\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{t32\_groeb\_2 (TMVbAXQM21tqGe78SvZi7164y9MWf1wgVP1)}
\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 $ v7\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l6\_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 $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k11\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m2\_ideal\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\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 $ r3\_groeb\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}}}$ be given.
Let $ r2\_groeb\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}$ be given.
Let $ k4\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k4\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_polynom7 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ v3\_polynom7 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k7\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k9\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ r1\_termord : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k3\_termord : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
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}\\
((\preoh v7\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~X2){\unicdcdcdhd}((v4\_vectsp\_1~X2){\unicdcdcdhd}((\\
v5\_vectsp\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l6\_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 .  ((\preoh v1\_xboole\_0~X4){\unicdcdcdhd}(m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(u1\_struct\_0~(k11\_polynom1~X0~X2))))){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
(m2\_ideal\_1~X5~(k11\_polynom1~X0~X2)~X4){\unicdbdjdcd}({\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}((r3\_groeb\_2~X0~X1~X2~\\
X3~X4~X5~X6){\unicdbdjded}((k4\_rlvect\_1~(k11\_polynom1~X0~X2)~X5=X3){\unicdcdcdhd}({\unicdcdadad} X7 .  \\
(m1\_subset\_1~X7~k5\_numbers){\unicdbdjdcd}(\preoh (X7 \in k4\_finseq\_1~X5){\unicdcdcdhd}({\unicdcdadad} X8 .  \\
((v1\_funct\_1~X8){\unicdcdcdhd}((v1\_funct\_2~X8~(k15\_pre\_poly~X0)~(u1\_struct\_0~\\
X2)){\unicdcdcdhd}((v1\_polynom7~X8~X0~X2){\unicdcdcdhd}((v3\_polynom7~X8~X0~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X8~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)))))))){\unicdbdjdcd}\\
({\unicdcdadad} X9 .  ((v1\_funct\_1~X9){\unicdcdcdhd}((v1\_funct\_2~X9~(k15\_pre\_poly~X0)~\\
(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom7~X9~X0~X2){\unicdcdcdhd}((v1\_polynom1~X9~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X9~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2)))))))){\unicdbdjdcd}(\preoh (X9 \in X4){\unicdcdcdhd}((k7\_partfun1~(u1\_struct\_0~\\
(k11\_polynom1~X0~X2))~X5~X7=k9\_polynom1~X0~X2~X8~X9){\unicdcdcdhd}(r1\_termord~\\
X0~X1~(k3\_termord~X0~X1~X2~(k9\_polynom1~X0~X2~X8~X9))~X6)))))))))))))))\end{array}
\label{hyp:eghdpfhgchpgfgcgpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v3\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v7\_struct\_0~X1){\unicdcdcdhd}\\
((v13\_algstr\_0~X1){\unicdcdcdhd}((v4\_vectsp\_1~X1){\unicdcdcdhd}((v5\_vectsp\_1~X1){\unicdcdcdhd}((v3\_rlvect\_1~\\
X1){\unicdcdcdhd}((v4\_rlvect\_1~X1){\unicdcdcdhd}(l6\_algstr\_0~X1))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  (\\
(v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~(k15\_pre\_poly~X0)~(u1\_struct\_0~\\
X1)){\unicdcdcdhd}((v1\_polynom1~X2~(k15\_pre\_poly~X0)~X1){\unicdcdcdhd}(m1\_subset\_1~X2~(\\
k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(u1\_struct\_0~X1))))))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  ((\preoh v1\_xboole\_0~X3){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~\\
(u1\_struct\_0~(k11\_polynom1~X0~X1))))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_ideal\_1~\\
X4~(k11\_polynom1~X0~X1)~X3){\unicdbdjdcd}((r2\_groeb\_2~X0~X1~X2~X3~X4){\unicdbdjded}((k4\_rlvect\_1~\\
(k11\_polynom1~X0~X1)~X4=X2){\unicdcdcdhd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~k5\_numbers){\unicdbdjdcd}\\
(\preoh (X5 \in k4\_finseq\_1~X4){\unicdcdcdhd}({\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}((v1\_funct\_2~\\
X6~(k15\_pre\_poly~X0)~(u1\_struct\_0~X1)){\unicdcdcdhd}((v3\_polynom7~X6~X0~X1){\unicdcdcdhd}\\
(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(\\
u1\_struct\_0~X1))))))){\unicdbdjdcd}({\unicdcdadad} X7 .  ((v1\_funct\_1~X7){\unicdcdcdhd}((v1\_funct\_2~\\
X7~(k15\_pre\_poly~X0)~(u1\_struct\_0~X1)){\unicdcdcdhd}((v1\_polynom1~X7~(k15\_pre\_poly~\\
X0)~X1){\unicdcdcdhd}(m1\_subset\_1~X7~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X1))))))){\unicdbdjdcd}(\preoh (X7 \in X3){\unicdcdcdhd}(k7\_partfun1~(u1\_struct\_0~\\
(k11\_polynom1~X0~X1))~X4~X5=k9\_polynom1~X0~X1~X6~X7))))))))))))\end{array}
\label{hyp:eggdpfhgchpgfgcgpfcd}
\end{equation}
\begin{theorem}\label{thm:ehddcdpfhgchpgfgcgpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v3\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_partfun1~X1~(k15\_pre\_poly~\\
X0)){\unicdcdcdhd}((v1\_relat\_2~X1){\unicdcdcdhd}((v4\_relat\_2~X1){\unicdcdcdhd}((v6\_relat\_2~X1){\unicdcdcdhd}((\\
v8\_relat\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(k15\_pre\_poly~X0))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}\\
((\preoh v7\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~X2){\unicdcdcdhd}((v4\_vectsp\_1~X2){\unicdcdcdhd}((\\
v5\_vectsp\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l6\_algstr\_0~\\
X2)))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X3~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}\\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(\\
u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X4 .  ((\preoh v1\_xboole\_0~X4){\unicdcdcdhd}(m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(u1\_struct\_0~(k11\_polynom1~X0~X2))))){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
(m2\_ideal\_1~X5~(k11\_polynom1~X0~X2)~X4){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_relat\_1~\\
X6){\unicdcdcdhd}((v4\_relat\_1~X6~X0){\unicdcdcdhd}((v1\_funct\_1~X6){\unicdcdcdhd}((v1\_partfun1~X6~X0){\unicdcdcdhd}\\
((v4\_valued\_0~X6){\unicdcdcdhd}(v2\_pre\_poly~X6)))))){\unicdbdjdcd}((r3\_groeb\_2~X0~X1~X2~\\
X3~X4~X5~X6){\unicdbdjdcd}(r2\_groeb\_2~X0~X2~X3~X4~X5))))))))\end{array}$$
\end{theorem}
\end{document}
