\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{t21\_ringcat1 (TMJYsbm3cniyAhNoQ75b7hGNSjX8yhdrXke)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_classes2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k17\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u1\_cat\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_graph\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_graph\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v5\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m3\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\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 $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ g1\_cat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v11\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_cat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_ringcat1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k9\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ l1\_cat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_graph\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k16\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k10\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k15\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k14\_ringcat1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u2\_graph\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_graph\_1 : {{\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 $ k1\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_ringcat1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v5\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m3\_ringcat1~X1~X0){\unicdbdjded}(m1\_subset\_1~X1~X0))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngddpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  (((v1\_funct\_1~\\
X2){\unicdcdcdhd}((v1\_funct\_2~X2~X1~X0){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X1~X0))))){\unicdcdcdhd}(((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~X1~X0){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X1~X0))))){\unicdcdcdhd}((v1\_funct\_1~X4){\unicdcdcdhd}(m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~X1~X1)~X1)))))){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
{\unicdcdadad} X6 .  {\unicdcdadad} X7 .  {\unicdcdadad} X8 .  {\unicdcdadad} X9 .  (g1\_cat\_1~X0~X1~X2~\\
X3~X4=g1\_cat\_1~X5~X6~X7~X8~X9){\unicdbdjdcd}((X0=X5){\unicdcdcdhd}((X1=X6){\unicdcdcdhd}((X2=X7){\unicdcdcdhd}((X3=\\
X8){\unicdcdcdhd}(X4=X9))))))\end{array}
\label{hyp:ggchfgfgpfhgbdpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}((\preoh v2\_struct\_0~\\
(k17\_ringcat1~X0)){\unicdcdcdhd}((\preoh v11\_struct\_0~(k17\_ringcat1~X0)){\unicdcdcdhd}(v1\_cat\_1~\\
(k17\_ringcat1~X0))))\end{array}
\label{hyp:ggdgfdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}(v4\_ringcat1~\\
(k9\_ringcat1~X0))\end{array}
\label{hyp:ggdgedpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}(\preoh v1\_xboole\_0~\\
(k9\_ringcat1~X0))\end{array}
\label{hyp:ggdgddpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_cat\_1~X0){\unicdbdjdcd}((v1\_funct\_1~(u1\_cat\_1~X0)){\unicdcdcdhd}(m1\_subset\_1~\\
(u1\_cat\_1~X0)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k2\_zfmisc\_1~(u4\_struct\_0~\\
X0)~(u4\_struct\_0~X0))~(u4\_struct\_0~X0)))))\end{array}
\label{hyp:egehpffhbdpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_cat\_1}~{X0}})}{\unicdbdjdcd}{({{l1\_graph\_1}~{X0}})}}}\label{hyp:egehpfmgbdpfdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}(l1\_cat\_1~(\\
k17\_ringcat1~X0))\end{array}
\label{hyp:egehpflgbdhdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}((v1\_funct\_1~\\
(k16\_ringcat1~X0)){\unicdcdcdhd}(m1\_subset\_1~(k16\_ringcat1~X0)~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(k2\_zfmisc\_1~(k10\_ringcat1~X0)~(k10\_ringcat1~X0))~\\
(k10\_ringcat1~X0)))))\end{array}
\label{hyp:egehpflgbdgdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}((v1\_funct\_1~\\
(k15\_ringcat1~X0)){\unicdcdcdhd}((v1\_funct\_2~(k15\_ringcat1~X0)~(k10\_ringcat1~\\
X0)~X0){\unicdcdcdhd}(m1\_subset\_1~(k15\_ringcat1~X0)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k10\_ringcat1~X0)~X0)))))\end{array}
\label{hyp:egehpflgbdfdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}((v1\_funct\_1~\\
(k14\_ringcat1~X0)){\unicdcdcdhd}((v1\_funct\_2~(k14\_ringcat1~X0)~(k10\_ringcat1~\\
X0)~X0){\unicdcdcdhd}(m1\_subset\_1~(k14\_ringcat1~X0)~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k10\_ringcat1~X0)~X0)))))\end{array}
\label{hyp:egehpflgbdedpfchjgoghgdgbgehbd}
\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 v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}(l1\_graph\_1~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u4\_struct\_0~X0)){\unicdbdjdcd}(k4\_graph\_1~\\
X0~X1=k3\_funct\_2~(u4\_struct\_0~X0)~(u1\_struct\_0~X0)~(u2\_graph\_1~\\
X0)~X1))\end{array}
\label{hyp:egedpfhgchbgahigpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((\preoh v11\_struct\_0~X0){\unicdcdcdhd}(l1\_graph\_1~\\
X0))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u4\_struct\_0~X0)){\unicdbdjdcd}(k3\_graph\_1~\\
X0~X1=k3\_funct\_2~(u4\_struct\_0~X0)~(u1\_struct\_0~X0)~(u1\_graph\_1~\\
X0)~X1))\end{array}
\label{hyp:egddpfhgchbgahigpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}(k17\_ringcat1~\\
X0=g1\_cat\_1~(k9\_ringcat1~X0)~(k10\_ringcat1~(k9\_ringcat1~X0))~\\
(k14\_ringcat1~(k9\_ringcat1~X0))~(k15\_ringcat1~(k9\_ringcat1~X0))~\\
(k16\_ringcat1~(k9\_ringcat1~X0)))\end{array}
\label{hyp:egcdedpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_funct\_1~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(\\
k2\_zfmisc\_1~(k10\_ringcat1~X0)~(k10\_ringcat1~X0))~(k10\_ringcat1~\\
X0))))){\unicdbdjdcd}((X1=k16\_ringcat1~X0){\unicdbdjded}(({\unicdcdadad} X2 .  (m3\_ringcat1~X2~\\
(k10\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m3\_ringcat1~X3~(k10\_ringcat1~\\
X0)){\unicdbdjdcd}((k4\_tarski~X2~X3 \in k1\_relset\_1~(k2\_zfmisc\_1~(k10\_ringcat1~\\
X0)~(k10\_ringcat1~X0))~X1){\unicdbdjded}(k11\_ringcat1~X0~X2=k12\_ringcat1~X0~\\
X3)))){\unicdcdcdhd}({\unicdcdadad} X2 .  (m3\_ringcat1~X2~(k10\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m3\_ringcat1~X3~(k10\_ringcat1~X0)){\unicdbdjdcd}((k4\_tarski~X2~X3 \in k1\_relset\_1~\\
(k2\_zfmisc\_1~(k10\_ringcat1~X0)~(k10\_ringcat1~X0))~X1){\unicdbdjdcd}(k1\_binop\_1~\\
X1~X2~X3=k6\_ringcat1~X2~X3)))))))\end{array}
\label{hyp:egcdddpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~X1~(k10\_ringcat1~X0)~X0){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k10\_ringcat1~X0)~X0))))){\unicdbdjdcd}((X1=\\
k15\_ringcat1~X0){\unicdbdjded}({\unicdcdadad} X2 .  (m3\_ringcat1~X2~(k10\_ringcat1~X0)){\unicdbdjdcd}\\
(k3\_funct\_2~(k10\_ringcat1~X0)~X0~X1~X2=k12\_ringcat1~X0~X2))))\end{array}
\label{hyp:egcdbdpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v4\_ringcat1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~X1~(k10\_ringcat1~X0)~X0){\unicdcdcdhd}(m1\_subset\_1~\\
X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k10\_ringcat1~X0)~X0))))){\unicdbdjdcd}((X1=\\
k14\_ringcat1~X0){\unicdbdjded}({\unicdcdadad} X2 .  (m3\_ringcat1~X2~(k10\_ringcat1~X0)){\unicdbdjdcd}\\
(k3\_funct\_2~(k10\_ringcat1~X0)~X0~X1~X2=k11\_ringcat1~X0~X2))))\end{array}
\label{hyp:egcdadpfchjgoghgdgbgehbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_cat\_1~X0){\unicdbdjdcd}((v1\_cat\_1~X0){\unicdbdjdcd}(X0=g1\_cat\_1~(u1\_struct\_0~\\
X0)~(u4\_struct\_0~X0)~(u1\_graph\_1~X0)~(u2\_graph\_1~X0)~(u1\_cat\_1~\\
X0)))\end{array}
\label{hyp:bgcgdhehchbgdgehogfgdhdhpfghbdpfdgbgehpfbd}
\end{equation}
\begin{theorem}\label{thm:ehcdbdpfchjgoghgdgbgehbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_classes2~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
(m1\_subset\_1~X1~(u4\_struct\_0~(k17\_ringcat1~X0))){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_subset\_1~X2~(u4\_struct\_0~(k17\_ringcat1~X0))){\unicdbdjdcd}((k4\_tarski~\\
X2~X1 \in k1\_relset\_1~(k2\_zfmisc\_1~(u4\_struct\_0~(k17\_ringcat1~X0))~\\
(u4\_struct\_0~(k17\_ringcat1~X0)))~(u1\_cat\_1~(k17\_ringcat1~X0))){\unicdbdjded}\\
(k3\_graph\_1~(k17\_ringcat1~X0)~X2=k4\_graph\_1~(k17\_ringcat1~X0)~\\
X1))))\end{array}$$
\end{theorem}
\end{document}
