\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{t8\_ringcat1 (TMFEv479ATbL2wiL8v8Ax8UgeE5UMPacEVA)}
\maketitle
Let $ v2\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v2\_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 $ v3\_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 $ m1\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r1\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ 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}((v3\_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}((v3\_group\_1~X1){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X1){\unicdcdcdhd}((v5\_vectsp\_1~X1){\unicdcdcdhd}(l6\_algstr\_0~X1))))))))){\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}((v3\_group\_1~X2){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X2){\unicdcdcdhd}((v5\_vectsp\_1~X2){\unicdcdcdhd}(l6\_algstr\_0~X2))))))))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_ringcat1~X3~X0~X1){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_ringcat1~X4~\\
X2~X0){\unicdbdjdcd}(((r1\_ringcat1~X2~X0){\unicdcdcdhd}(r1\_ringcat1~X0~X1)){\unicdbdjdcd}(m1\_ringcat1~\\
(k6\_ringcat1~X3~X4)~X2~X1))))))\end{array}
\label{hyp:ehgdpfchjgoghgdgbgehbd}
\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}((v3\_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}((v3\_group\_1~X1){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X1){\unicdcdcdhd}((v5\_vectsp\_1~X1){\unicdcdcdhd}(l6\_algstr\_0~X1))))))))){\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}((v3\_group\_1~X2){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X2){\unicdcdcdhd}((v5\_vectsp\_1~X2){\unicdcdcdhd}(l6\_algstr\_0~X2))))))))){\unicdbdjdcd}\\
(((r1\_ringcat1~X0~X1){\unicdcdcdhd}(r1\_ringcat1~X1~X2)){\unicdbdjdcd}(r1\_ringcat1~X0~X2))))\end{array}
\label{hyp:ehfdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v2\_ringcat1~X0){\unicdcdcdhd}((v3\_ringcat1~X0){\unicdcdcdhd}(l1\_ringcat1~\\
X0))){\unicdbdjdcd}((m1\_ringcat1~X0~(k1\_ringcat1~X0)~(k2\_ringcat1~X0)){\unicdcdcdhd}(r1\_ringcat1~\\
(k1\_ringcat1~X0)~(k2\_ringcat1~X0)))\end{array}
\label{hyp:ehddpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((\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}((v3\_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}((v3\_group\_1~X1){\unicdcdcdhd}((v4\_vectsp\_1~\\
X1){\unicdcdcdhd}((v5\_vectsp\_1~X1){\unicdcdcdhd}(l6\_algstr\_0~X1)))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_ringcat1~X2~X0~X1){\unicdbdjdcd}((v2\_ringcat1~X2){\unicdcdcdhd}((v3\_ringcat1~X2){\unicdcdcdhd}(\\
l1\_ringcat1~X2))))\end{array}
\label{hyp:egehpfngbdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_ringcat1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~(k2\_ringcat1~X0)){\unicdcdcdhd}\\
((v13\_algstr\_0~(k2\_ringcat1~X0)){\unicdcdcdhd}((v2\_rlvect\_1~(k2\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_rlvect\_1~(k2\_ringcat1~X0)){\unicdcdcdhd}((v4\_rlvect\_1~(k2\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_group\_1~(k2\_ringcat1~X0)){\unicdcdcdhd}((v4\_vectsp\_1~(k2\_ringcat1~\\
X0)){\unicdcdcdhd}((v5\_vectsp\_1~(k2\_ringcat1~X0)){\unicdcdcdhd}(l6\_algstr\_0~(k2\_ringcat1~\\
X0))))))))))\end{array}
\label{hyp:egehpflgcdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_ringcat1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~(k1\_ringcat1~X0)){\unicdcdcdhd}\\
((v13\_algstr\_0~(k1\_ringcat1~X0)){\unicdcdcdhd}((v2\_rlvect\_1~(k1\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_rlvect\_1~(k1\_ringcat1~X0)){\unicdcdcdhd}((v4\_rlvect\_1~(k1\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_group\_1~(k1\_ringcat1~X0)){\unicdcdcdhd}((v4\_vectsp\_1~(k1\_ringcat1~\\
X0)){\unicdcdcdhd}((v5\_vectsp\_1~(k1\_ringcat1~X0)){\unicdcdcdhd}(l6\_algstr\_0~(k1\_ringcat1~\\
X0))))))))))\end{array}
\label{hyp:egehpflgbdpfchjgoghgdgbgehbd}
\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}((v3\_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}((v3\_group\_1~X1){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X1){\unicdcdcdhd}((v5\_vectsp\_1~X1){\unicdcdcdhd}(l6\_algstr\_0~X1))))))))){\unicdbdjdcd}\\
((r1\_ringcat1~X0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v2\_ringcat1~X2){\unicdcdcdhd}((v3\_ringcat1~\\
X2){\unicdcdcdhd}(l1\_ringcat1~X2))){\unicdbdjdcd}((m1\_ringcat1~X2~X0~X1){\unicdbdjded}((k1\_ringcat1~\\
X2=X0){\unicdcdcdhd}(k2\_ringcat1~X2=X1))))))\end{array}
\label{hyp:egidpfchjgoghgdgbgehbd}
\end{equation}
\begin{theorem}\label{thm:ehidpfchjgoghgdgbgehbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((v2\_ringcat1~X0){\unicdcdcdhd}((v3\_ringcat1~X0){\unicdcdcdhd}(l1\_ringcat1~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v2\_ringcat1~X1){\unicdcdcdhd}((v3\_ringcat1~X1){\unicdcdcdhd}(l1\_ringcat1~\\
X1))){\unicdbdjdcd}((k1\_ringcat1~X1=k2\_ringcat1~X0){\unicdbdjdcd}((k1\_ringcat1~(k6\_ringcat1~\\
X1~X0)=k1\_ringcat1~X0){\unicdcdcdhd}(k2\_ringcat1~(k6\_ringcat1~X1~X0)=k2\_ringcat1~\\
X1))))\end{array}$$
\end{theorem}
\end{document}
