\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{t9\_vectmetr (TMG56hoE39wmLi8imssStDhuhfHrmoNhS2L)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ 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 $ k4\_vectmetr : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v3\_vectmetr : {{\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 $ k10\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v13\_algstr\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v8\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v9\_metric\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v7\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v8\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_vectmetr : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v5\_vectmetr : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v6\_vectmetr : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_vectmetr : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ r1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ u1\_metric\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k13\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_euclid : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m2\_finseq\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_binop\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ u1\_algstr\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k7\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ u1\_rlvect\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k9\_euclid : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v4\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}(v5\_relat\_1~X1~X0)){\unicdbdjdcd}(\\
k2\_relset\_1~X0~X1=k10\_xtuple\_0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfchfgmgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}((\preoh v2\_struct\_0~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v13\_algstr\_0~(k4\_vectmetr~X0)){\unicdcdcdhd}((v6\_metric\_1~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v7\_metric\_1~(k4\_vectmetr~X0)){\unicdcdcdhd}((v8\_metric\_1~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v9\_metric\_1~(k4\_vectmetr~X0)){\unicdcdcdhd}((v2\_rlvect\_1~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v3\_rlvect\_1~(k4\_vectmetr~X0)){\unicdcdcdhd}((v4\_rlvect\_1~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v5\_rlvect\_1~(k4\_vectmetr~X0)){\unicdcdcdhd}((v6\_rlvect\_1~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v7\_rlvect\_1~(k4\_vectmetr~X0)){\unicdcdcdhd}((v8\_rlvect\_1~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v4\_vectmetr~(k4\_vectmetr~X0)){\unicdcdcdhd}((v5\_vectmetr~(k4\_vectmetr~\\
X0)){\unicdcdcdhd}((v6\_vectmetr~(k4\_vectmetr~X0)){\unicdcdcdhd}(l1\_vectmetr~(k4\_vectmetr~\\
X0))))))))))))))))))\end{array}
\label{hyp:egehpflgedpfghfgdgehngfgehch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((\preoh v2\_struct\_0~\\
X1){\unicdcdcdhd}((v13\_algstr\_0~X1){\unicdcdcdhd}((v6\_metric\_1~X1){\unicdcdcdhd}((v7\_metric\_1~X1){\unicdcdcdhd}\\
((v8\_metric\_1~X1){\unicdcdcdhd}((v9\_metric\_1~X1){\unicdcdcdhd}((v2\_rlvect\_1~X1){\unicdcdcdhd}((v3\_rlvect\_1~\\
X1){\unicdcdcdhd}((v4\_rlvect\_1~X1){\unicdcdcdhd}((v5\_rlvect\_1~X1){\unicdcdcdhd}((v6\_rlvect\_1~X1){\unicdcdcdhd}\\
((v7\_rlvect\_1~X1){\unicdcdcdhd}((v8\_rlvect\_1~X1){\unicdcdcdhd}((v4\_vectmetr~X1){\unicdcdcdhd}((v5\_vectmetr~\\
X1){\unicdcdcdhd}((v6\_vectmetr~X1){\unicdcdcdhd}(l1\_vectmetr~X1))))))))))))))))){\unicdbdjdcd}((\\
X1=k4\_vectmetr~X0){\unicdbdjded}((u1\_struct\_0~X1=k1\_euclid~X0){\unicdcdcdhd}((r1\_funct\_2~\\
(k2\_zfmisc\_1~(u1\_struct\_0~X1)~(u1\_struct\_0~X1))~k1\_numbers~(k2\_zfmisc\_1~\\
(k1\_euclid~X0)~(k1\_euclid~X0))~k1\_numbers~(u1\_metric\_1~X1)~(k13\_euclid~\\
X0)){\unicdcdcdhd}((k4\_struct\_0~X1=k5\_euclid~X0){\unicdcdcdhd}(({\unicdcdadad} X2 .  (m2\_finseq\_2~\\
X2~k1\_numbers~(k1\_euclid~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_finseq\_2~X3~k1\_numbers~\\
(k1\_euclid~X0)){\unicdbdjdcd}(k1\_binop\_1~(u1\_algstr\_0~X1)~X2~X3=k7\_euclid~\\
X0~X2~X3))){\unicdcdcdhd}({\unicdcdadad} X2 .  (m2\_finseq\_2~X2~k1\_numbers~(k1\_euclid~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}(k1\_binop\_1~(u1\_rlvect\_1~\\
X1)~X3~X2=k9\_euclid~X0~X2~X3)))))))))\end{array}
\label{hyp:egidpfghfgdgehngfgehch}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X1){\unicdcdcdhd}(v5\_relat\_1~X1~X0)){\unicdbdjdcd}(\\
(v2\_funct\_2~X1~X0){\unicdbdjded}(k2\_relset\_1~X0~X1=X0))\end{array}
\label{hyp:egddpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~X0~X1))){\unicdbdjdcd}((v4\_relat\_1~X2~X0){\unicdcdcdhd}(v5\_relat\_1~X2~X1))\end{array}
\label{hyp:dgdgcdpfchfgmgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~X0~X1))){\unicdbdjdcd}(v1\_relat\_1~X2)\end{array}
\label{hyp:dgdgbdpfchfgmgdhfgehpfbd}
\end{equation}
\begin{theorem}\label{thm:ehjdpfghfgdgehngfgehch}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}((v1\_funct\_2~X1~(u1\_struct\_0~(k4\_vectmetr~X0))~(u1\_struct\_0~\\
(k4\_vectmetr~X0))){\unicdcdcdhd}((v2\_funct\_2~X1~(u1\_struct\_0~(k4\_vectmetr~\\
X0))){\unicdcdcdhd}((v3\_vectmetr~X1~(k4\_vectmetr~X0)){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~(u1\_struct\_0~(k4\_vectmetr~X0))~(u1\_struct\_0~(k4\_vectmetr~\\
X0))))))))){\unicdbdjdcd}(k10\_xtuple\_0~X1=k1\_euclid~X0))\end{array}$$
\end{theorem}
\end{document}
