\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{t18\_ring\_1 (TMNjjVAvb4reYdqyjszokSypsPKcmAgcYdt)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_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 $ 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 $ l6\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_ideal\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v2\_ideal\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v3\_ideal\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v2\_ring\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_ring\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v6\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_vectsp\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_ring\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v36\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l5\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l4\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l4\_struct\_0 : {{\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}((v3\_group\_1~\\
X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}\\
((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((v1\_ideal\_1~X1~X0){\unicdcdcdhd}((v2\_ideal\_1~\\
X1~X0){\unicdcdcdhd}((v3\_ideal\_1~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~\\
X0))))))){\unicdbdjdcd}((v1\_ring\_1~X1~X0){\unicdbdjded}(v1\_vectsp\_2~(k2\_ring\_1~X0~X1))))\end{array}
\label{hyp:ehbdhdpfchjgoghgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v3\_group\_1~\\
X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}\\
((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((v1\_ideal\_1~X1~X0){\unicdcdcdhd}((v2\_ideal\_1~\\
X1~X0){\unicdcdcdhd}((v3\_ideal\_1~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~\\
X0))))))){\unicdbdjdcd}((v1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjded}(\preoh v6\_struct\_0~\\
(k2\_ring\_1~X0~X1))))\end{array}
\label{hyp:ehbdgdpfchjgoghgpfbd}
\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}\\
((v3\_group\_1~X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~\\
X0){\unicdcdcdhd}((v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}\\
(l6\_algstr\_0~X0)))))))))){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((v1\_ideal\_1~\\
X1~X0){\unicdcdcdhd}((v2\_ideal\_1~X1~X0){\unicdcdcdhd}((v3\_ideal\_1~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~\\
(k1\_zfmisc\_1~(u1\_struct\_0~X0)))))))){\unicdbdjdcd}((v36\_algstr\_0~(k2\_ring\_1~\\
X0~X1)){\unicdcdcdhd}(v5\_group\_1~(k2\_ring\_1~X0~X1)))\end{array}
\label{hyp:ggdggdpfchjgoghgpfbd}
\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}\\
((v3\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdcdcdhd}\\
((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((v1\_ideal\_1~X1~X0){\unicdcdcdhd}((v2\_ideal\_1~X1~X0){\unicdcdcdhd}(\\
(v3\_ideal\_1~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~\\
X0)))))))){\unicdbdjdcd}((v13\_algstr\_0~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v36\_algstr\_0~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v3\_group\_1~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v4\_vectsp\_1~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v5\_vectsp\_1~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v2\_rlvect\_1~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v3\_rlvect\_1~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}(v4\_rlvect\_1~\\
(k2\_ring\_1~X0~X1)))))))))\end{array}
\label{hyp:ggdgfdpfchjgoghgpfbd}
\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}\\
((v3\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdcdcdhd}\\
((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((v1\_ideal\_1~X1~X0){\unicdcdcdhd}((v2\_ideal\_1~X1~X0){\unicdcdcdhd}(\\
(v3\_ideal\_1~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~\\
X0)))))))){\unicdbdjdcd}((\preoh v2\_struct\_0~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}(v36\_algstr\_0~\\
(k2\_ring\_1~X0~X1)))\end{array}
\label{hyp:ggdgedpfchjgoghgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l6\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{({{l2\_algstr\_0}~{X0}})}{\unicdcdcdhd}{({{l5\_algstr\_0}~{X0}})}})}}}\label{hyp:egehpfmggdpfbgmghgdhehchpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l5\_algstr\_0}~{X0}})}{\unicdbdjdcd}{({{({{l4\_algstr\_0}~{X0}})}{\unicdcdcdhd}{({{l4\_struct\_0}~{X0}})}})}}}\label{hyp:egehpfmgfdpfbgmghgdhehchpfad}
\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}\\
((v3\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdcdcdhd}\\
((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((v1\_ideal\_1~X1~X0){\unicdcdcdhd}((v2\_ideal\_1~X1~X0){\unicdcdcdhd}(\\
(v3\_ideal\_1~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~\\
X0)))))))){\unicdbdjdcd}((v36\_algstr\_0~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}(l6\_algstr\_0~(\\
k2\_ring\_1~X0~X1)))\end{array}
\label{hyp:egehpflgcdpfchjgoghgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l4\_algstr\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~X0))){\unicdbdjdcd}(((v1\_subset\_1~\\
X1~(u1\_struct\_0~X0)){\unicdcdcdhd}(v1\_ring\_1~X1~X0)){\unicdbdjdcd}(v2\_ring\_1~X1~X0)))\end{array}
\label{hyp:dgdgcdpfchjgoghgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l4\_algstr\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~(u1\_struct\_0~X0))){\unicdbdjdcd}((v2\_ring\_1~\\
X1~X0){\unicdbdjdcd}((v1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdcdcdhd}(v1\_ring\_1~X1~X0))))\end{array}
\label{hyp:dgdgbdpfchjgoghgpfbd}
\end{equation}
\begin{theorem}\label{thm:ehbdidpfchjgoghgpfbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v3\_group\_1~\\
X0){\unicdcdcdhd}((v5\_group\_1~X0){\unicdcdcdhd}((v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(\\
(v2\_rlvect\_1~X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~\\
X0)))))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}((v1\_ideal\_1~X1~\\
X0){\unicdcdcdhd}((v2\_ideal\_1~X1~X0){\unicdcdcdhd}((v3\_ideal\_1~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(\\
k1\_zfmisc\_1~(u1\_struct\_0~X0))))))){\unicdbdjdcd}((v2\_ring\_1~X1~X0){\unicdbdjded}((\preoh v2\_struct\_0~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((\preoh v6\_struct\_0~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v13\_algstr\_0~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v3\_group\_1~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v5\_group\_1~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v4\_vectsp\_1~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v5\_vectsp\_1~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v2\_rlvect\_1~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v3\_rlvect\_1~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v4\_rlvect\_1~(k2\_ring\_1~X0~X1)){\unicdcdcdhd}((v1\_vectsp\_2~\\
(k2\_ring\_1~X0~X1)){\unicdcdcdhd}(l6\_algstr\_0~(k2\_ring\_1~X0~X1)))))))))))))))\end{array}$$
\end{theorem}
\end{document}
