\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{t28\_bilinear (TMa8kL6GQbE7dZZyRe5a5vq8wdPQ1aQtRjw)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\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 $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_bilinear : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ v2\_bilinear : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\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 $ k1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ l2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l2\_algstr\_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 .  (m1\_subset\_1~X3~(u1\_struct\_0~\\
X1)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~(u1\_struct\_0~X2)){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
(m1\_subset\_1~X5~(u1\_struct\_0~X2)){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}\\
((v1\_funct\_2~X6~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~\\
(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~\\
X0)))))){\unicdbdjdcd}((v1\_bilinear~X6~X0~X1~X2){\unicdbdjdcd}(k2\_binop\_1~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X6~X3~(k1\_algstr\_0~X2~X4~X5)=\\
k1\_algstr\_0~X0~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(\\
u1\_struct\_0~X0)~X6~X3~X4)~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2)~(u1\_struct\_0~X0)~X6~X3~X5)))))))))\end{array}
\label{hyp:ehcdhdpfcgjgmgjgogfgbgch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l2\_algstr\_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 .  (m1\_subset\_1~X3~(u1\_struct\_0~\\
X1)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~(u1\_struct\_0~X1)){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
(m1\_subset\_1~X5~(u1\_struct\_0~X2)){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}\\
((v1\_funct\_2~X6~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~\\
(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~\\
X0)))))){\unicdbdjdcd}((v2\_bilinear~X6~X0~X1~X2){\unicdbdjdcd}(k2\_binop\_1~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X6~(k1\_algstr\_0~X1~X3~X4)~X5=\\
k1\_algstr\_0~X0~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(\\
u1\_struct\_0~X0)~X6~X3~X5)~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2)~(u1\_struct\_0~X0)~X6~X4~X5)))))))))\end{array}
\label{hyp:ehcdgdpfcgjgmgjgogfgbgch}
\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 .  (l1\_struct\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (l1\_vectsp\_1~X1~X0){\unicdbdjdcd}\\
(l2\_algstr\_0~X1))\end{array}
\label{hyp:egehpfmgbdpfghfgdgehdhahpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{l1\_struct\_0}~{X0}})}}}\label{hyp:egehpfmgbdpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((l1\_algstr\_0~X0){\unicdcdcdhd}((m1\_subset\_1~\\
X1~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~X0)))){\unicdbdjdcd}(m1\_subset\_1~\\
(k1\_algstr\_0~X0~X1~X2)~(u1\_struct\_0~X0))\end{array}
\label{hyp:egehpflgbdpfbgmghgdhehchpfad}
\end{equation}
\begin{theorem}\label{thm:ehcdidpfcgjgmgjgogfgbgch}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l2\_algstr\_0~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}(l1\_vectsp\_1~X1~X0)){\unicdbdjdcd}(\\
{\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}(l1\_vectsp\_1~X2~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~(u1\_struct\_0~X1)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~\\
(u1\_struct\_0~X1)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X2)){\unicdbdjdcd}\\
({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(u1\_struct\_0~X2)){\unicdbdjdcd}({\unicdcdadad} X7 .  ((\\
v1\_funct\_1~X7){\unicdcdcdhd}((v1\_funct\_2~X7~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~\\
(u1\_struct\_0~X2))~(u1\_struct\_0~X0)){\unicdcdcdhd}((v1\_bilinear~X7~X0~X1~X2){\unicdcdcdhd}\\
((v2\_bilinear~X7~X0~X1~X2){\unicdcdcdhd}(m1\_subset\_1~X7~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~\\
X0)))))))){\unicdbdjdcd}(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~\\
X0)~X7~(k1\_algstr\_0~X1~X3~X4)~(k1\_algstr\_0~X2~X5~X6)=k1\_algstr\_0~\\
X0~(k1\_algstr\_0~X0~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~\\
(u1\_struct\_0~X0)~X7~X3~X5)~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2)~(u1\_struct\_0~X0)~X7~X3~X6))~(k1\_algstr\_0~X0~(k2\_binop\_1~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X7~X4~X5)~(k2\_binop\_1~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X7~X4~X6))))))))))\end{array}$$
\end{theorem}
\end{document}
