\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{t26\_pdiff\_9 (TMKY3M58QE2HZvdMmp22Q3Nu226tLjsAyid)}
\maketitle
Let $ v1\_xboole\_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.
Let $ k5\_numbers : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_pdiff\_7 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ r2\_pdiff\_7 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}$ be given.
Let $ r4\_pdiff\_7 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k2\_pdiff\_7 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ r1\_pdiff\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ m2\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k12\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_pdiff\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k8\_pdiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k11\_binop\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_real\_ns1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r2\_ndiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ r3\_nfcont\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k16\_lopban\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_ndiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ v3\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_normsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_normsp\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}}$ be given.
Let $ u1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_real\_ns1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_real\_ns1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_normsp\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_real\_ns1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(m2\_subset\_1~X0~k1\_numbers~k5\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(m2\_subset\_1~X1~k1\_numbers~k5\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k1\_euclid~X0)~(k1\_euclid~X1))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}\\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~(k4\_real\_ns1~\\
X0))~(u1\_struct\_0~(k4\_real\_ns1~X1)))))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(k1\_euclid~X0))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~\\
(k1\_zfmisc\_1~(u1\_struct\_0~(k4\_real\_ns1~X0)))){\unicdbdjdcd}(((v1\_pdiff\_7~\\
X4~X0){\unicdcdcdhd}((r1\_tarski~X4~(k1\_relset\_1~(k1\_euclid~X0)~X2)){\unicdcdcdhd}((X3=X2){\unicdcdcdhd}\\
(X4=X5)))){\unicdbdjdcd}((((r2\_ndiff\_1~X5~(k4\_real\_ns1~X0)~(k4\_real\_ns1~X1)~\\
X3){\unicdcdcdhd}(r3\_nfcont\_1~(k4\_real\_ns1~X0)~(k16\_lopban\_1~(k4\_real\_ns1~\\
X0)~(k4\_real\_ns1~X1))~(k4\_ndiff\_1~(k4\_real\_ns1~X0)~(k4\_real\_ns1~\\
X1)~X3~X5)~X5)){\unicdbdjdcd}((r1\_pdiff\_6~X4~X1~X0~X2){\unicdcdcdhd}({\unicdcdadad} X6 .  (m2\_finseq\_2~\\
X6~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~X7~k1\_numbers){\unicdbdjdcd}\\
(\preoh (X6 \in X4){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X7~k6\_numbers){\unicdcdcdhd}({\unicdcdadad} X8 .  (m1\_subset\_1~\\
X8~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X8~k6\_numbers){\unicdcdcdhd}({\unicdcdadad} X9 .  (\\
m2\_finseq\_2~X9~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}((X9 \in X4){\unicdbdjdcd}((r1\_xxreal\_0~\\
X8~(k12\_euclid~(k8\_euclid~X0~X9~X6))){\unicdcdcdid}({\unicdcdadad} X10 .  (m2\_finseq\_2~\\
X10~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}(r1\_xxreal\_0~(k12\_euclid~(k8\_euclid~\\
X1~(k1\_pdiff\_6~X0~X1~(k8\_pdiff\_1~X0~X1~X2~X9)~X10)~(k1\_pdiff\_6~X0~X1~\\
(k8\_pdiff\_1~X0~X1~X2~X6)~X10)))~(k11\_binop\_2~X7~(k12\_euclid~X10))))))))))))))){\unicdcdcdhd}\\
(((r1\_pdiff\_6~X4~X1~X0~X2){\unicdcdcdhd}({\unicdcdadad} X6 .  (m2\_finseq\_2~X6~k1\_numbers~\\
(k1\_euclid~X0)){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~X7~k1\_numbers){\unicdbdjdcd}(\preoh (\\
X6 \in X4){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X7~k6\_numbers){\unicdcdcdhd}({\unicdcdadad} X8 .  (m1\_subset\_1~\\
X8~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X8~k6\_numbers){\unicdcdcdhd}({\unicdcdadad} X9 .  (\\
m2\_finseq\_2~X9~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}((X9 \in X4){\unicdbdjdcd}((r1\_xxreal\_0~\\
X8~(k12\_euclid~(k8\_euclid~X0~X9~X6))){\unicdcdcdid}({\unicdcdadad} X10 .  (m2\_finseq\_2~\\
X10~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}(r1\_xxreal\_0~(k12\_euclid~(k8\_euclid~\\
X1~(k1\_pdiff\_6~X0~X1~(k8\_pdiff\_1~X0~X1~X2~X9)~X10)~(k1\_pdiff\_6~X0~X1~\\
(k8\_pdiff\_1~X0~X1~X2~X6)~X10)))~(k11\_binop\_2~X7~(k12\_euclid~X10)))))))))))))){\unicdbdjdcd}\\
((r2\_ndiff\_1~X5~(k4\_real\_ns1~X0)~(k4\_real\_ns1~X1)~X3){\unicdcdcdhd}(r3\_nfcont\_1~\\
(k4\_real\_ns1~X0)~(k16\_lopban\_1~(k4\_real\_ns1~X0)~(k4\_real\_ns1~\\
X1))~(k4\_ndiff\_1~(k4\_real\_ns1~X0)~(k4\_real\_ns1~X1)~X3~X5)~X5))))))))))\end{array}
\label{hyp:ehcdfdpfahegjgggggpfjd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(m2\_subset\_1~X0~k1\_numbers~k5\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(m2\_subset\_1~X1~k1\_numbers~k5\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
(k1\_euclid~X0)~(k1\_euclid~X1))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}\\
(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(u1\_struct\_0~(k4\_real\_ns1~\\
X0))~(u1\_struct\_0~(k4\_real\_ns1~X1)))))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(k1\_euclid~X0))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~\\
(k1\_zfmisc\_1~(u1\_struct\_0~(k4\_real\_ns1~X0)))){\unicdbdjdcd}(((X4=X5){\unicdcdcdhd}((\\
v1\_pdiff\_7~X4~X0){\unicdcdcdhd}(X2=X3))){\unicdbdjdcd}(({\unicdcdadad} X6 .  (m2\_subset\_1~X6~k1\_numbers~\\
k5\_numbers){\unicdbdjdcd}(((r1\_xxreal\_0~np\_\_1~X6){\unicdcdcdhd}(r1\_xxreal\_0~X6~X0)){\unicdbdjdcd}(\\
(r2\_pdiff\_7~X0~X1~X6~X2~X4){\unicdcdcdhd}(r4\_pdiff\_7~X0~X1~(k2\_pdiff\_7~X0~X1~X6~\\
X2~X4)~X4)))){\unicdbdjded}((r2\_ndiff\_1~X5~(k4\_real\_ns1~X0)~(k4\_real\_ns1~X1)~\\
X3){\unicdcdcdhd}(r3\_nfcont\_1~(k4\_real\_ns1~X0)~(k16\_lopban\_1~(k4\_real\_ns1~\\
X0)~(k4\_real\_ns1~X1))~(k4\_ndiff\_1~(k4\_real\_ns1~X0)~(k4\_real\_ns1~\\
X1)~X3~X5)~X5)))))))))\end{array}
\label{hyp:ehcdedpfahegjgggggpfjd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}\\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~\\
X2~X0~X1){\unicdbdjded}(m1\_subset\_1~X2~X1))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngcdpfdhfhcgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{({\preoh{{{v1\_xboole\_0}~{k4\_ordinal1}}}})}{\unicdcdcdhd}{({{v3\_ordinal1}~{k4\_ordinal1}})}}\label{hyp:ggdggdpfpgchegjgogbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(v7\_ordinal1~X0)){\unicdbdjdcd}((\preoh v2\_struct\_0~\\
(k4\_real\_ns1~X0)){\unicdcdcdhd}((\preoh v7\_struct\_0~(k4\_real\_ns1~X0)){\unicdcdcdhd}(v1\_normsp\_1~\\
(k4\_real\_ns1~X0))))\end{array}
\label{hyp:ggdgcdpfchfgbgmgpfogdhbd}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{v1\_xboole\_0}~{k1\_numbers}}}}\label{hyp:ggdgbdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{{m1\_subset\_1}~{k5\_numbers}}~{({{k1\_zfmisc\_1}~{k1\_numbers}})}}\label{hyp:egehpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v7\_ordinal1~X0){\unicdbdjdcd}((\preoh v2\_struct\_0~(k4\_real\_ns1~X0)){\unicdcdcdhd}\\
((v1\_normsp\_1~(k4\_real\_ns1~X0)){\unicdcdcdhd}(l1\_normsp\_1~(k4\_real\_ns1~X0))))\end{array}
\label{hyp:egehpflgedpfchfgbgmgpfogdhbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v7\_ordinal1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~X1){\unicdcdcdhd}\\
((v1\_normsp\_1~X1){\unicdcdcdhd}(l1\_normsp\_1~X1))){\unicdbdjdcd}((X1=k4\_real\_ns1~X0){\unicdbdjded}\\
((u1\_struct\_0~X1=k1\_euclid~X0){\unicdcdcdhd}((k4\_struct\_0~X1=k5\_euclid~X0){\unicdcdcdhd}\\
((r1\_funct\_2~(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X1))~\\
(u1\_struct\_0~X1)~(k2\_zfmisc\_1~(k1\_euclid~X0)~(k1\_euclid~X0))~(\\
k1\_euclid~X0)~(u1\_algstr\_0~X1)~(k1\_real\_ns1~X0)){\unicdcdcdhd}((r1\_funct\_2~\\
(k2\_zfmisc\_1~k1\_numbers~(u1\_struct\_0~X1))~(u1\_struct\_0~X1)~(k2\_zfmisc\_1~\\
k1\_numbers~(k1\_euclid~X0))~(k1\_euclid~X0)~(u1\_rlvect\_1~X1)~(k2\_real\_ns1~\\
X0)){\unicdcdcdhd}(r1\_funct\_2~(u1\_struct\_0~X1)~k1\_numbers~(k1\_euclid~X0)~k1\_numbers~\\
(u1\_normsp\_0~X1)~(k3\_real\_ns1~X0))))))))\end{array}
\label{hyp:egedpfchfgbgmgpfogdhbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k4\_ordinal1}})}{\unicdbdjdcd}{({{v7\_ordinal1}~{X0}})}}}\label{hyp:dgdgidpfpgchegjgogbgmgbd}
\end{equation}
\begin{theorem}\label{thm:ehcdgdpfahegjgggggpfjd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}(m2\_subset\_1~X0~k1\_numbers~k5\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}(m2\_subset\_1~X1~k1\_numbers~k5\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~(k1\_euclid~X0))){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
((v1\_funct\_1~X3){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(\\
k1\_euclid~X0)~(k1\_euclid~X1))))){\unicdbdjdcd}(((v1\_pdiff\_7~X2~X0){\unicdcdcdhd}(r1\_tarski~\\
X2~(k1\_relset\_1~(k1\_euclid~X0)~X3))){\unicdbdjdcd}(({\unicdcdadad} X4 .  (m2\_subset\_1~\\
X4~k1\_numbers~k5\_numbers){\unicdbdjdcd}(((r1\_xxreal\_0~np\_\_1~X4){\unicdcdcdhd}(r1\_xxreal\_0~\\
X4~X0)){\unicdbdjdcd}((r2\_pdiff\_7~X0~X1~X4~X3~X2){\unicdcdcdhd}(r4\_pdiff\_7~X0~X1~(k2\_pdiff\_7~\\
X0~X1~X4~X3~X2)~X2)))){\unicdbdjded}((r1\_pdiff\_6~X2~X1~X0~X3){\unicdcdcdhd}({\unicdcdadad} X4 .  (m2\_finseq\_2~\\
X4~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~k1\_numbers){\unicdbdjdcd}\\
(\preoh (X4 \in X2){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X5~k6\_numbers){\unicdcdcdhd}({\unicdcdadad} X6 .  (m1\_subset\_1~\\
X6~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X6~k6\_numbers){\unicdcdcdhd}({\unicdcdadad} X7 .  (\\
m2\_finseq\_2~X7~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}((X7 \in X2){\unicdbdjdcd}((r1\_xxreal\_0~\\
X6~(k12\_euclid~(k8\_euclid~X0~X7~X4))){\unicdcdcdid}({\unicdcdadad} X8 .  (m2\_finseq\_2~\\
X8~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}(r1\_xxreal\_0~(k12\_euclid~(k8\_euclid~\\
X1~(k1\_pdiff\_6~X0~X1~(k8\_pdiff\_1~X0~X1~X3~X7)~X8)~(k1\_pdiff\_6~X0~X1~\\
(k8\_pdiff\_1~X0~X1~X3~X4)~X8)))~(k11\_binop\_2~X5~(k12\_euclid~X8)))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
