\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{t10\_ranknull (TMYgtxvT54Sn5YHCNSrJry2937dJbg8pVX2)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v33\_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 $ v3\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_group\_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 $ v8\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v9\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v10\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v11\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{{\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 $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v13\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ v1\_mod\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\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 $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_struct\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_ranknull : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k3\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_vectsp\_4 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ l2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l5\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}\\
((v3\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdcdcdhd}\\
((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v8\_vectsp\_1~X1~X0){\unicdcdcdhd}\\
((v9\_vectsp\_1~X1~X0){\unicdcdcdhd}((v10\_vectsp\_1~X1~X0){\unicdcdcdhd}((v11\_vectsp\_1~X1~\\
X0){\unicdcdcdhd}((v2\_rlvect\_1~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~X1){\unicdcdcdhd}\\
(l1\_vectsp\_1~X1~X0))))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_vectsp\_4~X2~X0~\\
X1){\unicdbdjdcd}((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~X2){\unicdcdcdhd}((v8\_vectsp\_1~X2~\\
X0){\unicdcdcdhd}((v9\_vectsp\_1~X2~X0){\unicdcdcdhd}((v10\_vectsp\_1~X2~X0){\unicdcdcdhd}((v11\_vectsp\_1~\\
X2~X0){\unicdcdcdhd}((v2\_rlvect\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}\\
(l1\_vectsp\_1~X2~X0)))))))))))\end{array}
\label{hyp:egehpfngbdpfghfgdgehdhahpfed}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l6\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{({{l2\_algstr\_0}~{X0}})}{\unicdcdcdhd}{({{l5\_algstr\_0}~{X0}})}})}}}\label{hyp:egehpfmggdpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l2\_struct\_0}~{X0}})}{\unicdbdjdcd}{({{l1\_struct\_0}~{X0}})}}}\label{hyp:egehpfmgcdpfdhehchfhdgehpfad}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (((\preoh v2\_struct\_0~\\
X0){\unicdcdcdhd}((\preoh v6\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v33\_algstr\_0~X0){\unicdcdcdhd}\\
((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v3\_group\_1~\\
X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(\\
l6\_algstr\_0~X0)))))))))))){\unicdcdcdhd}(((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~\\
X1){\unicdcdcdhd}((v2\_rlvect\_1~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~X1){\unicdcdcdhd}\\
((v8\_vectsp\_1~X1~X0){\unicdcdcdhd}((v9\_vectsp\_1~X1~X0){\unicdcdcdhd}((v10\_vectsp\_1~X1~X0){\unicdcdcdhd}\\
((v11\_vectsp\_1~X1~X0){\unicdcdcdhd}(l1\_vectsp\_1~X1~X0)))))))))){\unicdcdcdhd}(((\preoh v2\_struct\_0~\\
X2){\unicdcdcdhd}((v13\_algstr\_0~X2){\unicdcdcdhd}((v2\_rlvect\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}\\
((v4\_rlvect\_1~X2){\unicdcdcdhd}((v8\_vectsp\_1~X2~X0){\unicdcdcdhd}((v9\_vectsp\_1~X2~X0){\unicdcdcdhd}\\
((v10\_vectsp\_1~X2~X0){\unicdcdcdhd}((v11\_vectsp\_1~X2~X0){\unicdcdcdhd}(l1\_vectsp\_1~X2~X0)))))))))){\unicdcdcdhd}\\
((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2)){\unicdcdcdhd}((v13\_vectsp\_1~X3~X1~X2){\unicdcdcdhd}((v1\_mod\_2~X3~X0~X1~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2))))))))))){\unicdbdjdcd}\\
((v7\_vectsp\_1~(k1\_ranknull~X0~X1~X2~X3)~X0){\unicdcdcdhd}(m1\_vectsp\_4~(k1\_ranknull~\\
X0~X1~X2~X3)~X0~X1))\end{array}
\label{hyp:egehpflgbdpfchbgoglgogfhmgmg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_struct\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (r1\_struct\_0~X0~X1){\unicdbdjded}\\
(X1 \in u1\_struct\_0~X0))\end{array}
\label{hyp:egfdpfdhehchfhdgehpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_struct\_0}~{X0}})}{\unicdbdjdcd}{({{{k2\_struct\_0}~{X0}}={{u1\_struct\_0}~{X0}}})}}}\label{hyp:egddpfdhehchfhdgehpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v6\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~\\
X0){\unicdcdcdhd}((v33\_algstr\_0~X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}\\
((v4\_rlvect\_1~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v2\_rlvect\_1~X1){\unicdcdcdhd}((\\
v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~X1){\unicdcdcdhd}((v8\_vectsp\_1~X1~X0){\unicdcdcdhd}((v9\_vectsp\_1~\\
X1~X0){\unicdcdcdhd}((v10\_vectsp\_1~X1~X0){\unicdcdcdhd}((v11\_vectsp\_1~X1~X0){\unicdcdcdhd}(l1\_vectsp\_1~\\
X1~X0)))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~\\
X2){\unicdcdcdhd}((v2\_rlvect\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}\\
((v8\_vectsp\_1~X2~X0){\unicdcdcdhd}((v9\_vectsp\_1~X2~X0){\unicdcdcdhd}((v10\_vectsp\_1~X2~X0){\unicdcdcdhd}\\
((v11\_vectsp\_1~X2~X0){\unicdcdcdhd}(l1\_vectsp\_1~X2~X0)))))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2)){\unicdcdcdhd}((v13\_vectsp\_1~X3~X1~X2){\unicdcdcdhd}((v1\_mod\_2~X3~X0~X1~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)))))))){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  ((v7\_vectsp\_1~X4~X0){\unicdcdcdhd}(m1\_vectsp\_4~X4~X0~X1)){\unicdbdjdcd}((X4=\\
k1\_ranknull~X0~X1~X2~X3){\unicdbdjded}(k2\_struct\_0~X4=ReplSep~(toset~({\uniddcgcg} X5 : \\
\iota .  m1\_subset\_1~X5~(u1\_struct\_0~X1)))~({\uniddcgcg} X5 : \iota .  k3\_funct\_2~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~X3~X5=k4\_struct\_0~X2)~({\uniddcgcg} X5 : \\
\iota .  X5)))))))\end{array}
\label{hyp:egbdpfchbgoglgogfhmgmg}
\end{equation}
\begin{theorem}\label{thm:ehbdadpfchbgoglgogfhmgmg}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v6\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~\\
X0){\unicdcdcdhd}((v33\_algstr\_0~X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}\\
((v4\_rlvect\_1~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v2\_rlvect\_1~X1){\unicdcdcdhd}((\\
v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~X1){\unicdcdcdhd}((v8\_vectsp\_1~X1~X0){\unicdcdcdhd}((v9\_vectsp\_1~\\
X1~X0){\unicdcdcdhd}((v10\_vectsp\_1~X1~X0){\unicdcdcdhd}((v11\_vectsp\_1~X1~X0){\unicdcdcdhd}(l1\_vectsp\_1~\\
X1~X0)))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~\\
X2){\unicdcdcdhd}((v2\_rlvect\_1~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}\\
((v8\_vectsp\_1~X2~X0){\unicdcdcdhd}((v9\_vectsp\_1~X2~X0){\unicdcdcdhd}((v10\_vectsp\_1~X2~X0){\unicdcdcdhd}\\
((v11\_vectsp\_1~X2~X0){\unicdcdcdhd}(l1\_vectsp\_1~X2~X0)))))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2)){\unicdcdcdhd}((v13\_vectsp\_1~X3~X1~X2){\unicdcdcdhd}((v1\_mod\_2~X3~X0~X1~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)))))))){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  (m1\_subset\_1~X4~(u1\_struct\_0~X1)){\unicdbdjdcd}((r1\_struct\_0~\\
(k1\_ranknull~X0~X1~X2~X3)~X4){\unicdbdjded}(k3\_funct\_2~(u1\_struct\_0~X1)~(u1\_struct\_0~\\
X2)~X3~X4=k4\_struct\_0~X2))))))\end{array}$$
\end{theorem}
\end{document}
