\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{l69\_anproj\_2 (TMEhTsf1oQ5z4P51ZYthL7pFCvRTkAHb528)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\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 $ v5\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v8\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_rlvect\_1 : {{\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 $ k5\_anproj\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_collsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ r1\_anproj\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r2\_anproj\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ v9\_struct\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_anproj\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v5\_rlvect\_1~X0){\unicdcdcdhd}\\
((v6\_rlvect\_1~X0){\unicdcdcdhd}((v7\_rlvect\_1~X0){\unicdcdcdhd}((v8\_rlvect\_1~X0){\unicdcdcdhd}(l1\_rlvect\_1~\\
X0)))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X6 .  \\
(m1\_subset\_1~X6~(u1\_struct\_0~X0)){\unicdbdjdcd}(((r1\_anproj\_1~X0~X1~X2){\unicdcdcdhd}((\\
r1\_anproj\_1~X0~X3~X4){\unicdcdcdhd}((r1\_anproj\_1~X0~X5~X6){\unicdcdcdhd}(r2\_anproj\_1~X0~X1~\\
X3~X5)))){\unicdbdjdcd}(r2\_anproj\_1~X0~X2~X4~X6))))))))\end{array}
\label{hyp:ehedpfbgogahchpgkgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((\preoh v7\_struct\_0~X1){\unicdcdcdhd}\\
((v13\_algstr\_0~X1){\unicdcdcdhd}((v2\_rlvect\_1~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~\\
X1){\unicdcdcdhd}((v5\_rlvect\_1~X1){\unicdcdcdhd}((v6\_rlvect\_1~X1){\unicdcdcdhd}((v7\_rlvect\_1~X1){\unicdcdcdhd}\\
((v8\_rlvect\_1~X1){\unicdcdcdhd}(l1\_rlvect\_1~X1))))))))))){\unicdbdjdcd}((m1\_subset\_1~\\
X0~(u1\_struct\_0~(k5\_anproj\_1~X1))){\unicdbdjded}({\unicdcdaddd} X2 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~X1)){\unicdcdcdhd}((\preoh v9\_struct\_0~X2~X1){\unicdcdcdhd}(X0=k2\_anproj\_1~X1~\\
X2))))\end{array}
\label{hyp:ehcdgdpfbgogahchpgkgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v7\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~\\
X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}\\
((v5\_rlvect\_1~X0){\unicdcdcdhd}((v6\_rlvect\_1~X0){\unicdcdcdhd}((v7\_rlvect\_1~X0){\unicdcdcdhd}((v8\_rlvect\_1~\\
X0){\unicdcdcdhd}(l1\_rlvect\_1~X0))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}((r1\_collsp~(k5\_anproj\_1~X0)~\\
X1~X2~X3){\unicdbdjded}({\unicdcdaddd} X4 .  (m1\_subset\_1~X4~(u1\_struct\_0~X0)){\unicdcdcdhd}({\unicdcdaddd} X5 .  \\
(m1\_subset\_1~X5~(u1\_struct\_0~X0)){\unicdcdcdhd}({\unicdcdaddd} X6 .  (m1\_subset\_1~X6~\\
(u1\_struct\_0~X0)){\unicdcdcdhd}((X1=k2\_anproj\_1~X0~X4){\unicdcdcdhd}((X2=k2\_anproj\_1~\\
X0~X5){\unicdcdcdhd}((X3=k2\_anproj\_1~X0~X6){\unicdcdcdhd}((\preoh v9\_struct\_0~X4~X0){\unicdcdcdhd}((\preoh v9\_struct\_0~\\
X5~X0){\unicdcdcdhd}((\preoh v9\_struct\_0~X6~X0){\unicdcdcdhd}(r2\_anproj\_1~X0~X4~X5~X6))))))))))))))\end{array}
\label{hyp:ehcdddpfbgogahchpgkgpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v7\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~\\
X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}\\
((v5\_rlvect\_1~X0){\unicdcdcdhd}((v6\_rlvect\_1~X0){\unicdcdcdhd}((v7\_rlvect\_1~X0){\unicdcdcdhd}((v8\_rlvect\_1~\\
X0){\unicdcdcdhd}(l1\_rlvect\_1~X0))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~\\
(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}\\
(\preoh (\preoh v9\_struct\_0~X1~X0){\unicdcdcdhd}((\preoh v9\_struct\_0~X2~X0){\unicdcdcdhd}(\preoh (k2\_anproj\_1~\\
X0~X1=k2\_anproj\_1~X0~X2){\unicdbdjded}(r1\_anproj\_1~X0~X1~X2))))))\end{array}
\label{hyp:ehcdcdpfbgogahchpgkgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v5\_rlvect\_1~X0){\unicdcdcdhd}\\
((v6\_rlvect\_1~X0){\unicdcdcdhd}((v7\_rlvect\_1~X0){\unicdcdcdhd}((v8\_rlvect\_1~X0){\unicdcdcdhd}(l1\_rlvect\_1~\\
X0)))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~(u1\_struct\_0~X0)){\unicdbdjdcd}(\preoh (\preoh r2\_anproj\_1~\\
X0~X1~X2~X3){\unicdcdcdhd}((r2\_anproj\_1~X0~X1~X2~X4){\unicdcdcdhd}((r2\_anproj\_1~X0~X2~X3~X5){\unicdcdcdhd}\\
({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(u1\_struct\_0~X0)){\unicdbdjdcd}(\preoh (r2\_anproj\_1~\\
X0~X1~X3~X6){\unicdcdcdhd}((r2\_anproj\_1~X0~X4~X5~X6){\unicdcdcdhd}(\preoh v9\_struct\_0~X6~X0))))))))))))\end{array}
\label{hyp:ehbdfdpfbgogahchpgkgpfbd}
\end{equation}
\begin{theorem}\label{thm:mggdjdpfbgogahchpgkgpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v7\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~\\
X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}\\
((v5\_rlvect\_1~X0){\unicdcdcdhd}((v6\_rlvect\_1~X0){\unicdcdcdhd}((v7\_rlvect\_1~X0){\unicdcdcdhd}((v8\_rlvect\_1~\\
X0){\unicdcdcdhd}(l1\_rlvect\_1~X0))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~\\
(u1\_struct\_0~(k5\_anproj\_1~X0))){\unicdbdjdcd}(\preoh (\preoh r1\_collsp~(k5\_anproj\_1~\\
X0)~X1~X2~X3){\unicdcdcdhd}((r1\_collsp~(k5\_anproj\_1~X0)~X1~X2~X4){\unicdcdcdhd}((r1\_collsp~\\
(k5\_anproj\_1~X0)~X2~X3~X5){\unicdcdcdhd}({\unicdcdadad} X6 .  (m1\_subset\_1~X6~(u1\_struct\_0~\\
(k5\_anproj\_1~X0))){\unicdbdjdcd}(\preoh (r1\_collsp~(k5\_anproj\_1~X0)~X1~X3~X6){\unicdcdcdhd}(r1\_collsp~\\
(k5\_anproj\_1~X0)~X4~X5~X6)))))))))))\end{array}$$
\end{theorem}
\end{document}
