\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{l32\_groeb\_3 (TMafhcWB3Utp32rqq8JGQ5mcygRuKJEjDYP)}
\maketitle
Let $ v3\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k15\_pre\_poly : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_relat\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_relat\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_relat\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v8\_relat\_2 : {{\iota}{\unicdbdjdcd}{o}}$ 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 $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l2\_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 $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_card\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k2\_groeb\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k3\_groeb\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k4\_groeb\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k4\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ l2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v4\_valued\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_pre\_poly : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ l1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_finset\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_termord : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{{r1\_tarski}~{({{{k4\_xboole\_0}~{X0}}~{X1}})}}~{X0}}}}\label{hyp:ehddgdpfihcgpgpgmgfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{{r1\_tarski}~{X0}}~{X1}})}{\unicdbdjdcd}{({{{{k3\_xboole\_0}~{X0}}~{X1}}={X0}})}}}}\label{hyp:ehcdidpfihcgpgpgmgfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}(l2\_struct\_0~X1)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~(k15\_pre\_poly~X0)~\\
(u1\_struct\_0~X1)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k15\_pre\_poly~X0)~(u1\_struct\_0~X1)))))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k15\_pre\_poly~X0))){\unicdbdjdcd}((k2\_polynom1~(k15\_pre\_poly~\\
X0)~X1~(k2\_groeb\_3~X0~X1~X2~X3)=k3\_xboole\_0~(k2\_polynom1~(k15\_pre\_poly~\\
X0)~X1~X2)~X3){\unicdcdcdhd}({\unicdcdadad} X4 .  ((v1\_relat\_1~X4){\unicdcdcdhd}((v4\_relat\_1~X4~X0){\unicdcdcdhd}\\
((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_partfun1~X4~X0){\unicdcdcdhd}((v4\_valued\_0~X4){\unicdcdcdhd}(v2\_pre\_poly~\\
X4)))))){\unicdbdjdcd}((X4 \in k2\_polynom1~(k15\_pre\_poly~X0)~X1~(k2\_groeb\_3~X0~\\
X1~X2~X3)){\unicdbdjdcd}(k3\_polynom1~X0~X1~(k2\_groeb\_3~X0~X1~X2~X3)~X4=k3\_polynom1~\\
X0~X1~X2~X4))))))\end{array}
\label{hyp:ehbdgdpfhgchpgfgcgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l2\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{({{l2\_struct\_0}~{X0}})}{\unicdcdcdhd}{({{l1\_algstr\_0}~{X0}})}})}}}\label{hyp:egehpfmgcdpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  ((v3\_ordinal1~\\
X0){\unicdcdcdhd}(((v1\_partfun1~X1~(k15\_pre\_poly~X0)){\unicdcdcdhd}((v1\_relat\_2~X1){\unicdcdcdhd}(\\
(v4\_relat\_2~X1){\unicdcdcdhd}((v6\_relat\_2~X1){\unicdcdcdhd}((v8\_relat\_2~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(k15\_pre\_poly~\\
X0))))))))){\unicdcdcdhd}(((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~X2){\unicdcdcdhd}((v3\_rlvect\_1~\\
X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l2\_algstr\_0~X2))))){\unicdcdcdhd}(((v1\_funct\_1~X3){\unicdcdcdhd}\\
((v1\_funct\_2~X3~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~\\
X3~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k15\_pre\_poly~X0)~(u1\_struct\_0~X2))))))){\unicdcdcdhd}(m1\_subset\_1~X4~k5\_numbers))))){\unicdbdjdcd}\\
((v1\_finset\_1~(k4\_groeb\_3~X0~X1~X2~X3~X4)){\unicdcdcdhd}(m1\_subset\_1~(k4\_groeb\_3~\\
X0~X1~X2~X3~X4)~(k1\_zfmisc\_1~(k15\_pre\_poly~X0))))\end{array}
\label{hyp:egehpflgedpfhgchpgfgcgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  ((v3\_ordinal1~\\
X0){\unicdcdcdhd}(((v1\_partfun1~X1~(k15\_pre\_poly~X0)){\unicdcdcdhd}((v1\_relat\_2~X1){\unicdcdcdhd}(\\
(v4\_relat\_2~X1){\unicdcdcdhd}((v6\_relat\_2~X1){\unicdcdcdhd}((v8\_relat\_2~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(k15\_pre\_poly~\\
X0))))))))){\unicdcdcdhd}(((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~X2){\unicdcdcdhd}((v3\_rlvect\_1~\\
X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l2\_algstr\_0~X2))))){\unicdcdcdhd}(((v1\_funct\_1~X3){\unicdcdcdhd}\\
((v1\_funct\_2~X3~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~\\
X3~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k15\_pre\_poly~X0)~(u1\_struct\_0~X2))))))){\unicdcdcdhd}(m1\_subset\_1~X4~k5\_numbers))))){\unicdbdjdcd}\\
((v1\_finset\_1~(k3\_groeb\_3~X0~X1~X2~X3~X4)){\unicdcdcdhd}(m1\_subset\_1~(k3\_groeb\_3~\\
X0~X1~X2~X3~X4)~(k1\_zfmisc\_1~(k15\_pre\_poly~X0))))\end{array}
\label{hyp:egehpflgddpfhgchpgfgcgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v3\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_partfun1~X1~(k15\_pre\_poly~\\
X0)){\unicdcdcdhd}((v1\_relat\_2~X1){\unicdcdcdhd}((v4\_relat\_2~X1){\unicdcdcdhd}((v6\_relat\_2~X1){\unicdcdcdhd}((\\
v8\_relat\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(k15\_pre\_poly~X0))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}\\
((v13\_algstr\_0~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l2\_algstr\_0~\\
X2))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X3~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}\\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(\\
u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k5\_numbers){\unicdbdjdcd}\\
(k4\_groeb\_3~X0~X1~X2~X3~X4=k4\_xboole\_0~(k2\_polynom1~(k15\_pre\_poly~\\
X0)~X2~X3)~(k3\_groeb\_3~X0~X1~X2~X3~X4))))))\end{array}
\label{hyp:egddpfhgchpgfgcgpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v3\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_partfun1~X1~(k15\_pre\_poly~\\
X0)){\unicdcdcdhd}((v1\_relat\_2~X1){\unicdcdcdhd}((v4\_relat\_2~X1){\unicdcdcdhd}((v6\_relat\_2~X1){\unicdcdcdhd}((\\
v8\_relat\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(k15\_pre\_poly~X0))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}\\
((v13\_algstr\_0~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l2\_algstr\_0~\\
X2))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X3~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}\\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(\\
u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k5\_numbers){\unicdbdjdcd}\\
((r1\_xxreal\_0~X4~(k5\_card\_1~(k2\_polynom1~(k15\_pre\_poly~X0)~X2~\\
X3))){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v1\_finset\_1~X5){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~\\
(k15\_pre\_poly~X0)))){\unicdbdjdcd}((X5=k3\_groeb\_3~X0~X1~X2~X3~X4){\unicdbdjded}((r1\_tarski~\\
X5~(k2\_polynom1~(k15\_pre\_poly~X0)~X2~X3)){\unicdcdcdhd}((k5\_card\_1~X5=X4){\unicdcdcdhd}\\
({\unicdcdadad} X6 .  ((v1\_relat\_1~X6){\unicdcdcdhd}((v4\_relat\_1~X6~X0){\unicdcdcdhd}((v1\_funct\_1~\\
X6){\unicdcdcdhd}((v1\_partfun1~X6~X0){\unicdcdcdhd}((v4\_valued\_0~X6){\unicdcdcdhd}(v2\_pre\_poly~X6)))))){\unicdbdjdcd}\\
({\unicdcdadad} X7 .  ((v1\_relat\_1~X7){\unicdcdcdhd}((v4\_relat\_1~X7~X0){\unicdcdcdhd}((v1\_funct\_1~\\
X7){\unicdcdcdhd}((v1\_partfun1~X7~X0){\unicdcdcdhd}((v4\_valued\_0~X7){\unicdcdcdhd}(v2\_pre\_poly~X7)))))){\unicdbdjdcd}\\
(((X6 \in X5){\unicdcdcdhd}((X7 \in k2\_polynom1~(k15\_pre\_poly~X0)~X2~X3){\unicdcdcdhd}(r1\_termord~\\
X0~X1~X6~X7))){\unicdbdjdcd}(X7 \in X5)))))))))))))\end{array}
\label{hyp:egcdpfhgchpgfgcgpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{{{k3\_xboole\_0}~{X0}}~{X1}}={{{k3\_xboole\_0}~{X1}}~{X0}}}}}\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgddpfihcgpgpgmgfgpfad}
\end{equation}
\begin{theorem}\label{thm:mgddcdpfhgchpgfgcgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v3\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_partfun1~X1~(k15\_pre\_poly~\\
X0)){\unicdcdcdhd}((v1\_relat\_2~X1){\unicdcdcdhd}((v4\_relat\_2~X1){\unicdcdcdhd}((v6\_relat\_2~X1){\unicdcdcdhd}((\\
v8\_relat\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(k15\_pre\_poly~X0))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}\\
((v13\_algstr\_0~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l2\_algstr\_0~\\
X2))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X3~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}\\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(\\
u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~k5\_numbers){\unicdbdjdcd}\\
((r1\_xxreal\_0~X4~(k5\_card\_1~(k2\_polynom1~(k15\_pre\_poly~X0)~X2~\\
X3))){\unicdbdjdcd}((k2\_polynom1~(k15\_pre\_poly~X0)~X2~(k2\_groeb\_3~X0~X2~X3~(\\
k3\_groeb\_3~X0~X1~X2~X3~X4))=k3\_groeb\_3~X0~X1~X2~X3~X4){\unicdcdcdhd}(k2\_polynom1~\\
(k15\_pre\_poly~X0)~X2~(k2\_groeb\_3~X0~X2~X3~(k4\_groeb\_3~X0~X1~X2~X3~X4))=\\
k4\_groeb\_3~X0~X1~X2~X3~X4)))))))\end{array}$$
\end{theorem}
\end{document}
