\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{t41\_bilinear (TMKJLuSk9CUmHPVkWeRffkASZ8SZLjDtwSF)}
\maketitle
Let $ l2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_vectsp\_1 : {{\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 $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k10\_bilinear : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k12\_bilinear : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k11\_bilinear : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k2\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ k4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (r1\_tarski~X0~X1){\unicdbdjded}({\unicdcdadad} X2 .  (X2 \in X0){\unicdbdjdcd}\\
(X2 \in X1))\end{array}
\label{hyp:egddpfehbgchdhlgjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l2\_struct\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}\\
(l1\_vectsp\_1~X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X1))~(u1\_struct\_0~\\
X0)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X1))~(u1\_struct\_0~X0)))))){\unicdbdjdcd}(k12\_bilinear~\\
X0~X1~X2=ReplSep~(toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~(u1\_struct\_0~\\
X1)))~({\uniddcgcg} X3 : \iota .  k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X1)~\\
(u1\_struct\_0~X0)~X2~X3~X3=k4\_struct\_0~X0)~({\uniddcgcg} X3 : \iota .  X3))))\end{array}
\label{hyp:egbdhdpfcgjgmgjgogfgbgch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l2\_struct\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}\\
(l1\_vectsp\_1~X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}(l1\_vectsp\_1~\\
X2~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2))~(u1\_struct\_0~X0)))))){\unicdbdjdcd}(k11\_bilinear~X0~X1~X2~X3=ReplSep~(\\
toset~({\uniddcgcg} X4 : \iota .  m1\_subset\_1~X4~(u1\_struct\_0~X2)))~({\uniddcgcg} X4 : \\
\iota .  {\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X1)){\unicdbdjdcd}(k2\_binop\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X3~X5~X4=k4\_struct\_0~\\
X0))~({\uniddcgcg} X4 : \iota .  X4)))))\end{array}
\label{hyp:egbdgdpfcgjgmgjgogfgbgch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l2\_struct\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}\\
(l1\_vectsp\_1~X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}(l1\_vectsp\_1~\\
X2~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2))~(u1\_struct\_0~X0)))))){\unicdbdjdcd}(k10\_bilinear~X0~X1~X2~X3=ReplSep~(\\
toset~({\uniddcgcg} X4 : \iota .  m1\_subset\_1~X4~(u1\_struct\_0~X1)))~({\uniddcgcg} X4 : \\
\iota .  {\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X2)){\unicdbdjdcd}(k2\_binop\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X3~X4~X5=k4\_struct\_0~\\
X0))~({\uniddcgcg} X4 : \iota .  X4)))))\end{array}
\label{hyp:egbdfdpfcgjgmgjgogfgbgch}
\end{equation}
\begin{theorem}\label{thm:ehedbdpfcgjgmgjgogfgbgch}
$$\begin{array}{c}{\unicdcdadad} X0 .  (l2\_struct\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}\\
(l1\_vectsp\_1~X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X1))~(u1\_struct\_0~\\
X0)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X1))~(u1\_struct\_0~X0)))))){\unicdbdjdcd}((r1\_tarski~\\
(k10\_bilinear~X0~X1~X1~X2)~(k12\_bilinear~X0~X1~X2)){\unicdcdcdhd}(r1\_tarski~(\\
k11\_bilinear~X0~X1~X1~X2)~(k12\_bilinear~X0~X1~X2)))))\end{array}$$
\end{theorem}
\end{document}
