\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{t20\_ringcat1 (TMVTAtsyUxs9ZcbZtUzbSzsH7Qczy2DS7jg)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m3\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k10\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k11\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k12\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\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 $ r1\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m2\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m1\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ v2\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k7\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ v5\_ringcat1 : {{\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}\\
(((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 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m3\_ringcat1~X1~(k10\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m3\_ringcat1~\\
X2~(k10\_ringcat1~X0)){\unicdbdjdcd}(\preoh (k11\_ringcat1~X0~X1=k12\_ringcat1~X0~X2){\unicdcdcdhd}\\
({\unicdcdadad} X3 .  (m2\_ringcat1~X3~X0){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_ringcat1~X4~X0){\unicdbdjdcd}\\
({\unicdcdadad} X5 .  (m2\_ringcat1~X5~X0){\unicdbdjdcd}(\preoh (r1\_ringcat1~X3~X4){\unicdcdcdhd}((r1\_ringcat1~\\
X4~X5){\unicdcdcdhd}((m1\_ringcat1~X1~X4~X5){\unicdcdcdhd}(m1\_ringcat1~X2~X3~X4))))))))))\end{array}
\label{hyp:ehbdjdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m2\_ringcat1~X1~X0){\unicdbdjdcd}((\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))))))))))\end{array}
\label{hyp:egehpfngcdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  (((\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}(((\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))))))))){\unicdcdcdhd}\\
((m1\_ringcat1~X3~X1~X2){\unicdcdcdhd}(m1\_ringcat1~X4~X0~X1))))){\unicdbdjdcd}((v2\_ringcat1~\\
(k7\_ringcat1~X0~X1~X2~X3~X4)){\unicdcdcdhd}(m1\_ringcat1~(k7\_ringcat1~X0~X1~X2~\\
X3~X4)~X0~X2))\end{array}
\label{hyp:egehpflghdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}((\preoh v1\_xboole\_0~\\
(k10\_ringcat1~X0)){\unicdcdcdhd}(v5\_ringcat1~(k10\_ringcat1~X0)))\end{array}
\label{hyp:egehpflgbdadpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(v5\_ringcat1~X1)){\unicdbdjdcd}((X1=k10\_ringcat1~X0){\unicdbdjded}\\
({\unicdcdadad} X2 .  (X2 \in X1){\unicdbdjded}({\unicdcdaddd} X3 .  (m2\_ringcat1~X3~X0){\unicdcdcdhd}({\unicdcdaddd} X4 .  \\
(m2\_ringcat1~X4~X0){\unicdcdcdhd}((r1\_ringcat1~X3~X4){\unicdcdcdhd}(m1\_ringcat1~X2~X3~X4)))))))\end{array}
\label{hyp:egbdhdpfchjgoghgdgbgehbd}
\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}\\
({\unicdcdadad} X3 .  (m1\_ringcat1~X3~X1~X2){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_ringcat1~X4~\\
X0~X1){\unicdbdjdcd}(((r1\_ringcat1~X0~X1){\unicdcdcdhd}(r1\_ringcat1~X1~X2)){\unicdbdjdcd}(k7\_ringcat1~\\
X0~X1~X2~X3~X4=k6\_ringcat1~X3~X4))))))\end{array}
\label{hyp:egbdadpfchjgoghgdgbgehbd}
\end{equation}
\begin{theorem}\label{thm:ehcdadpfchjgoghgdgbgehbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m3\_ringcat1~X1~(k10\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m3\_ringcat1~\\
X2~(k10\_ringcat1~X0)){\unicdbdjdcd}((k11\_ringcat1~X0~X1=k12\_ringcat1~X0~X2){\unicdbdjdcd}\\
(k6\_ringcat1~X1~X2 \in k10\_ringcat1~X0))))\end{array}$$
\end{theorem}
\end{document}
