\_monoid\_1~X0~X1)){\unicdcdcdhd}(l4\_algstr\_0~(k11\_monoid\_1~\\
X0~X1))))))\end{array}
\label{hyp:egehpflgbdbdpfngpgogpgjgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l4\_algstr\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (l4\_algstr\_0~X1){\unicdbdjdcd}((\\
m3\_monoid\_0~X1~X0){\unicdbdjded}((r1\_tarski~(u2\_algstr\_0~X1)~(u2\_algstr\_0~\\
X0)){\unicdcdcdhd}(k5\_struct\_0~X1=k5\_struct\_0~X0))))\end{array}
\label{hyp:egcdfdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l3\_algstr\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (l3\_algstr\_0~X1){\unicdbdjdcd}((\\
m2\_monoid\_0~X1~X0){\unicdbdjded}(r1\_tarski~(u2\_algstr\_0~X1)~(u2\_algstr\_0~X0))))\end{array}
\label{hyp:egcdddpfngpgogpgjgegpfad}
\end{equation}
\begin{theorem}\label{thm:ehcdfdpfngpgogpgjgegpfbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v1\_group\_1~X1){\unicdcdcdhd}(\\
l3\_algstr\_0~X1))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}(m2\_monoid\_0~\\
X2~X1)){\unicdbdjdcd}((k4\_binop\_1~(u1\_struct\_0~X1)~(u2\_algstr\_0~X1) \in u1\_struct\_0~\\
X2){\unicdbdjdcd}(m3\_monoid\_0~(k9\_monoid\_1~X2~X0)~(k11\_monoid\_1~X1~X0))))\end{array}$$
\end{theorem}
\end{document}
