\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{t26\_monoid\_1 (TMGjaagnSGgkenZisi95HnCT2T7QhSoFC5q)}
\maketitle
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k13\_monoid\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k9\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k5\_numbers : {\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 $ u2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_monoid\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k47\_binop\_2 : {\iota}$ 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 $ k6\_numbers : {\iota}$ be given.
Let $ k5\_monoid\_0 : {\iota}$ be given.
Let $ g4\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_monoid\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_monoid\_0 : {\iota}$ be given.
Let $ k4\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ l3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k11\_monoid\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ k9\_monoid\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_funcop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_monoid\_0 : {\iota}$ be given.
Let $ k2\_gr\_cy\_1 : {\iota}$ be given.
Let $ v3\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v16\_monoid\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v17\_monoid\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k12\_monoid\_1 : {{\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 $ g3\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v3\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v15\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_monoid\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m5\_monoid\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ m1\_monoid\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ l4\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l3\_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 $ k4\_monoid\_0 : {\iota}$ be given.
Let $ u3\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{k5\_monoid\_0}={{{{g4\_algstr\_0}~{k5\_numbers}}~{k47\_binop\_2}}~{k6\_numbers}}}\label{hyp:ehedgdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v1\_group\_1~X0){\unicdcdcdhd}(m2\_monoid\_0~\\
X0~k2\_monoid\_0))){\unicdbdjdcd}(k4\_binop\_1~(u1\_struct\_0~X0)~(u2\_algstr\_0~\\
X0)=k6\_numbers)\end{array}
\label{hyp:ehedadpfngpgogpgjgegpfad}
\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}
{{{m1\_subset\_1}~{k1\_xboole\_0}}~{k4\_ordinal1}}\label{hyp:ehbdpfogfhngfgchbgmgdh}
\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}
{{k6\_numbers}={k1\_xboole\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflggdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\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}
{{k3\_monoid\_0}={k2\_gr\_cy\_1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfngpgogpgjgegpfad}
\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}(\\
(v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}((v16\_monoid\_0~X0){\unicdcdcdhd}((v17\_monoid\_0~\\
X0){\unicdcdcdhd}(l3\_algstr\_0~X0))))))){\unicdbdjdcd}(k12\_monoid\_1~X0~X1=k9\_monoid\_1~\\
X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdcdpfngpgogpgjgegpfbd}
\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 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((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)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
{\unicdcdadad} X4 .  {\unicdcdadad} X5 .  (g4\_algstr\_0~X0~X1~X2=g4\_algstr\_0~X3~X4~X5){\unicdbdjdcd}\\
((X0=X3){\unicdcdcdhd}((X1=X4){\unicdcdcdhd}(X2=X5))))\end{array}
\label{hyp:ggchfgfgpfhgedpfbgmghgdhehchpfad}
\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}
{{({\preoh{{{v1\_xboole\_0}~{k4\_ordinal1}}}})}{\unicdcdcdhd}{({{v3\_ordinal1}~{k4\_ordinal1}})}}\label{hyp:ggdggdpfpgchegjgogbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\preoh v2\_struct\_0~k2\_monoid\_0){\unicdcdcdhd}((v15\_algstr\_0~k2\_monoid\_0){\unicdcdcdhd}((\\
v1\_group\_1~k2\_monoid\_0){\unicdcdcdhd}((v3\_group\_1~k2\_monoid\_0){\unicdcdcdhd}((v5\_group\_1~\\
k2\_monoid\_0){\unicdcdcdhd}((v13\_monoid\_0~k2\_monoid\_0){\unicdcdcdhd}(v16\_monoid\_0~k2\_monoid\_0))))))\end{array}
\label{hyp:ggdgbdpfngpgogpgjgegpfad}
\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 .  {\unicdcdadad} X1 .  ((l3\_algstr\_0~X0){\unicdcdcdhd}(m2\_monoid\_0~X1~X0)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m5\_monoid\_0~X2~X0~X1){\unicdbdjdcd}(m2\_monoid\_0~X2~X0))\end{array}
\label{hyp:egehpfngfdpfngpgogpgjgegpfad}
\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}{\unicdcdadad} X0 .  (l3\_algstr\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_monoid\_0~X1~X0){\unicdbdjdcd}\\
(l4\_algstr\_0~X1))\end{array}
\label{hyp:egehpfngbdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l4\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{({{l3\_struct\_0}~{X0}})}{\unicdcdcdhd}{({{l3\_algstr\_0}~{X0}})}})}}}\label{hyp:egehpfmgedpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\preoh v2\_struct\_0~k5\_monoid\_0){\unicdcdcdhd}((v22\_algstr\_0~k5\_monoid\_0){\unicdcdcdhd}((\\
v4\_vectsp\_1~k5\_monoid\_0){\unicdcdcdhd}(m1\_monoid\_0~k5\_monoid\_0~k4\_monoid\_0)))\end{array}
\label{hyp:egehpflgfdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(\preoh v2\_struct\_0~k4\_monoid\_0){\unicdcdcdhd}((v15\_algstr\_0~k4\_monoid\_0){\unicdcdcdhd}((\\
v1\_group\_1~k4\_monoid\_0){\unicdcdcdhd}((v17\_monoid\_0~k4\_monoid\_0){\unicdcdcdhd}(m5\_monoid\_0~\\
k4\_monoid\_0~k2\_monoid\_0~k3\_monoid\_0))))\end{array}
\label{hyp:egehpflgedpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k47\_binop\_2){\unicdcdcdhd}((v1\_funct\_2~k47\_binop\_2~(k2\_zfmisc\_1~\\
k5\_numbers~k5\_numbers)~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~k47\_binop\_2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k5\_numbers)~\\
k5\_numbers))))\end{array}
\label{hyp:egehpflgedhdpfcgjgogpgahpfcd}
\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}
{{\unicdcdadad} X0 .  {{{k13\_monoid\_1}~{X0}}={{{k12\_monoid\_1}~{k5\_monoid\_0}}~{X0}}}}\label{hyp:egedpfngpgogpgjgegpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k47\_binop\_2}={{u2\_algstr\_0}~{k4\_monoid\_0}}}\label{hyp:egcdjdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l3\_algstr\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (l4\_algstr\_0~X1){\unicdbdjdcd}((\\
m1\_monoid\_0~X1~X0){\unicdbdjded}(g3\_algstr\_0~(u1\_struct\_0~X1)~(u2\_algstr\_0~\\
X1)=g3\_algstr\_0~(u1\_struct\_0~X0)~(u2\_algstr\_0~X0))))\end{array}
\label{hyp:egcdcdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v16\_monoid\_0~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_monoid\_0~X1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~X1){\unicdbdjdcd}((\\
\preoh v2\_struct\_0~X1){\unicdcdcdhd}(v16\_monoid\_0~X1))))\end{array}
\label{hyp:dgdgddbdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_monoid\_0~X1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~X1){\unicdbdjdcd}((\\
\preoh v2\_struct\_0~X1){\unicdcdcdhd}(v3\_group\_1~X1))))\end{array}
\label{hyp:dgdgcdhdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_monoid\_0~X1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~X1){\unicdbdjdcd}((\\
\preoh v2\_struct\_0~X1){\unicdcdcdhd}(v5\_group\_1~X1))))\end{array}
\label{hyp:dgdgcdfdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v17\_monoid\_0~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_monoid\_0~X1~X0){\unicdbdjdcd}(v17\_monoid\_0~X1))\end{array}
\label{hyp:dgdgbdjdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v16\_monoid\_0~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_monoid\_0~X1~X0){\unicdbdjdcd}(v16\_monoid\_0~X1))\end{array}
\label{hyp:dgdgbdidpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_monoid\_0~X1~X0){\unicdbdjdcd}(v5\_group\_1~X1))\end{array}
\label{hyp:dgdgbdgdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_monoid\_0~X1~X0){\unicdbdjdcd}(v3\_group\_1~X1))\end{array}
\label{hyp:dgdgbdfdpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v1\_group\_1~X0){\unicdcdcdhd}(l3\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_monoid\_0~X1~X0){\unicdbdjdcd}(v1\_group\_1~X1))\end{array}
\label{hyp:dgdgbdedpfngpgogpgjgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l4\_algstr\_0~X0){\unicdbdjdcd}((v22\_algstr\_0~X0){\unicdbdjdcd}(X0=g4\_algstr\_0~\\
(u1\_struct\_0~X0)~(u2\_algstr\_0~X0)~(u3\_struct\_0~X0)))\end{array}
\label{hyp:bgcgdhehchbgdgehogfgdhdhpfghcdcdpfbgmghgdhehchpfad}
\end{equation}
\begin{theorem}\label{thm:ehcdgdpfngpgogpgjgegpfbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (u1\_struct\_0~(k13\_monoid\_1~X0)=k9\_funct\_2~X0~k5\_numbers){\unicdcdcdhd}\\
((r1\_funct\_2~(k2\_zfmisc\_1~(u1\_struct\_0~(k13\_monoid\_1~X0))~(u1\_struct\_0~\\
(k13\_monoid\_1~X0)))~(u1\_struct\_0~(k13\_monoid\_1~X0))~(k2\_zfmisc\_1~\\
(k9\_funct\_2~X0~k5\_numbers)~(k9\_funct\_2~X0~k5\_numbers))~(k9\_funct\_2~\\
X0~k5\_numbers)~(u2\_algstr\_0~(k13\_monoid\_1~X0))~(k8\_monoid\_1~k5\_numbers~\\
k5\_numbers~k5\_numbers~k47\_binop\_2~X0)){\unicdcdcdhd}(k5\_struct\_0~(k13\_monoid\_1~\\
X0)=k5\_monoid\_1~k5\_numbers~X0~k6\_numbers))\end{array}$$
\end{theorem}
\end{document}
