\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{t17\_hilbasis (TMTZgM9nokhyCWgygo2ht32j3x4hvkbivcv)}
\maketitle
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 $ v4\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l6\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xboole\_0 : {{\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 $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k14\_polynom3 : {{\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 $ k5\_numbers : {\iota}$ be given.
Let $ v1\_algseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_hilbasis : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_algseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(X2 \in X1){\unicdbdjdcd}(X2 \in X0))\end{array}
\label{hyp:mgcdpfdhfhcgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}\\
((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~\\
X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(u1\_struct\_0~(k14\_polynom3~X0)))))){\unicdbdjdcd}((\preoh v1\_xboole\_0~\\
(k4\_hilbasis~X0~X1)){\unicdcdcdhd}(m1\_subset\_1~(k4\_hilbasis~X0~X1)~(k1\_zfmisc\_1~\\
X1)))\end{array}
\label{hyp:egehpflgedpfigjgmgcgbgdhjgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v3\_rlvect\_1~\\
X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}\\
(l6\_algstr\_0~X0))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(u1\_struct\_0~(k14\_polynom3~X0))))){\unicdbdjdcd}(k4\_hilbasis~\\
X0~X1=ReplSep~(toset~({\uniddcgcg} X2 : \iota .  m2\_subset\_1~X2~(u1\_struct\_0~\\
(k14\_polynom3~X0))~X1))~({\uniddcgcg} X2 : \iota .  {\unicdcdadad} X3 .  ((v1\_funct\_1~\\
X3){\unicdcdcdhd}((v1\_funct\_2~X3~k5\_numbers~(u1\_struct\_0~X0)){\unicdcdcdhd}((v1\_algseq\_1~\\
X3~X0){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~(\\
u1\_struct\_0~X0))))))){\unicdbdjdcd}({\unicdcdadad} X4 .  ((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~\\
X4~k5\_numbers~(u1\_struct\_0~X0)){\unicdcdcdhd}((v1\_algseq\_1~X4~X0){\unicdcdcdhd}(m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~(u1\_struct\_0~X0))))))){\unicdbdjdcd}\\
(((X3=X2){\unicdcdcdhd}(X4 \in X1)){\unicdbdjdcd}(r1\_xxreal\_0~(k1\_algseq\_1~X0~X3)~(k1\_algseq\_1~\\
X0~X4)))))~({\uniddcgcg} X2 : \iota .  X2)))\end{array}
\label{hyp:egedpfigjgmgcgbgdhjgdh}
\end{equation}
\begin{theorem}\label{thm:ehbdhdpfigjgmgcgbgdhjgdh}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v3\_rlvect\_1~\\
X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}\\
(l6\_algstr\_0~X0))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(u1\_struct\_0~(k14\_polynom3~X0))))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~k5\_numbers~(u1\_struct\_0~X0)){\unicdcdcdhd}\\
((v1\_algseq\_1~X2~X0){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~(u1\_struct\_0~X0))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~\\
X3){\unicdcdcdhd}((v1\_funct\_2~X3~k5\_numbers~(u1\_struct\_0~X0)){\unicdcdcdhd}((v1\_algseq\_1~\\
X3~X0){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~(\\
u1\_struct\_0~X0))))))){\unicdbdjdcd}(((X2 \in k4\_hilbasis~X0~X1){\unicdcdcdhd}(X3 \in X1)){\unicdbdjdcd}((\\
X2 \in X1){\unicdcdcdhd}(r1\_xxreal\_0~(k1\_algseq\_1~X0~X2)~(k1\_algseq\_1~X0~X3)))))))\end{array}$$
\end{theorem}
\end{document}
