\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{t16\_ringcat1 (TMbBbjHoLWVrfxu5272mF1rpVaZ5TgGTkdZ)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_classes2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r2\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k18\_mod\_2 : {\iota}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k3\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v33\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v36\_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 $ v4\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_vectsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v12\_vectsp\_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 $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l6\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k4\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}((u1\_struct\_0~\\
k18\_mod\_2 \in X0){\unicdcdcdhd}((u1\_algstr\_0~k18\_mod\_2 \in X0){\unicdcdcdhd}((k5\_vectsp\_1~k18\_mod\_2 \in \\
X0){\unicdcdcdhd}((k4\_struct\_0~k18\_mod\_2 \in X0){\unicdcdcdhd}((u2\_algstr\_0~k18\_mod\_2 \in X0){\unicdcdcdhd}\\
(k5\_struct\_0~k18\_mod\_2 \in X0))))))\end{array}
\label{hyp:ehcdjdpfngpgegpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{X0} \in {X1}})}{\unicdbdjdcd}{({{{m1\_subset\_1}~{X0}}~{X1}})}}}}\label{hyp:ehbdpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~X0){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~X0){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_subset\_1~X3~X0){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~X4~X0){\unicdbdjdcd}((k3\_xtuple\_0~\\
X1~X2~X3 \in X0){\unicdcdcdhd}(k6\_xtuple\_0~X1~X2~X3~X4 \in X0))))))\end{array}
\label{hyp:ehbdpfhgchdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v13\_algstr\_0~k18\_mod\_2){\unicdcdcdhd}((v33\_algstr\_0~k18\_mod\_2){\unicdcdcdhd}((v36\_algstr\_0~\\
k18\_mod\_2){\unicdcdcdhd}((v3\_group\_1~k18\_mod\_2){\unicdcdcdhd}((v5\_group\_1~k18\_mod\_2){\unicdcdcdhd}\\
((v4\_vectsp\_1~k18\_mod\_2){\unicdcdcdhd}((v5\_vectsp\_1~k18\_mod\_2){\unicdcdcdhd}((v12\_vectsp\_1~\\
k18\_mod\_2){\unicdcdcdhd}((v2\_rlvect\_1~k18\_mod\_2){\unicdcdcdhd}((v3\_rlvect\_1~k18\_mod\_2){\unicdcdcdhd}\\
(v4\_rlvect\_1~k18\_mod\_2))))))))))\end{array}
\label{hyp:ggdghdpfngpgegpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{({\preoh{{{v2\_struct\_0}~{k18\_mod\_2}}}})}{\unicdcdcdhd}{({{v36\_algstr\_0}~{k18\_mod\_2}})}}\label{hyp:ggdgedpfngpgegpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{({{v36\_algstr\_0}~{k18\_mod\_2}})}{\unicdcdcdhd}{({{l6\_algstr\_0}~{k18\_mod\_2}})}}\label{hyp:egehpflgbdidpfngpgegpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  k6\_xtuple\_0~X0~X1~\\
X2~X3=k4\_tarski~(k3\_xtuple\_0~X0~X1~X2)~X3\end{array}
\label{hyp:egidpfihehfhahmgfgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  k3\_xtuple\_0~X0~X1~X2=k4\_tarski~\\
(k4\_tarski~X0~X1)~X2\end{array}
\label{hyp:egedpfihehfhahmgfgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (r2\_ringcat1~X0~X1){\unicdbdjded}({\unicdcdaddd} X2 .  {\unicdcdaddd} X3 .  \\
{\unicdcdaddd} X4 .  {\unicdcdaddd} X5 .  {\unicdcdaddd} X6 .  {\unicdcdaddd} X7 .  (X0=k3\_xtuple\_0~\\
(k6\_xtuple\_0~X2~X3~X4~X5)~X6~X7){\unicdcdcdhd}({\unicdcdaddd} X8 .  ((\preoh v2\_struct\_0~X8){\unicdcdcdhd}\\
((v13\_algstr\_0~X8){\unicdcdcdhd}((v36\_algstr\_0~X8){\unicdcdcdhd}((v2\_rlvect\_1~X8){\unicdcdcdhd}((\\
v3\_rlvect\_1~X8){\unicdcdcdhd}((v4\_rlvect\_1~X8){\unicdcdcdhd}((v3\_group\_1~X8){\unicdcdcdhd}((v4\_vectsp\_1~\\
X8){\unicdcdcdhd}((v5\_vectsp\_1~X8){\unicdcdcdhd}(l6\_algstr\_0~X8)))))))))){\unicdcdcdhd}((X1=X8){\unicdcdcdhd}\\
((X2=u1\_struct\_0~X8){\unicdcdcdhd}((X3=u1\_algstr\_0~X8){\unicdcdcdhd}((X4=k5\_vectsp\_1~\\
X8){\unicdcdcdhd}((X5=k4\_struct\_0~X8){\unicdcdcdhd}((X6=u2\_algstr\_0~X8){\unicdcdcdhd}(X7=k5\_struct\_0~\\
X8)))))))))\end{array}
\label{hyp:egbdfdpfchjgoghgdgbgehbd}
\end{equation}
\begin{theorem}\label{thm:ehbdgdpfchjgoghgdgbgehbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}({\unicdcdaddd} X1 .  \\
(X1 \in X0){\unicdcdcdhd}(r2\_ringcat1~X1~k18\_mod\_2))\end{array}$$
\end{theorem}
\end{document}
