\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{l38\_scmring2 (TMKZNkcfYGorcs1dXkaTHoLWb9AHTny8mSd)}
\maketitle
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\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_compos\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_scmring2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v2\_extpro\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ k3\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ v1\_ami\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_scmring2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k4\_scmring2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k5\_scmring2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_scmring2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k8\_scmring2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k9\_scmring2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k7\_scmring2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ 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 .  (m1\_subset\_1~X1~(u1\_compos\_1~(k1\_scmring2~X0))){\unicdbdjded}\\
(\preoh (X1{\unicdcdgdad}k3\_xtuple\_0~k6\_numbers~k1\_xboole\_0~k1\_xboole\_0){\unicdcdcdhd}(({\unicdcdadad} X2 .  \\
((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~(k1\_scmring2~X0)))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  ((v1\_ami\_2~X3){\unicdcdcdhd}(m1\_subset\_1~X3~(u1\_struct\_0~(k1\_scmring2~\\
X0)))){\unicdbdjdcd}(X1{\unicdcdgdad}k3\_scmring2~X0~X2~X3))){\unicdcdcdhd}(({\unicdcdadad} X2 .  ((v1\_ami\_2~X2){\unicdcdcdhd}\\
(m1\_subset\_1~X2~(u1\_struct\_0~(k1\_scmring2~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
((v1\_ami\_2~X3){\unicdcdcdhd}(m1\_subset\_1~X3~(u1\_struct\_0~(k1\_scmring2~X0)))){\unicdbdjdcd}\\
(X1{\unicdcdgdad}k4\_scmring2~X0~X2~X3))){\unicdcdcdhd}(({\unicdcdadad} X2 .  ((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(u1\_struct\_0~(k1\_scmring2~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_ami\_2~X3){\unicdcdcdhd}\\
(m1\_subset\_1~X3~(u1\_struct\_0~(k1\_scmring2~X0)))){\unicdbdjdcd}(X1{\unicdcdgdad}k5\_scmring2~\\
X0~X2~X3))){\unicdcdcdhd}(({\unicdcdadad} X2 .  ((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~\\
(k1\_scmring2~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_ami\_2~X3){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(u1\_struct\_0~(k1\_scmring2~X0)))){\unicdbdjdcd}(X1{\unicdcdgdad}k6\_scmring2~X0~X2~X3))){\unicdcdcdhd}\\
(({\unicdcdadad} X2 .  (m1\_subset\_1~X2~k5\_numbers){\unicdbdjdcd}(X1{\unicdcdgdad}k8\_scmring2~X0~\\
X2)){\unicdcdcdhd}(({\unicdcdadad} X2 .  ((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~\\
(k1\_scmring2~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k5\_numbers){\unicdbdjdcd}\\
(X1{\unicdcdgdad}k9\_scmring2~X0~X3~X2))){\unicdcdcdhd}({\unicdcdadad} X2 .  ((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(u1\_struct\_0~(k1\_scmring2~X0)))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~X0)){\unicdbdjdcd}(X1{\unicdcdgdad}k7\_scmring2~X0~X2~X3)))))))))))\end{array}
\label{hyp:ehhdpfdhdgngchjgoghgcd}
\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 .  (m1\_subset\_1~X1~k5\_numbers){\unicdbdjdcd}(\preoh v2\_extpro\_1~(k8\_scmring2~\\
X0~X1)~np\_\_2~(k1\_scmring2~X0)))\end{array}
\label{hyp:mgddfdpfdhdgngchjgoghgcd}
\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}(((v1\_ami\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(u1\_struct\_0~(k1\_scmring2~\\
X0)))){\unicdcdcdhd}(m1\_subset\_1~X2~k5\_numbers))){\unicdbdjdcd}(\preoh v2\_extpro\_1~(k9\_scmring2~\\
X0~X2~X1)~np\_\_2~(k1\_scmring2~X0))\end{array}
\label{hyp:ggdgjdpfdhdgngchjgoghgcd}
\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}(((v1\_ami\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(u1\_struct\_0~(k1\_scmring2~\\
X0)))){\unicdcdcdhd}((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~(k1\_scmring2~\\
X0)))))){\unicdbdjdcd}(\preoh v2\_extpro\_1~(k6\_scmring2~X0~X1~X2)~np\_\_2~(k1\_scmring2~\\
X0))\end{array}
\label{hyp:ggdghdpfdhdgngchjgoghgcd}
\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}(((v1\_ami\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(u1\_struct\_0~(k1\_scmring2~\\
X0)))){\unicdcdcdhd}((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~(k1\_scmring2~\\
X0)))))){\unicdbdjdcd}(\preoh v2\_extpro\_1~(k5\_scmring2~X0~X1~X2)~np\_\_2~(k1\_scmring2~\\
X0))\end{array}
\label{hyp:ggdggdpfdhdgngchjgoghgcd}
\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}(((v1\_ami\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(u1\_struct\_0~(k1\_scmring2~\\
X0)))){\unicdcdcdhd}((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~(k1\_scmring2~\\
X0)))))){\unicdbdjdcd}(\preoh v2\_extpro\_1~(k4\_scmring2~X0~X1~X2)~np\_\_2~(k1\_scmring2~\\
X0))\end{array}
\label{hyp:ggdgfdpfdhdgngchjgoghgcd}
\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}(((v1\_ami\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(u1\_struct\_0~(k1\_scmring2~\\
X0)))){\unicdcdcdhd}((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~(k1\_scmring2~\\
X0)))))){\unicdbdjdcd}(\preoh v2\_extpro\_1~(k3\_scmring2~X0~X1~X2)~np\_\_2~(k1\_scmring2~\\
X0))\end{array}
\label{hyp:ggdgedpfdhdgngchjgoghgcd}
\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}(((v1\_ami\_2~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(u1\_struct\_0~(k1\_scmring2~\\
X0)))){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~X0)))){\unicdbdjdcd}(\preoh v2\_extpro\_1~(k7\_scmring2~\\
X0~X1~X2)~np\_\_2~(k1\_scmring2~X0))\end{array}
\label{hyp:ggdgbdadpfdhdgngchjgoghgcd}
\end{equation}
\begin{theorem}\label{thm:mgddidpfdhdgngchjgoghgcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v2\_rlvect\_1~\\
X0){\unicdcdcdhd}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}((v3\_group\_1~X0){\unicdcdcdhd}(\\
(v4\_vectsp\_1~X0){\unicdcdcdhd}((v5\_vectsp\_1~X0){\unicdcdcdhd}(l6\_algstr\_0~X0))))))))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_compos\_1~(k1\_scmring2~X0))){\unicdbdjdcd}\\
((v2\_extpro\_1~X1~np\_\_2~(k1\_scmring2~X0)){\unicdbdjdcd}(X1=k3\_xtuple\_0~k6\_numbers~\\
k1\_xboole\_0~k1\_xboole\_0)))\end{array}$$
\end{theorem}
\end{document}
