\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{t57\_groeb\_3 (TMH1Hq1AAeaMJ47wF5G7EnRX3CsP1NW2SSC)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ 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 $ v2\_bagorder : {{\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 $ v6\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v33\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_rlvect\_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 $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k11\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_groeb\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\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 $ v1\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r1\_groeb\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k3\_termord : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ r1\_rewrite1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k3\_polyred : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k3\_groeb\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k7\_polynom1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r4\_rewrite1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r8\_pboole : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\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}((v2\_bagorder~X1~X0){\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 v6\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~\\
X2){\unicdcdcdhd}((v33\_algstr\_0~X2){\unicdcdcdhd}((v3\_group\_1~X2){\unicdcdcdhd}((v5\_group\_1~X2){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X2){\unicdcdcdhd}((v5\_vectsp\_1~X2){\unicdcdcdhd}((v2\_rlvect\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~\\
X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l6\_algstr\_0~X2)))))))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(u1\_struct\_0~(k11\_polynom1~X0~X2)))){\unicdbdjdcd}\\
(({\unicdcdadad} X4 .  ((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~X4~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X4~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}\\
(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(\\
u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v1\_funct\_1~X5){\unicdcdcdhd}((v1\_funct\_2~\\
X5~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X5~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}((v1\_funct\_2~\\
X6~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X6~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}(((X4 \in X3){\unicdcdcdhd}((X5 \in X3){\unicdcdcdhd}(r4\_rewrite1~\\
(k3\_polyred~X0~X1~X2~X3)~(k3\_groeb\_2~X0~X1~X2~X4~X5)~X6))){\unicdbdjdcd}(r8\_pboole~\\
(k15\_pre\_poly~X0)~X6~(k7\_polynom1~X0~X2)))))){\unicdbdjdcd}({\unicdcdadad} X4 .  ((v1\_funct\_1~\\
X4){\unicdcdcdhd}((v1\_funct\_2~X4~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~\\
X4~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k15\_pre\_poly~X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v1\_funct\_1~\\
X5){\unicdcdcdhd}((v1\_funct\_2~X5~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~\\
X5~(k15\_pre\_poly~X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k15\_pre\_poly~X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}(((X4 \in X3){\unicdcdcdhd}(X5 \in X3)){\unicdbdjdcd}\\
(r1\_rewrite1~(k3\_polyred~X0~X1~X2~X3)~(k3\_groeb\_2~X0~X1~X2~X4~X5)~(\\
k7\_polynom1~X0~X2)))))))))\end{array}
\label{hyp:ehcdedpfhgchpgfgcgpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\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}((v2\_bagorder~X1~X0){\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 v6\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~\\
X2){\unicdcdcdhd}((v33\_algstr\_0~X2){\unicdcdcdhd}((v3\_group\_1~X2){\unicdcdcdhd}((v5\_group\_1~X2){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X2){\unicdcdcdhd}((v5\_vectsp\_1~X2){\unicdcdcdhd}((v2\_rlvect\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~\\
X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l6\_algstr\_0~X2)))))))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(u1\_struct\_0~(k11\_polynom1~X0~X2)))){\unicdbdjdcd}\\
((r1\_groeb\_1~X0~X1~X2~X3){\unicdbdjdcd}({\unicdcdadad} X4 .  ((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~\\
X4~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X4~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v1\_funct\_1~X5){\unicdcdcdhd}((v1\_funct\_2~\\
X5~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X5~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}((v1\_funct\_2~\\
X6~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X6~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}(((X4 \in X3){\unicdcdcdhd}((X5 \in X3){\unicdcdcdhd}(r4\_rewrite1~\\
(k3\_polyred~X0~X1~X2~X3)~(k3\_groeb\_2~X0~X1~X2~X4~X5)~X6))){\unicdbdjdcd}(r8\_pboole~\\
(k15\_pre\_poly~X0)~X6~(k7\_polynom1~X0~X2))))))))))\end{array}
\label{hyp:ehcdddpfhgchpgfgcgpfcd}
\end{equation}
\begin{theorem}\label{thm:ehfdhdpfhgchpgfgcgpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_partfun1~\\
X1~(k15\_pre\_poly~X0)){\unicdcdcdhd}((v1\_relat\_2~X1){\unicdcdcdhd}((v4\_relat\_2~X1){\unicdcdcdhd}((v6\_relat\_2~\\
X1){\unicdcdcdhd}((v8\_relat\_2~X1){\unicdcdcdhd}((v2\_bagorder~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k15\_pre\_poly~X0)~(k15\_pre\_poly~X0)))))))))){\unicdbdjdcd}(\\
{\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((\preoh v6\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~\\
X2){\unicdcdcdhd}((v33\_algstr\_0~X2){\unicdcdcdhd}((v3\_group\_1~X2){\unicdcdcdhd}((v5\_group\_1~X2){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X2){\unicdcdcdhd}((v5\_vectsp\_1~X2){\unicdcdcdhd}((v2\_rlvect\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~\\
X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l6\_algstr\_0~X2)))))))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(u1\_struct\_0~(k11\_polynom1~X0~X2)))){\unicdbdjdcd}\\
((r1\_groeb\_1~X0~X1~X2~X3){\unicdbdjdcd}({\unicdcdadad} X4 .  ((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~\\
X4~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X4~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v1\_funct\_1~X5){\unicdcdcdhd}((v1\_funct\_2~\\
X5~(k15\_pre\_poly~X0)~(u1\_struct\_0~X2)){\unicdcdcdhd}((v1\_polynom1~X5~(k15\_pre\_poly~\\
X0)~X2){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k15\_pre\_poly~\\
X0)~(u1\_struct\_0~X2))))))){\unicdbdjdcd}(((X4 \in X3){\unicdcdcdhd}(X5 \in X3)){\unicdbdjdcd}((r1\_groeb\_2~\\
X0~(k3\_termord~X0~X1~X2~X4)~(k3\_termord~X0~X1~X2~X5)){\unicdcdcdid}(r1\_rewrite1~\\
(k3\_polyred~X0~X1~X2~X3)~(k3\_groeb\_2~X0~X1~X2~X4~X5)~(k7\_polynom1~X0~\\
X2))))))))))\end{array}$$
\end{theorem}
\end{document}
