\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{t23\_matrix\_4 (TMLGVGxPAzm5Ybw5S1ePkjc21hPGQQyNKy7)}
\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 $ v3\_group\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_group\_1 : {{\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 $ 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 $ v1\_matrix\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k3\_finseq\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_matrix\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_matrix\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_matrix\_4 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ m1\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_matrix\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v7\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k4\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_matrix\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_matrix\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k3\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k4\_algstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
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}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m2\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_matrix\_1~X2){\unicdcdcdhd}(m2\_finseq\_1~X2~(k3\_finseq\_2~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_matrix\_1~X3){\unicdcdcdhd}(m2\_finseq\_1~\\
X3~(k3\_finseq\_2~(u1\_struct\_0~X0)))){\unicdbdjdcd}(((k3\_finseq\_1~X1=k3\_finseq\_1~\\
X2){\unicdcdcdhd}(k1\_matrix\_1~X1=k1\_matrix\_1~X2)){\unicdbdjdcd}(k3\_matrix\_3~X0~(k3\_matrix\_3~\\
X0~X1~X2)~X3=k3\_matrix\_3~X0~X1~(k3\_matrix\_3~X0~X2~X3))))))\end{array}
\label{hyp:ehddpfngbgehchjgihpfdd}
\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}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m2\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_matrix\_1~X2){\unicdcdcdhd}(m2\_finseq\_1~X2~(k3\_finseq\_2~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}(((k3\_finseq\_1~X1=k3\_finseq\_1~X2){\unicdcdcdhd}(k1\_matrix\_1~\\
X1=k1\_matrix\_1~X2)){\unicdbdjdcd}(k3\_matrix\_3~X0~X1~X2=k3\_matrix\_3~X0~X2~X1))))\end{array}
\label{hyp:ehcdpfngbgehchjgihpfdd}
\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}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m2\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_matrix\_1~X2){\unicdcdcdhd}(m2\_finseq\_1~X2~(k3\_finseq\_2~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}(((k3\_finseq\_1~X1=k3\_finseq\_1~X2){\unicdcdcdhd}(k1\_matrix\_1~\\
X1=k1\_matrix\_1~X2)){\unicdbdjdcd}(X1=k3\_matrix\_3~X0~X1~(k1\_matrix\_4~X0~X2~X2)))))\end{array}
\label{hyp:ehcdadpfngbgehchjgihpfed}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{{m2\_finseq\_1}~{X1}}~{X0}})}{\unicdbdjded}{({{{m1\_finseq\_1}~{X1}}~{X0}})}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpfngcdpfggjgogdhfgbhpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v6\_struct\_0~\\
X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v33\_algstr\_0~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}\\
((v5\_group\_1~X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~\\
X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdcdcdhd}\\
(((v1\_matrix\_1~X1){\unicdcdcdhd}(m1\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdcdcdhd}((v1\_matrix\_1~X2){\unicdcdcdhd}(m1\_finseq\_1~X2~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))))){\unicdbdjdcd}((v1\_matrix\_1~(k3\_matrix\_3~X0~X1~X2)){\unicdcdcdhd}(m2\_finseq\_1~\\
(k3\_matrix\_3~X0~X1~X2)~(k3\_finseq\_2~(u1\_struct\_0~X0))))\end{array}
\label{hyp:egehpflgddpfngbgehchjgihpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v6\_struct\_0~X0){\unicdcdcdhd}\\
((v13\_algstr\_0~X0){\unicdcdcdhd}((v33\_algstr\_0~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~\\
X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}\\
((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdcdcdhd}\\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m1\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0))))){\unicdbdjdcd}((v1\_matrix\_1~(k2\_matrix\_3~X0~X1)){\unicdcdcdhd}(m2\_finseq\_1~(k2\_matrix\_3~\\
X0~X1)~(k3\_finseq\_2~(u1\_struct\_0~X0))))\end{array}
\label{hyp:egehpflgcdpfngbgehchjgihpfdd}
\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}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m2\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_matrix\_1~X2){\unicdcdcdhd}(m2\_finseq\_1~X2~(k3\_finseq\_2~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_matrix\_1~X3){\unicdcdcdhd}(m2\_finseq\_1~\\
X3~(k3\_finseq\_2~(u1\_struct\_0~X0)))){\unicdbdjdcd}((X3=k3\_matrix\_3~X0~X1~X2){\unicdbdjded}\\
((k3\_finseq\_1~X3=k3\_finseq\_1~X1){\unicdcdcdhd}((k1\_matrix\_1~X3=k1\_matrix\_1~\\
X1){\unicdcdcdhd}({\unicdcdadad} X4 .  (v7\_ordinal1~X4){\unicdbdjdcd}({\unicdcdadad} X5 .  (v7\_ordinal1~X5){\unicdbdjdcd}\\
((k4\_tarski~X4~X5 \in k2\_matrix\_1~X1){\unicdbdjdcd}(k3\_matrix\_1~(u1\_struct\_0~\\
X0)~X3~X4~X5=k3\_rlvect\_1~X0~(k3\_matrix\_1~(u1\_struct\_0~X0)~X1~X4~X5)~\\
(k3\_matrix\_1~(u1\_struct\_0~X0)~X2~X4~X5)))))))))))\end{array}
\label{hyp:egddpfngbgehchjgihpfdd}
\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}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m2\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_matrix\_1~X2){\unicdcdcdhd}(m2\_finseq\_1~X2~(k3\_finseq\_2~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}((X2=k2\_matrix\_3~X0~X1){\unicdbdjded}((k3\_finseq\_1~X2=\\
k3\_finseq\_1~X1){\unicdcdcdhd}((k1\_matrix\_1~X2=k1\_matrix\_1~X1){\unicdcdcdhd}({\unicdcdadad} X3 .  \\
(v7\_ordinal1~X3){\unicdbdjdcd}({\unicdcdadad} X4 .  (v7\_ordinal1~X4){\unicdbdjdcd}((k4\_tarski~X3~\\
X4 \in k2\_matrix\_1~X1){\unicdbdjdcd}(k3\_matrix\_1~(u1\_struct\_0~X0)~X2~X3~X4=k4\_algstr\_0~\\
X0~(k3\_matrix\_1~(u1\_struct\_0~X0)~X1~X3~X4))))))))))\end{array}
\label{hyp:egcdpfngbgehchjgihpfdd}
\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}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m2\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_matrix\_1~X2){\unicdcdcdhd}(m2\_finseq\_1~X2~(k3\_finseq\_2~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}(k1\_matrix\_4~X0~X1~X2=k3\_matrix\_3~X0~X1~(k2\_matrix\_3~\\
X0~X2))))\end{array}
\label{hyp:egbdpfngbgehchjgihpfed}
\end{equation}
\begin{theorem}\label{thm:ehcdddpfngbgehchjgihpfed}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v6\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~\\
X0){\unicdcdcdhd}((v33\_algstr\_0~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~\\
X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0)))))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_matrix\_1~X1){\unicdcdcdhd}(m2\_finseq\_1~X1~(k3\_finseq\_2~(u1\_struct\_0~\\
X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_matrix\_1~X2){\unicdcdcdhd}(m2\_finseq\_1~X2~(k3\_finseq\_2~\\
(u1\_struct\_0~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_matrix\_1~X3){\unicdcdcdhd}(m2\_finseq\_1~\\
X3~(k3\_finseq\_2~(u1\_struct\_0~X0)))){\unicdbdjdcd}(((k3\_finseq\_1~X1=k3\_finseq\_1~\\
X2){\unicdcdcdhd}((k3\_finseq\_1~X2=k3\_finseq\_1~X3){\unicdcdcdhd}((k1\_matrix\_1~X1=k1\_matrix\_1~\\
X2){\unicdcdcdhd}(k1\_matrix\_1~X2=k1\_matrix\_1~X3)))){\unicdbdjdcd}(k3\_matrix\_3~X0~X1~X3=\\
k3\_matrix\_3~X0~(k3\_matrix\_3~X0~X1~X2)~(k1\_matrix\_4~X0~X3~X2))))))\end{array}$$
\end{theorem}
\end{document}
