\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{l24\_scmp\_gcd (TMUYGJJjD567VNczDNzTb4jUDxtD6CaNof5)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_scmpds\_2 : {\iota}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_memstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ v1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v5\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_compos\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k4\_scmp\_gcd : {\iota}$ be given.
Let $ k5\_memstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ np\_\_5 : {\iota}$ be given.
Let $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_scmp\_gcd : {\iota}$ be given.
Let $ k2\_scmp\_gcd : {\iota}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k2\_scmpds\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_3 : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_extpro\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ np\_\_7 : {\iota}$ be given.
Let $ k2\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_8 : {\iota}$ be given.
Let $ k2\_extpro\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k3\_scmpds\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_4 : {\iota}$ be given.
Let $ k1\_scmp\_gcd : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_int\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_6 : {\iota}$ be given.
Let $ np\_\_11 : {\iota}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k5\_numbers){\unicdbdjdcd}(k2\_scmpds\_2~X0~X1=k1\_scmp\_gcd~(k2\_nat\_1~X0~X1)))\end{array}
\label{hyp:ehbdpfdhdgngahpfhgdgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_3){\unicdcdcdhd}(m2\_subset\_1~np\_\_3~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_3~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_3~k1\_numbers))\end{array}
\label{hyp:dhahdgddpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_2){\unicdcdcdhd}(m2\_subset\_1~np\_\_2~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_2~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_2~k1\_numbers))\end{array}
\label{hyp:dhahdgcdpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_relat\_1~\\
X1){\unicdcdcdhd}((v4\_relat\_1~X1~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~\\
X1){\unicdcdcdhd}((v5\_funct\_1~X1~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}(v1\_partfun1~\\
X1~(u1\_struct\_0~k1\_scmpds\_2)))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_relat\_1~\\
X2){\unicdcdcdhd}((v4\_relat\_1~X2~k5\_numbers){\unicdcdcdhd}((v5\_relat\_1~X2~(u1\_compos\_1~\\
k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}(v1\_partfun1~X2~k5\_numbers))))){\unicdbdjdcd}\\
(((r1\_tarski~k4\_scmp\_gcd~X2){\unicdcdcdhd}((k5\_memstr\_0~np\_\_2~k1\_scmpds\_2~\\
X1=np\_\_5){\unicdcdcdhd}((X0=k1\_funct\_1~X1~k3\_scmp\_gcd){\unicdcdcdhd}(k1\_funct\_1~X1~k2\_scmp\_gcd=\\
k6\_numbers)))){\unicdbdjdcd}((r1\_xxreal\_0~(k1\_funct\_1~X1~(k2\_scmpds\_2~(k1\_funct\_1~\\
X1~k3\_scmp\_gcd)~np\_\_3))~k6\_numbers){\unicdcdcdid}((k5\_memstr\_0~np\_\_2~k1\_scmpds\_2~\\
(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X2~X1~np\_\_7)=k2\_nat\_1~np\_\_5~np\_\_7){\unicdcdcdhd}\\
((k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X2~X1~np\_\_8=k2\_extpro\_1~np\_\_2~\\
k1\_scmpds\_2~(k3\_scmpds\_2~(k4\_xcmplx\_0~np\_\_7))~(k5\_extpro\_1~np\_\_2~\\
k1\_scmpds\_2~X2~X1~np\_\_7)){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~\\
X2~X1~np\_\_7)~k3\_scmp\_gcd=k2\_nat\_1~X0~np\_\_4){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~\\
np\_\_2~k1\_scmpds\_2~X2~X1~np\_\_7)~k2\_scmp\_gcd=k6\_numbers){\unicdcdcdhd}((k1\_funct\_1~\\
(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X2~X1~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~\\
X0~np\_\_7))=k6\_int\_1~(k1\_funct\_1~X1~(k2\_scmpds\_2~(k1\_funct\_1~X1~\\
k3\_scmp\_gcd)~np\_\_2))~(k1\_funct\_1~X1~(k2\_scmpds\_2~(k1\_funct\_1~\\
X1~k3\_scmp\_gcd)~np\_\_3))){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~\\
X2~X1~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_6))=k1\_funct\_1~X1~(k2\_scmpds\_2~\\
(k1\_funct\_1~X1~k3\_scmp\_gcd)~np\_\_3)){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~\\
np\_\_2~k1\_scmpds\_2~X2~X1~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_4))=\\
X0){\unicdcdcdhd}(k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X2~X1~np\_\_7)~(k1\_scmp\_gcd~\\
(k2\_nat\_1~X0~np\_\_5))=np\_\_11))))))))))))\end{array}
\label{hyp:mgbdjdpfdhdgngahpfhgdgeg}
\end{equation}
\begin{theorem}\label{thm:mgcdedpfdhdgngahpfhgdgeg}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_relat\_1~\\
X1){\unicdcdcdhd}((v4\_relat\_1~X1~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~\\
X1){\unicdcdcdhd}((v5\_funct\_1~X1~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}(v1\_partfun1~\\
X1~(u1\_struct\_0~k1\_scmpds\_2)))))){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_relat\_1~\\
X2){\unicdcdcdhd}((v4\_relat\_1~X2~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~\\
X2){\unicdcdcdhd}((v5\_funct\_1~X2~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}(v1\_partfun1~\\
X2~(u1\_struct\_0~k1\_scmpds\_2)))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_relat\_1~\\
X3){\unicdcdcdhd}((v4\_relat\_1~X3~k5\_numbers){\unicdcdcdhd}((v5\_relat\_1~X3~(u1\_compos\_1~\\
k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X3){\unicdcdcdhd}(v1\_partfun1~X3~k5\_numbers))))){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  ((v1\_relat\_1~X4){\unicdcdcdhd}((v4\_relat\_1~X4~k5\_numbers){\unicdcdcdhd}((\\
v5\_relat\_1~X4~(u1\_compos\_1~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X4){\unicdcdcdhd}(\\
v1\_partfun1~X4~k5\_numbers))))){\unicdbdjdcd}(((r1\_tarski~k4\_scmp\_gcd~X3){\unicdcdcdhd}\\
((r1\_tarski~k4\_scmp\_gcd~X4){\unicdcdcdhd}((k5\_memstr\_0~np\_\_2~k1\_scmpds\_2~\\
X1=np\_\_5){\unicdcdcdhd}((X0=k1\_funct\_1~X1~k3\_scmp\_gcd){\unicdcdcdhd}((k1\_funct\_1~X1~k2\_scmp\_gcd=\\
k6\_numbers){\unicdcdcdhd}((k5\_memstr\_0~np\_\_2~k1\_scmpds\_2~X2=k5\_memstr\_0~\\
np\_\_2~k1\_scmpds\_2~X1){\unicdcdcdhd}((k1\_funct\_1~X2~k3\_scmp\_gcd=k1\_funct\_1~\\
X1~k3\_scmp\_gcd){\unicdcdcdhd}((k1\_funct\_1~X2~k2\_scmp\_gcd=k6\_numbers){\unicdcdcdhd}((\\
k1\_funct\_1~X2~(k2\_scmpds\_2~(k1\_funct\_1~X1~k3\_scmp\_gcd)~np\_\_2)=\\
k1\_funct\_1~X1~(k2\_scmpds\_2~(k1\_funct\_1~X1~k3\_scmp\_gcd)~np\_\_2)){\unicdcdcdhd}\\
(k1\_funct\_1~X2~(k2\_scmpds\_2~(k1\_funct\_1~X1~k3\_scmp\_gcd)~np\_\_3)=\\
k1\_funct\_1~X1~(k2\_scmpds\_2~(k1\_funct\_1~X1~k3\_scmp\_gcd)~np\_\_3))))))))))){\unicdbdjdcd}\\
((r1\_xxreal\_0~(k1\_funct\_1~X1~(k2\_scmpds\_2~(k1\_funct\_1~X1~k3\_scmp\_gcd)~\\
np\_\_3))~k6\_numbers){\unicdcdcdid}((k5\_memstr\_0~np\_\_2~k1\_scmpds\_2~(k5\_extpro\_1~\\
np\_\_2~k1\_scmpds\_2~X3~X1~np\_\_7)=k2\_nat\_1~np\_\_5~np\_\_7){\unicdcdcdhd}((k5\_extpro\_1~\\
np\_\_2~k1\_scmpds\_2~X3~X1~np\_\_8=k2\_extpro\_1~np\_\_2~k1\_scmpds\_2~(k3\_scmpds\_2~\\
(k4\_xcmplx\_0~np\_\_7))~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X3~X1~np\_\_7)){\unicdcdcdhd}\\
((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X3~X1~np\_\_7)~k3\_scmp\_gcd=\\
k2\_nat\_1~X0~np\_\_4){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~\\
X3~X1~np\_\_7)~k2\_scmp\_gcd=k6\_numbers){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~\\
np\_\_2~k1\_scmpds\_2~X3~X1~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_7))=\\
k6\_int\_1~(k1\_funct\_1~X1~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_2)))~(k1\_funct\_1~\\
X1~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_3)))){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~\\
np\_\_2~k1\_scmpds\_2~X3~X1~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_6))=\\
k1\_funct\_1~X1~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_3))){\unicdcdcdhd}((k5\_memstr\_0~\\
np\_\_2~k1\_scmpds\_2~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X4~X2~np\_\_7)=\\
k2\_nat\_1~np\_\_5~np\_\_7){\unicdcdcdhd}((k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X4~X2~np\_\_8=\\
k2\_extpro\_1~np\_\_2~k1\_scmpds\_2~(k3\_scmpds\_2~(k4\_xcmplx\_0~np\_\_7))~\\
(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X4~X2~np\_\_7)){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~\\
np\_\_2~k1\_scmpds\_2~X4~X2~np\_\_7)~k3\_scmp\_gcd=k2\_nat\_1~X0~np\_\_4){\unicdcdcdhd}\\
((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X4~X2~np\_\_7)~k2\_scmp\_gcd=\\
k6\_numbers){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X4~X2~\\
np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_7))=k6\_int\_1~(k1\_funct\_1~\\
X1~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_2)))~(k1\_funct\_1~X1~(k1\_scmp\_gcd~\\
(k2\_nat\_1~X0~np\_\_3)))){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~\\
X4~X2~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_6))=k1\_funct\_1~X1~(k1\_scmp\_gcd~\\
(k2\_nat\_1~X0~np\_\_3))){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~\\
X3~X1~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_4))=X0){\unicdcdcdhd}((k1\_funct\_1~\\
(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X3~X1~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~\\
X0~np\_\_5))=np\_\_11){\unicdcdcdhd}((k1\_funct\_1~(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~\\
X4~X2~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~X0~np\_\_4))=X0){\unicdcdcdhd}(k1\_funct\_1~\\
(k5\_extpro\_1~np\_\_2~k1\_scmpds\_2~X4~X2~np\_\_7)~(k1\_scmp\_gcd~(k2\_nat\_1~\\
X0~np\_\_5))=np\_\_11))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
