\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{t13\_bilinear (TMWLuLAC5scJf3XsHsLuA5gtBLKqaHRNu8k)}
\maketitle
Let $ v2\_struct\_0 : {{\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 $ 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 $ r2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k7\_bilinear : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k2\_bilinear : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k3\_hahnban1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_funct\_2 : {{\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 $ l2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l1\_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}({\unicdcdadad} X4 .  \\
(m1\_subset\_1~X4~(u1\_struct\_0~X1)){\unicdbdjdcd}((k1\_relset\_1~(u1\_struct\_0~\\
X2)~(k7\_bilinear~X0~X1~X2~X3~X4)=u1\_struct\_0~X2){\unicdcdcdhd}((r1\_tarski~(k2\_relset\_1~\\
(u1\_struct\_0~X0)~(k7\_bilinear~X0~X1~X2~X3~X4))~(u1\_struct\_0~X0)){\unicdcdcdhd}\\
({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X2)){\unicdbdjdcd}(k3\_funct\_2~(u1\_struct\_0~\\
X2)~(u1\_struct\_0~X0)~(k7\_bilinear~X0~X1~X2~X3~X4)~X5=k2\_binop\_1~(\\
u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X3~X4~X5))))))))\end{array}
\label{hyp:ehidpfcgjgmgjgogfgbgch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (((v1\_funct\_1~X2){\unicdcdcdhd}\\
((v1\_funct\_2~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X1))))){\unicdcdcdhd}((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~X0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1)))))){\unicdbdjdcd}((r2\_funct\_2~X0~X1~X2~\\
X3){\unicdbdjded}(X2=X3))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchcdpfggfhogdgehpfcd}
\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}
{{\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 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  (((\preoh v2\_struct\_0~\\
X0){\unicdcdcdhd}(l1\_struct\_0~X0)){\unicdcdcdhd}(((\preoh v2\_struct\_0~X1){\unicdcdcdhd}(l1\_vectsp\_1~X1~X0)){\unicdcdcdhd}\\
(((\preoh v2\_struct\_0~X2){\unicdcdcdhd}(l1\_vectsp\_1~X2~X0)){\unicdcdcdhd}(((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)))))){\unicdcdcdhd}(m1\_subset\_1~X4~(u1\_struct\_0~X1)))))){\unicdbdjdcd}((v1\_funct\_1~\\
(k7\_bilinear~X0~X1~X2~X3~X4)){\unicdcdcdhd}((v1\_funct\_2~(k7\_bilinear~X0~X1~X2~\\
X3~X4)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~(k7\_bilinear~\\
X0~X1~X2~X3~X4)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X2)~(u1\_struct\_0~\\
X0))))))\end{array}
\label{hyp:egehpflghdpfcgjgmgjgogfgbgch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  (((\preoh v2\_struct\_0~\\
X0){\unicdcdcdhd}(l2\_algstr\_0~X0)){\unicdcdcdhd}(((\preoh v2\_struct\_0~X1){\unicdcdcdhd}(l1\_vectsp\_1~X1~X0)){\unicdcdcdhd}\\
(((\preoh v2\_struct\_0~X2){\unicdcdcdhd}(l1\_vectsp\_1~X2~X0)){\unicdcdcdhd}(((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)))))){\unicdcdcdhd}((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~X4~(k2\_zfmisc\_1~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~\\
(u1\_struct\_0~X0)))))))))){\unicdbdjdcd}((v1\_funct\_1~(k2\_bilinear~X0~X1~X2~\\
X3~X4)){\unicdcdcdhd}((v1\_funct\_2~(k2\_bilinear~X0~X1~X2~X3~X4)~(k2\_zfmisc\_1~(\\
u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~\\
(k2\_bilinear~X0~X1~X2~X3~X4)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~X0))))))\end{array}
\label{hyp:egehpflgcdpfcgjgmgjgogfgbgch}
\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 .  ((v1\_funct\_1~\\
X2){\unicdcdcdhd}((v1\_funct\_2~X2~(u1\_struct\_0~X1)~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X0)))))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(u1\_struct\_0~X1)~\\
(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X0)))))){\unicdbdjdcd}({\unicdcdadad} X4 .  ((v1\_funct\_1~\\
X4){\unicdcdcdhd}((v1\_funct\_2~X4~(u1\_struct\_0~X1)~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X0)))))){\unicdbdjdcd}\\
((X4=k3\_hahnban1~X0~X1~X2~X3){\unicdbdjded}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~\\
X1)){\unicdbdjdcd}(k3\_funct\_2~(u1\_struct\_0~X1)~(u1\_struct\_0~X0)~X4~X5=k1\_algstr\_0~\\
X0~(k3\_funct\_2~(u1\_struct\_0~X1)~(u1\_struct\_0~X0)~X2~X5)~(k3\_funct\_2~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X0)~X3~X5))))))))\end{array}
\label{hyp:egddpfigbgigogcgbgogbd}
\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 .  ((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}({\unicdcdadad} X4 .  \\
((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~X4~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~\\
(u1\_struct\_0~X2))~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~\\
(u1\_struct\_0~X0)))))){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v1\_funct\_1~X5){\unicdcdcdhd}((v1\_funct\_2~\\
X5~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~\\
X0)){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~X0)))))){\unicdbdjdcd}((X5=\\
k2\_bilinear~X0~X1~X2~X3~X4){\unicdbdjded}({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(u1\_struct\_0~\\
X1)){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~X7~(u1\_struct\_0~X2)){\unicdbdjdcd}(k2\_binop\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~X5~X6~X7=k1\_algstr\_0~\\
X0~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~\\
X0)~X3~X6~X7)~(k2\_binop\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~(u1\_struct\_0~\\
X0)~X4~X6~X7))))))))))\end{array}
\label{hyp:egcdpfcgjgmgjgogfgbgch}
\end{equation}
\begin{theorem}\label{thm:ehbdddpfcgjgmgjgogfgbgch}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l2\_algstr\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((\preoh v2\_struct\_0~X1){\unicdcdcdhd}(l1\_vectsp\_1~X1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~\\
X2){\unicdcdcdhd}(l1\_vectsp\_1~X2~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~\\
X3~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~\\
X0)){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~(u1\_struct\_0~X0)))))){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
((v1\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~X4~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~\\
(u1\_struct\_0~X2))~(u1\_struct\_0~X0)){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))~\\
(u1\_struct\_0~X0)))))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~\\
X1)){\unicdbdjdcd}(r2\_funct\_2~(u1\_struct\_0~X2)~(u1\_struct\_0~X0)~(k7\_bilinear~\\
X0~X1~X2~(k2\_bilinear~X0~X1~X2~X3~X4)~X5)~(k3\_hahnban1~X0~X2~(k7\_bilinear~\\
X0~X1~X2~X3~X5)~(k7\_bilinear~X0~X1~X2~X4~X5))))))))\end{array}$$
\end{theorem}
\end{document}
