\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{l10\_ringcat1 (TMJhcvUZGMbw3MPWLJU4X3hQg2F35o76bX2)}
\maketitle
Let $ v3\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ g1\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ u1\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u2\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u3\_ringcat1 : {{\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 $ 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 $ 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 $ v2\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k3\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\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))))))))){\unicdcdcdhd}\\
((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~(u1\_struct\_0~X0)~(u1\_struct\_0~\\
X1)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~\\
X0)~(u1\_struct\_0~X1)))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  {\unicdcdadad} X4 .  {\unicdcdadad} X5 .  \\
(g1\_ringcat1~X0~X1~X2=g1\_ringcat1~X3~X4~X5){\unicdbdjdcd}((X0=X3){\unicdcdcdhd}((X1=X4){\unicdcdcdhd}\\
(X2=X5))))\end{array}
\label{hyp:ggchfgfgpfhgbdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_ringcat1~X0){\unicdbdjdcd}((v1\_funct\_1~(u3\_ringcat1~X0)){\unicdcdcdhd}\\
((v1\_funct\_2~(u3\_ringcat1~X0)~(u1\_struct\_0~(u1\_ringcat1~X0))~\\
(u1\_struct\_0~(u2\_ringcat1~X0))){\unicdcdcdhd}(m1\_subset\_1~(u3\_ringcat1~X0)~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~(u1\_ringcat1~X0))~(u1\_struct\_0~\\
(u2\_ringcat1~X0)))))))\end{array}
\label{hyp:egehpffhddpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_ringcat1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~(u2\_ringcat1~X0)){\unicdcdcdhd}\\
((v13\_algstr\_0~(u2\_ringcat1~X0)){\unicdcdcdhd}((v2\_rlvect\_1~(u2\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_rlvect\_1~(u2\_ringcat1~X0)){\unicdcdcdhd}((v4\_rlvect\_1~(u2\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_group\_1~(u2\_ringcat1~X0)){\unicdcdcdhd}((v4\_vectsp\_1~(u2\_ringcat1~\\
X0)){\unicdcdcdhd}((v5\_vectsp\_1~(u2\_ringcat1~X0)){\unicdcdcdhd}(l6\_algstr\_0~(u2\_ringcat1~\\
X0))))))))))\end{array}
\label{hyp:egehpffhcdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_ringcat1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~(u1\_ringcat1~X0)){\unicdcdcdhd}\\
((v13\_algstr\_0~(u1\_ringcat1~X0)){\unicdcdcdhd}((v2\_rlvect\_1~(u1\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_rlvect\_1~(u1\_ringcat1~X0)){\unicdcdcdhd}((v4\_rlvect\_1~(u1\_ringcat1~\\
X0)){\unicdcdcdhd}((v3\_group\_1~(u1\_ringcat1~X0)){\unicdcdcdhd}((v4\_vectsp\_1~(u1\_ringcat1~\\
X0)){\unicdcdcdhd}((v5\_vectsp\_1~(u1\_ringcat1~X0)){\unicdcdcdhd}(l6\_algstr\_0~(u1\_ringcat1~\\
X0))))))))))\end{array}
\label{hyp:egehpffhbdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\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))))))))){\unicdcdcdhd}\\
((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~(u1\_struct\_0~X0)~(u1\_struct\_0~\\
X1)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~\\
X0)~(u1\_struct\_0~X1)))))))){\unicdbdjdcd}((v2\_ringcat1~(g1\_ringcat1~X0~X1~\\
X2)){\unicdcdcdhd}(l1\_ringcat1~(g1\_ringcat1~X0~X1~X2)))\end{array}
\label{hyp:egehpfhgbdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_ringcat1~X0){\unicdbdjdcd}((v3\_ringcat1~X0){\unicdbdjded}(v1\_ringcat1~\\
(k3\_ringcat1~X0)~(u1\_ringcat1~X0)~(u2\_ringcat1~X0)))\end{array}
\label{hyp:egfdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_ringcat1}~{X0}})}{\unicdbdjdcd}{({{{k3\_ringcat1}~{X0}}={{u3\_ringcat1}~{X0}}})}}}\label{hyp:egedpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_ringcat1~X0){\unicdbdjdcd}((v2\_ringcat1~X0){\unicdbdjdcd}(X0=g1\_ringcat1~\\
(u1\_ringcat1~X0)~(u2\_ringcat1~X0)~(u3\_ringcat1~X0)))\end{array}
\label{hyp:bgcgdhehchbgdgehogfgdhdhpfghcdpfchjgoghgdgbgehbd}
\end{equation}
\begin{theorem}\label{thm:mgbdadpfchjgoghgdgbgehbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((v3\_ringcat1~X0){\unicdcdcdhd}(l1\_ringcat1~X0)){\unicdbdjdcd}(v3\_ringcat1~\\
(g1\_ringcat1~(u1\_ringcat1~X0)~(u2\_ringcat1~X0)~(u3\_ringcat1~X0)))\end{array}$$
\end{theorem}
\end{document}
