\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{t28\_jordan12 (TMbDKHQhFw5h7fXE9j1DkX4kGwjfWEZDW6b)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k15\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ k1\_rltopsp1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k17\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k18\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k19\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_sppol\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_sppol\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ v2\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}((r1\_xxreal\_0~X0~X1){\unicdbdjdcd}(ReplSep~(toset~({\uniddcgcg} X3 : \\
\iota .  m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X3 : \\
\iota .  (k17\_euclid~X3=X2){\unicdcdcdhd}((r1\_xxreal\_0~X0~(k18\_euclid~X3)){\unicdcdcdhd}(r1\_xxreal\_0~\\
(k18\_euclid~X3)~X1)))~({\uniddcgcg} X3 : \iota .  X3)=k1\_rltopsp1~(k15\_euclid~\\
np\_\_2)~(k19\_euclid~X2~X0)~(k19\_euclid~X2~X1)))))\end{array}
\label{hyp:ehjdpfehpgahchfgbgmgdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((r1\_xxreal\_0~(k17\_euclid~X0)~(k17\_euclid~X1)){\unicdcdcdhd}((r1\_xxreal\_0~\\
(k17\_euclid~X1)~(k17\_euclid~X2)){\unicdcdcdhd}((k18\_euclid~X0=k18\_euclid~\\
X1){\unicdcdcdhd}(k18\_euclid~X1=k18\_euclid~X2)))){\unicdbdjdcd}(X1 \in k1\_rltopsp1~(k15\_euclid~\\
np\_\_2)~X0~X2))))\end{array}
\label{hyp:ehidpfhgpgcgpgbgcheghd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((k17\_euclid~X0=k17\_euclid~X1){\unicdcdcdhd}((k17\_euclid~X1=k17\_euclid~\\
X2){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X0)~(k18\_euclid~X1)){\unicdcdcdhd}(r1\_xxreal\_0~\\
(k18\_euclid~X1)~(k18\_euclid~X2))))){\unicdbdjdcd}(X1 \in k1\_rltopsp1~(k15\_euclid~\\
np\_\_2)~X0~X2))))\end{array}
\label{hyp:ehhdpfhgpgcgpgbgcheghd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{\preoh{{({{X0} \in {X1}})}}}{\unicdcdcdhd}{({{v1\_xboole\_0}~{X1}})}}}}\label{hyp:ehhdpfcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((k17\_euclid~X0=k17\_euclid~X1){\unicdcdcdhd}(k18\_euclid~X0=k18\_euclid~X1)){\unicdbdjdcd}\\
(X0=X1)))\end{array}
\label{hyp:ehgdpfehpgahchfgbgmgdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((X0 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X1~X2){\unicdcdcdhd}(k18\_euclid~X1=k18\_euclid~\\
X2)){\unicdbdjdcd}(k18\_euclid~X0=k18\_euclid~X2))))\end{array}
\label{hyp:ehgdpfhgpgcgpgbgcheghd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((v2\_sppol\_1~(k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X1)){\unicdcdcdhd}((v2\_sppol\_1~\\
(k1\_rltopsp1~(k15\_euclid~np\_\_2)~X2~X3)){\unicdcdcdhd}((k17\_euclid~X0=k17\_euclid~\\
X2){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X0)~(k18\_euclid~X2)){\unicdcdcdhd}((r1\_xxreal\_0~\\
(k18\_euclid~X2)~(k18\_euclid~X3)){\unicdcdcdhd}(r1\_xxreal\_0~(k18\_euclid~X3)~\\
(k18\_euclid~X1))))))){\unicdbdjdcd}(r1\_tarski~(k1\_rltopsp1~(k15\_euclid~np\_\_2)~\\
X2~X3)~(k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X1))))))\end{array}
\label{hyp:ehgdddpfhgpgcgpgbgcheghd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((X0 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X1~X2){\unicdcdcdhd}(k17\_euclid~X1=k17\_euclid~\\
X2)){\unicdbdjdcd}(k17\_euclid~X0=k17\_euclid~X2))))\end{array}
\label{hyp:ehfdpfhgpgcgpgbgcheghd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(X0=k19\_euclid~(k17\_euclid~X0)~(k18\_euclid~X0))\end{array}
\label{hyp:ehfdddpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((r1\_xxreal\_0~(k18\_euclid~X0)~(k18\_euclid~X1)){\unicdcdcdhd}(X2 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X0~X1)){\unicdbdjdcd}((r1\_xxreal\_0~(k18\_euclid~X0)~(k18\_euclid~\\
X2)){\unicdcdcdhd}(r1\_xxreal\_0~(k18\_euclid~X2)~(k18\_euclid~X1))))))\end{array}
\label{hyp:ehedpfehpgahchfgbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((X0 \in X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~\\
X2))){\unicdbdjdcd}(m1\_subset\_1~X0~X2)\end{array}
\label{hyp:ehedpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((v2\_sppol\_1~(k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X1)){\unicdcdcdhd}(X2 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X0~X1)){\unicdbdjdcd}(k17\_euclid~X0=k17\_euclid~X2))))\end{array}
\label{hyp:ehedbdpfdhahahpgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (X0 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X1~X2){\unicdcdcdhd}((X0 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X1~X3){\unicdcdcdhd}((X1{\unicdcdgdad}X0){\unicdcdcdhd}((\preoh X2 \in k1\_rltopsp1~(k15\_euclid~\\
np\_\_2)~X1~X3){\unicdcdcdhd}(\preoh X3 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X1~X2))))))))\end{array}
\label{hyp:ehedbdpfkgpgchegbgoged}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((v1\_sppol\_1~(k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X1)){\unicdcdcdhd}(X2 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X0~X1)){\unicdbdjdcd}(k18\_euclid~X0=k18\_euclid~X2))))\end{array}
\label{hyp:ehedadpfdhahahpgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(((r1\_xxreal\_0~(k17\_euclid~X0)~(k17\_euclid~X1)){\unicdcdcdhd}(X2 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X0~X1)){\unicdbdjdcd}((r1\_xxreal\_0~(k17\_euclid~X0)~(k17\_euclid~\\
X2)){\unicdcdcdhd}(r1\_xxreal\_0~(k17\_euclid~X2)~(k17\_euclid~X1))))))\end{array}
\label{hyp:ehddpfehpgahchfgbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_subset\_1~X0~(k1\_zfmisc\_1~X1)){\unicdbdjded}(r1\_tarski~\\
X0~X1)\end{array}
\label{hyp:ehddpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xxreal\_0~X2){\unicdbdjdcd}(((r1\_xxreal\_0~X0~X1){\unicdcdcdhd}(r1\_xxreal\_0~X1~X2)){\unicdbdjdcd}\\
(r1\_xxreal\_0~X0~X2))))\end{array}
\label{hyp:ehcdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_subset\_1~X0~X1){\unicdbdjdcd}((v1\_xboole\_0~X1){\unicdcdcdid}\\
(X0 \in X1))\end{array}
\label{hyp:ehcdpfdhfhcgdhfgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (\preoh X0 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X1~X2){\unicdcdcdhd}((k17\_euclid~X1=\\
k17\_euclid~X2){\unicdcdcdhd}((k17\_euclid~X2=k17\_euclid~X0){\unicdcdcdhd}((\preoh X1 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X0~X2){\unicdcdcdhd}(\preoh X2 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~\\
X0~X1)))))))\end{array}
\label{hyp:ehcdgdpfkgpgchegbgogbdcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (\preoh X0 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X1~X2){\unicdcdcdhd}((k18\_euclid~X1=\\
k18\_euclid~X2){\unicdcdcdhd}((k18\_euclid~X2=k18\_euclid~X0){\unicdcdcdhd}((\preoh X1 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X0~X2){\unicdcdcdhd}(\preoh X2 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~\\
X0~X1)))))))\end{array}
\label{hyp:ehcdfdpfkgpgchegbgogbdcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (X1 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X2){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~\\
X1)~(k18\_euclid~X0)){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X1)~(k18\_euclid~\\
X2)){\unicdcdcdhd}((X0{\unicdcdgdad}X1){\unicdcdcdhd}((X1{\unicdcdgdad}X2){\unicdcdcdhd}(\preoh (k18\_euclid~X0=k18\_euclid~X1){\unicdcdcdhd}(k18\_euclid~\\
X2=k18\_euclid~X1)))))))))\end{array}
\label{hyp:ehcdadpfdhahchfgdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (X1 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X2){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~\\
X1)~(k17\_euclid~X0)){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~X1)~(k17\_euclid~\\
X2)){\unicdcdcdhd}((X0{\unicdcdgdad}X1){\unicdcdcdhd}((X1{\unicdcdgdad}X2){\unicdcdcdhd}(\preoh (k17\_euclid~X0=k17\_euclid~X1){\unicdcdcdhd}(k17\_euclid~\\
X2=k17\_euclid~X1)))))))))\end{array}
\label{hyp:ehbdjdpfdhahchfgdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (X1 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X2){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~\\
X0)~(k18\_euclid~X1)){\unicdcdcdhd}((r1\_xxreal\_0~(k18\_euclid~X2)~(k18\_euclid~\\
X1)){\unicdcdcdhd}((X0{\unicdcdgdad}X1){\unicdcdcdhd}((X1{\unicdcdgdad}X2){\unicdcdcdhd}(\preoh (k18\_euclid~X0=k18\_euclid~X1){\unicdcdcdhd}(k18\_euclid~\\
X2=k18\_euclid~X1)))))))))\end{array}
\label{hyp:ehbdidpfdhahchfgdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (X1 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X0~X2){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~\\
X0)~(k17\_euclid~X1)){\unicdcdcdhd}((r1\_xxreal\_0~(k17\_euclid~X2)~(k17\_euclid~\\
X1)){\unicdcdcdhd}((X0{\unicdcdgdad}X1){\unicdcdcdhd}((X1{\unicdcdgdad}X2){\unicdcdcdhd}(\preoh (k17\_euclid~X0=k17\_euclid~X1){\unicdcdcdhd}(k17\_euclid~\\
X2=k17\_euclid~X1)))))))))\end{array}
\label{hyp:ehbdhdpfdhahchfgdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
((k17\_euclid~X0=k17\_euclid~X1){\unicdbdjded}(v2\_sppol\_1~(k1\_rltopsp1~(k15\_euclid~\\
np\_\_2)~X0~X1))))\end{array}
\label{hyp:ehbdgdpfdhahahpgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
((k18\_euclid~X0=k18\_euclid~X1){\unicdbdjded}(v1\_sppol\_1~(k1\_rltopsp1~(k15\_euclid~\\
np\_\_2)~X0~X1))))\end{array}
\label{hyp:ehbdfdpfdhahahpgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  (v1\_xreal\_0~X2){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(v1\_xreal\_0~X3){\unicdbdjdcd}((X0 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~(k19\_euclid~\\
X1~X2)~(k19\_euclid~X3~X2)){\unicdbdjdcd}(k18\_euclid~X0=X2)))))\end{array}
\label{hyp:ehbdcdpfehpgahchfgbgmgdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  (v1\_xreal\_0~X2){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(v1\_xreal\_0~X3){\unicdbdjdcd}((X0 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~(k19\_euclid~\\
X1~X2)~(k19\_euclid~X1~X3)){\unicdbdjdcd}(k17\_euclid~X0=X1)))))\end{array}
\label{hyp:ehbdbdpfehpgahchfgbgmgdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xreal\_0~X2){\unicdbdjdcd}((r1\_xxreal\_0~X0~X1){\unicdbdjdcd}(ReplSep~(toset~({\uniddcgcg} X3 : \\
\iota .  m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))))~({\uniddcgcg} X3 : \\
\iota .  (k18\_euclid~X3=X2){\unicdcdcdhd}((r1\_xxreal\_0~X0~(k17\_euclid~X3)){\unicdcdcdhd}(r1\_xxreal\_0~\\
(k17\_euclid~X3)~X1)))~({\uniddcgcg} X3 : \iota .  X3)=k1\_rltopsp1~(k15\_euclid~\\
np\_\_2)~(k19\_euclid~X0~X2)~(k19\_euclid~X1~X2)))))\end{array}
\label{hyp:ehbdadpfehpgahchfgbgmgdd}
\end{equation}
Assume the following.
\begin{equation}
{{v3\_membered}~{k1\_numbers}}\label{hyp:ggdgddpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(m1\_subset\_1~(k18\_euclid~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdidpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(m1\_subset\_1~(k17\_euclid~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdhdpffgfhdgmgjgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(\\
(r1\_xxreal\_0~X0~X1){\unicdcdcdid}(r1\_xxreal\_0~X1~X0))\end{array}
\label{hyp:dgpgogogfgdgehfgegogfgdhdhpfchbdpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v3\_membered}~{X0}})}{\unicdbdjdcd}{({{v2\_membered}~{X0}})}}}\label{hyp:dgdgedpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v2\_membered~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
(v1\_xxreal\_0~X1))\end{array}
\label{hyp:dgdgbdddpfngfgngcgfgchfgeg}
\end{equation}
\begin{theorem}\label{thm:ehcdidpfkgpgchegbgogbdcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~(k15\_euclid~np\_\_2))){\unicdbdjdcd}\\
(\preoh (\preoh X3 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~X1~X2){\unicdcdcdhd}((X0 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X1~X2){\unicdcdcdhd}((X0{\unicdcdgdad}X1){\unicdcdcdhd}((X0{\unicdcdgdad}X2){\unicdcdcdhd}((((k17\_euclid~\\
X1=k17\_euclid~X2){\unicdcdcdhd}(k17\_euclid~X2=k17\_euclid~X3)){\unicdcdcdid}((k18\_euclid~\\
X1=k18\_euclid~X2){\unicdcdcdhd}(k18\_euclid~X2=k18\_euclid~X3))){\unicdcdcdhd}((\preoh X1 \in k1\_rltopsp1~\\
(k15\_euclid~np\_\_2)~X3~X0){\unicdcdcdhd}(\preoh X2 \in k1\_rltopsp1~(k15\_euclid~np\_\_2)~\\
X3~X0))))))))))\end{array}$$
\end{theorem}
\end{document}
