\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{t14\_scmpds\_8 (TMUXC3xcE9GZcN43cRVeAY8FMh2vFjwJdg1)}
\maketitle
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ v5\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_compos\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_scmpds\_2 : {\iota}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ 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 $ v5\_memstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_finset\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_afinsq\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_compos\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v3\_scmpds\_4 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_ami\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v1\_int\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k4\_card\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_recdef\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_scmpds\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_scmpds\_4 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ r1\_scmpds\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r2\_scmpds\_6 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k8\_memstr\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k1\_scmpds\_8 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ 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 .  ((v1\_ami\_2~X0){\unicdcdcdhd}(m1\_subset\_1~X0~(u1\_struct\_0~k1\_scmpds\_2))){\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}(k1\_funct\_1~\\
(k8\_memstr\_0~np\_\_2~k1\_scmpds\_2~X1)~X0=k1\_funct\_1~X1~X0))\end{array}
\label{hyp:ehbdfdpfdhdgngahegdhpffd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 : \iota{\unicdbdjdcd}o .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  {\unicdcdadad} X4 .  \\
{\unicdcdadad} X5 .  {\unicdcdadad} X6 : \iota{\unicdbdjdcd}\iota .  (({\unicdcdadad} X7 .  m2\_subset\_1~(X6~X7)~\\
k1\_numbers~k5\_numbers){\unicdcdcdhd}(((v1\_relat\_1~X5){\unicdcdcdhd}((v4\_relat\_1~X5~(u1\_struct\_0~\\
k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X5){\unicdcdcdhd}((v5\_funct\_1~X5~(k2\_memstr\_0~\\
np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_partfun1~X5~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}\\
(v5\_memstr\_0~X5~np\_\_2~k1\_scmpds\_2~k6\_numbers)))))){\unicdcdcdhd}(((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))))){\unicdcdcdhd}\\
(((\preoh v1\_xboole\_0~X3){\unicdcdcdhd}((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\_finset\_1~X3){\unicdcdcdhd}((v1\_afinsq\_1~X3){\unicdcdcdhd}((v2\_compos\_1~X3~k1\_scmpds\_2){\unicdcdcdhd}\\
(v3\_scmpds\_4~X3))))))))){\unicdcdcdhd}(((v1\_ami\_2~X2){\unicdcdcdhd}(m1\_subset\_1~X2~(u1\_struct\_0~\\
k1\_scmpds\_2))){\unicdcdcdhd}(v1\_int\_1~X1)))))){\unicdbdjdcd}(((\preoh r1\_xxreal\_0~k6\_numbers~\\
(k1\_funct\_1~X5~(k2\_scmpds\_2~(k1\_funct\_1~X5~X2)~X1))){\unicdcdcdhd}(({\unicdcdadad} X7 .  \\
((v1\_relat\_1~X7){\unicdcdcdhd}((v4\_relat\_1~X7~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}\\
((v1\_funct\_1~X7){\unicdcdcdhd}((v5\_funct\_1~X7~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}\\
((v1\_partfun1~X7~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}(v5\_memstr\_0~X7~\\
np\_\_2~k1\_scmpds\_2~k6\_numbers)))))){\unicdbdjdcd}(((X0~X7){\unicdcdcdhd}(X6~X7=k6\_numbers)){\unicdbdjdcd}\\
(r1\_xxreal\_0~k6\_numbers~(k1\_funct\_1~X7~(k2\_scmpds\_2~(k1\_funct\_1~\\
X5~X2)~X1))))){\unicdcdcdhd}((X0~X5){\unicdcdcdhd}({\unicdcdadad} X7 .  ((v1\_relat\_1~X7){\unicdcdcdhd}((v4\_relat\_1~\\
X7~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X7){\unicdcdcdhd}((v5\_funct\_1~\\
X7~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_partfun1~X7~(u1\_struct\_0~\\
k1\_scmpds\_2)){\unicdcdcdhd}(v5\_memstr\_0~X7~np\_\_2~k1\_scmpds\_2~k6\_numbers)))))){\unicdbdjdcd}\\
({\unicdcdadad} X8 .  ((v1\_relat\_1~X8){\unicdcdcdhd}((v4\_relat\_1~X8~k5\_numbers){\unicdcdcdhd}((\\
v5\_relat\_1~X8~(u1\_compos\_1~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X8){\unicdcdcdhd}(\\
v1\_partfun1~X8~k5\_numbers))))){\unicdbdjdcd}(((X0~X7){\unicdcdcdhd}(k1\_funct\_1~X7~X2=k1\_funct\_1~\\
X5~X2)){\unicdbdjdcd}((r1\_xxreal\_0~k6\_numbers~(k1\_funct\_1~X7~(k2\_scmpds\_2~\\
(k1\_funct\_1~X5~X2)~X1))){\unicdcdcdid}((k1\_funct\_1~(k6\_scmpds\_4~X3~X7~X8)~X2=\\
k1\_funct\_1~X7~X2){\unicdcdcdhd}((r1\_scmpds\_6~X3~X7~X8){\unicdcdcdhd}((r2\_scmpds\_6~X3~X7~X8){\unicdcdcdhd}\\
((\preoh r1\_xxreal\_0~(X6~X7)~(X6~(k8\_memstr\_0~np\_\_2~k1\_scmpds\_2~(k6\_scmpds\_4~\\
X3~X7~X8)))){\unicdcdcdhd}(X0~(k8\_memstr\_0~np\_\_2~k1\_scmpds\_2~(k6\_scmpds\_4~X3~\\
X7~X8)))))))))))))){\unicdbdjdcd}(k6\_scmpds\_4~(k1\_scmpds\_8~X2~X1~X3)~X5~X4=\\
k6\_scmpds\_4~(k1\_scmpds\_8~X2~X1~X3)~(k8\_memstr\_0~np\_\_2~k1\_scmpds\_2~\\
(k6\_scmpds\_4~X3~X5~X4))~X4))\end{array}
\label{hyp:dhcdpfdhdgngahegdhpfid}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((v1\_relat\_1~\\
X0){\unicdcdcdhd}((v4\_relat\_1~X0~k5\_numbers){\unicdcdcdhd}((v5\_relat\_1~X0~(u1\_compos\_1~\\
k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_finset\_1~X0){\unicdcdcdhd}(v1\_afinsq\_1~\\
X0))))))){\unicdcdcdhd}(((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)))))){\unicdcdcdhd}((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}\\
((v1\_relat\_1~(k6\_scmpds\_4~X0~X1~X2)){\unicdcdcdhd}((v4\_relat\_1~(k6\_scmpds\_4~\\
X0~X1~X2)~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~(k6\_scmpds\_4~\\
X0~X1~X2)){\unicdcdcdhd}((v5\_funct\_1~(k6\_scmpds\_4~X0~X1~X2)~(k2\_memstr\_0~np\_\_2~\\
k1\_scmpds\_2)){\unicdcdcdhd}(v1\_partfun1~(k6\_scmpds\_4~X0~X1~X2)~(u1\_struct\_0~\\
k1\_scmpds\_2))))))\end{array}
\label{hyp:egehpflggdpfdhdgngahegdhpfed}
\end{equation}
\begin{theorem}\label{thm:ehbdedpfdhdgngahegdhpfid}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_relat\_1~X0){\unicdcdcdhd}((v4\_relat\_1~X0~k5\_numbers){\unicdcdcdhd}((v5\_relat\_1~\\
X0~(u1\_compos\_1~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X0){\unicdcdcdhd}(v1\_partfun1~\\
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)){\unicdcdcdhd}(v5\_memstr\_0~X1~np\_\_2~k1\_scmpds\_2~k6\_numbers)))))){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  ((\preoh v1\_xboole\_0~X2){\unicdcdcdhd}((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\_finset\_1~X2){\unicdcdcdhd}((v1\_afinsq\_1~X2){\unicdcdcdhd}((v2\_compos\_1~\\
X2~k1\_scmpds\_2){\unicdcdcdhd}(v3\_scmpds\_4~X2))))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_ami\_2~\\
X3){\unicdcdcdhd}(m1\_subset\_1~X3~(u1\_struct\_0~k1\_scmpds\_2))){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
(v1\_int\_1~X4){\unicdbdjdcd}({\unicdcdadad} X5 .  {\unicdcdadad} X6 .  ((v1\_funct\_1~X6){\unicdcdcdhd}((v1\_funct\_2~\\
X6~(k4\_card\_3~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2))~k5\_numbers){\unicdcdcdhd}(\\
m1\_subset\_1~X6~(k1\_zfmisc\_1~(k2\_zfmisc\_1~(k4\_card\_3~(k2\_memstr\_0~\\
np\_\_2~k1\_scmpds\_2))~k5\_numbers))))){\unicdbdjdcd}((({\unicdcdadad} X7 .  ((v1\_relat\_1~\\
X7){\unicdcdcdhd}((v4\_relat\_1~X7~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~\\
X7){\unicdcdcdhd}((v5\_funct\_1~X7~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}((v1\_partfun1~\\
X7~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}(v5\_memstr\_0~X7~np\_\_2~k1\_scmpds\_2~\\
k6\_numbers)))))){\unicdbdjdcd}((k1\_recdef\_1~X6~X7=k6\_numbers){\unicdbdjdcd}(r1\_xxreal\_0~\\
k6\_numbers~(k1\_funct\_1~X7~(k2\_scmpds\_2~(k1\_funct\_1~X1~X3)~X4))))){\unicdcdcdhd}\\
({\unicdcdadad} X7 .  ((v1\_relat\_1~X7){\unicdcdcdhd}((v4\_relat\_1~X7~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}\\
((v1\_funct\_1~X7){\unicdcdcdhd}((v5\_funct\_1~X7~(k2\_memstr\_0~np\_\_2~k1\_scmpds\_2)){\unicdcdcdhd}\\
((v1\_partfun1~X7~(u1\_struct\_0~k1\_scmpds\_2)){\unicdcdcdhd}(v5\_memstr\_0~X7~\\
np\_\_2~k1\_scmpds\_2~k6\_numbers)))))){\unicdbdjdcd}({\unicdcdadad} X8 .  ((v1\_relat\_1~\\
X8){\unicdcdcdhd}((v4\_relat\_1~X8~k5\_numbers){\unicdcdcdhd}((v5\_relat\_1~X8~(u1\_compos\_1~\\
k1\_scmpds\_2)){\unicdcdcdhd}((v1\_funct\_1~X8){\unicdcdcdhd}(v1\_partfun1~X8~k5\_numbers))))){\unicdbdjdcd}\\
((({\unicdcdadad} X9 .  ((v1\_ami\_2~X9){\unicdcdcdhd}(m1\_subset\_1~X9~(u1\_struct\_0~k1\_scmpds\_2))){\unicdbdjdcd}\\
((X9 \in X5){\unicdbdjdcd}(k1\_funct\_1~X7~X9=k1\_funct\_1~X1~X9))){\unicdcdcdhd}(k1\_funct\_1~X7~\\
X3=k1\_funct\_1~X1~X3)){\unicdbdjdcd}((r1\_xxreal\_0~k6\_numbers~(k1\_funct\_1~X7~\\
(k2\_scmpds\_2~(k1\_funct\_1~X1~X3)~X4))){\unicdcdcdid}((k1\_funct\_1~(k6\_scmpds\_4~\\
X2~X7~X8)~X3=k1\_funct\_1~X7~X3){\unicdcdcdhd}((r1\_scmpds\_6~X2~X7~X8){\unicdcdcdhd}((r2\_scmpds\_6~\\
X2~X7~X8){\unicdcdcdhd}((\preoh r1\_xxreal\_0~(k1\_recdef\_1~X6~X7)~(k1\_recdef\_1~X6~(k8\_memstr\_0~\\
np\_\_2~k1\_scmpds\_2~(k6\_scmpds\_4~X2~X7~X8)))){\unicdcdcdhd}({\unicdcdadad} X9 .  ((v1\_ami\_2~\\
X9){\unicdcdcdhd}(m1\_subset\_1~X9~(u1\_struct\_0~k1\_scmpds\_2))){\unicdbdjdcd}((X9 \in X5){\unicdbdjdcd}(\\
k1\_funct\_1~(k6\_scmpds\_4~X2~X7~X8)~X9=k1\_funct\_1~X7~X9)))))))))))){\unicdbdjdcd}\\
((r1\_xxreal\_0~k6\_numbers~(k1\_funct\_1~X1~(k2\_scmpds\_2~(k1\_funct\_1~\\
X1~X3)~X4))){\unicdcdcdid}(k6\_scmpds\_4~(k1\_scmpds\_8~X3~X4~X2)~X1~X0=k6\_scmpds\_4~\\
(k1\_scmpds\_8~X3~X4~X2)~(k8\_memstr\_0~np\_\_2~k1\_scmpds\_2~(k6\_scmpds\_4~\\
X2~X1~X0))~X0))))))))\end{array}$$
\end{theorem}
\end{document}
