\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\_grcat\_1 (TMRzB7mtG3jVP6dhHnYYAhA6k8rw4MV3CNx)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l2\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_grcat\_1 : {{\iota}{\unicdbdjdcd}{{\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 $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ g1\_grcat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k14\_grcat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ k13\_grcat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v2\_grcat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_grcat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_grcat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k7\_grcat\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_grcat\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_grcat\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u2\_grcat\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u3\_grcat\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
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}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}\\
(l2\_algstr\_0~X0))))){\unicdcdcdhd}(((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}\\
((v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~X1){\unicdcdcdhd}(l2\_algstr\_0~X1))))){\unicdcdcdhd}(\\
((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((v13\_algstr\_0~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((\\
v4\_rlvect\_1~X2){\unicdcdcdhd}(l2\_algstr\_0~X2))))){\unicdcdcdhd}((m1\_grcat\_1~X3~X1~X2){\unicdcdcdhd}\\
(m1\_grcat\_1~X4~X0~X1))))){\unicdbdjdcd}(k14\_grcat\_1~X0~X1~X2~X3~X4=k13\_grcat\_1~\\
X3~X4)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdedpfhgchdgbgehpfbd}
\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}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l2\_algstr\_0~X0))))){\unicdcdcdhd}\\
(((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}(\\
(v4\_rlvect\_1~X1){\unicdcdcdhd}(l2\_algstr\_0~X1))))){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~(u1\_struct\_0~X0)~(u1\_struct\_0~X1)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~X0)~(u1\_struct\_0~X1)))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
{\unicdcdadad} X4 .  {\unicdcdadad} X5 .  (g1\_grcat\_1~X0~X1~X2=g1\_grcat\_1~X3~X4~X5){\unicdbdjdcd}\\
((X0=X3){\unicdcdcdhd}((X1=X4){\unicdcdcdhd}(X2=X5))))\end{array}
\label{hyp:ggchfgfgpfhgbdpfhgchdgbgehpfbd}
\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\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l2\_algstr\_0~X0))))){\unicdcdcdhd}(\\
(\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~\\
X1){\unicdcdcdhd}(l2\_algstr\_0~X1)))))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_grcat\_1~X2~X0~X1){\unicdbdjdcd}\\
((v2\_grcat\_1~X2){\unicdcdcdhd}(l1\_grcat\_1~X2)))\end{array}
\label{hyp:egehpfngbdpfhgchdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((v2\_grcat\_1~X0){\unicdcdcdhd}(l1\_grcat\_1~X0)){\unicdcdcdhd}((\\
v2\_grcat\_1~X1){\unicdcdcdhd}(l1\_grcat\_1~X1))){\unicdbdjdcd}((v1\_grcat\_1~(k13\_grcat\_1~\\
X0~X1)){\unicdcdcdhd}((v2\_grcat\_1~(k13\_grcat\_1~X0~X1)){\unicdcdcdhd}(l1\_grcat\_1~(k13\_grcat\_1~\\
X0~X1))))\end{array}
\label{hyp:egehpflgbdddpfhgchdgbgehpfbd}
\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}((v3\_rlvect\_1~X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l2\_algstr\_0~X0))))){\unicdcdcdhd}\\
(((\preoh v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}(\\
(v4\_rlvect\_1~X1){\unicdcdcdhd}(l2\_algstr\_0~X1))))){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~(u1\_struct\_0~X0)~(u1\_struct\_0~X1)){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~X0)~(u1\_struct\_0~X1)))))))){\unicdbdjdcd}((v1\_grcat\_1~\\
(g1\_grcat\_1~X0~X1~X2)){\unicdcdcdhd}(l1\_grcat\_1~(g1\_grcat\_1~X0~X1~X2)))\end{array}
\label{hyp:egehpfhgbdpfhgchdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_grcat\_1}~{X0}})}{\unicdbdjdcd}{({{{k7\_grcat\_1}~{X0}}={{u1\_grcat\_1}~{X0}}})}}}\label{hyp:egjdpfhgchdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v2\_grcat\_1~X0){\unicdcdcdhd}(l1\_grcat\_1~X0)){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\\
v2\_grcat\_1~X1){\unicdcdcdhd}(l1\_grcat\_1~X1)){\unicdbdjdcd}((k7\_grcat\_1~X0=k8\_grcat\_1~\\
X1){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_grcat\_1~X2){\unicdcdcdhd}((v2\_grcat\_1~X2){\unicdcdcdhd}(l1\_grcat\_1~\\
X2))){\unicdbdjdcd}((X2=k13\_grcat\_1~X0~X1){\unicdbdjded}({\unicdcdadad} X3 .  ((\preoh v2\_struct\_0~X3){\unicdcdcdhd}\\
((v13\_algstr\_0~X3){\unicdcdcdhd}((v3\_rlvect\_1~X3){\unicdcdcdhd}((v4\_rlvect\_1~X3){\unicdcdcdhd}(l2\_algstr\_0~\\
X3))))){\unicdbdjdcd}({\unicdcdadad} X4 .  ((\preoh v2\_struct\_0~X4){\unicdcdcdhd}((v13\_algstr\_0~X4){\unicdcdcdhd}\\
((v3\_rlvect\_1~X4){\unicdcdcdhd}((v4\_rlvect\_1~X4){\unicdcdcdhd}(l2\_algstr\_0~X4))))){\unicdbdjdcd}(\\
{\unicdcdadad} X5 .  ((\preoh v2\_struct\_0~X5){\unicdcdcdhd}((v13\_algstr\_0~X5){\unicdcdcdhd}((v3\_rlvect\_1~\\
X5){\unicdcdcdhd}((v4\_rlvect\_1~X5){\unicdcdcdhd}(l2\_algstr\_0~X5))))){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~\\
X6){\unicdcdcdhd}((v1\_funct\_2~X6~(u1\_struct\_0~X4)~(u1\_struct\_0~X5)){\unicdcdcdhd}(m1\_subset\_1~\\
X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X4)~(u1\_struct\_0~X5)))))){\unicdbdjdcd}\\
({\unicdcdadad} X7 .  ((v1\_funct\_1~X7){\unicdcdcdhd}((v1\_funct\_2~X7~(u1\_struct\_0~X3)~\\
(u1\_struct\_0~X4)){\unicdcdcdhd}(m1\_subset\_1~X7~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X3)~(u1\_struct\_0~X4)))))){\unicdbdjdcd}(((g1\_grcat\_1~(u1\_grcat\_1~\\
X0)~(u2\_grcat\_1~X0)~(u3\_grcat\_1~X0)=g1\_grcat\_1~X4~X5~X6){\unicdcdcdhd}(g1\_grcat\_1~\\
(u1\_grcat\_1~X1)~(u2\_grcat\_1~X1)~(u3\_grcat\_1~X1)=g1\_grcat\_1~X3~\\
X4~X7)){\unicdbdjdcd}(X2=g1\_grcat\_1~X3~X5~(k1\_partfun1~(u1\_struct\_0~X3)~(u1\_struct\_0~\\
X4)~(u1\_struct\_0~X4)~(u1\_struct\_0~X5)~X7~X6))))))))))))\end{array}
\label{hyp:egbdgdpfhgchdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v3\_rlvect\_1~\\
X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l2\_algstr\_0~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh \\
v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~\\
X1){\unicdcdcdhd}(l2\_algstr\_0~X1))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v2\_grcat\_1~X2){\unicdcdcdhd}(l1\_grcat\_1~\\
X2)){\unicdbdjdcd}((m1\_grcat\_1~X2~X0~X1){\unicdbdjded}((k7\_grcat\_1~X2=X0){\unicdcdcdhd}(k8\_grcat\_1~\\
X2=X1)))))\end{array}
\label{hyp:egbdedpfhgchdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{l1\_grcat\_1}~{X0}})}{\unicdbdjdcd}{({{{k8\_grcat\_1}~{X0}}={{u2\_grcat\_1}~{X0}}})}}}\label{hyp:egbdadpfhgchdgbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (l1\_grcat\_1~X0){\unicdbdjdcd}((v1\_grcat\_1~X0){\unicdbdjdcd}(X0=g1\_grcat\_1~\\
(u1\_grcat\_1~X0)~(u2\_grcat\_1~X0)~(u3\_grcat\_1~X0)))\end{array}
\label{hyp:bgcgdhehchbgdgehogfgdhdhpfghbdpfhgchdgbgehpfbd}
\end{equation}
\begin{theorem}\label{thm:ehbdidpfhgchdgbgehpfbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v13\_algstr\_0~X0){\unicdcdcdhd}((v3\_rlvect\_1~\\
X0){\unicdcdcdhd}((v4\_rlvect\_1~X0){\unicdcdcdhd}(l2\_algstr\_0~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh \\
v2\_struct\_0~X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v3\_rlvect\_1~X1){\unicdcdcdhd}((v4\_rlvect\_1~\\
X1){\unicdcdcdhd}(l2\_algstr\_0~X1))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((\preoh v2\_struct\_0~X2){\unicdcdcdhd}((\\
v13\_algstr\_0~X2){\unicdcdcdhd}((v3\_rlvect\_1~X2){\unicdcdcdhd}((v4\_rlvect\_1~X2){\unicdcdcdhd}(l2\_algstr\_0~\\
X2))))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_grcat\_1~X3~X1~X2){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_grcat\_1~\\
X4~X0~X1){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v1\_funct\_1~X5){\unicdcdcdhd}((v1\_funct\_2~X5~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X2)){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(u1\_struct\_0~X1)~(u1\_struct\_0~X2)))))){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v1\_funct\_1~\\
X6){\unicdcdcdhd}((v1\_funct\_2~X6~(u1\_struct\_0~X0)~(u1\_struct\_0~X1)){\unicdcdcdhd}(m1\_subset\_1~\\
X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~X0)~(u1\_struct\_0~X1)))))){\unicdbdjdcd}\\
(((X3=g1\_grcat\_1~X1~X2~X5){\unicdcdcdhd}(X4=g1\_grcat\_1~X0~X1~X6)){\unicdbdjdcd}(k14\_grcat\_1~\\
X0~X1~X2~X3~X4=g1\_grcat\_1~X0~X2~(k1\_partfun1~(u1\_struct\_0~X0)~(u1\_struct\_0~\\
X1)~(u1\_struct\_0~X1)~(u1\_struct\_0~X2)~X6~X5)))))))))\end{array}$$
\end{theorem}
\end{document}
