\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{l60\_scmpds\_2 (TMLbkkQjqgG5aPiFRnKRxWXkE298vPhTiUH)}
\maketitle
Let $ m1\_subset\_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 $ k1\_tarski : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ k4\_numbers : {\iota}$ be given.
Let $ np\_\_14 : {\iota}$ be given.
Let $ k9\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_ami\_2 : {\iota}$ be given.
Let $ k2\_ami\_2 : {\iota}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k12\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k7\_card\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_15 : {\iota}$ be given.
Let $ k3\_scmpds\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ np\_\_3 : {\iota}$ be given.
Let $ k11\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k3\_enumset1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ np\_\_4 : {\iota}$ be given.
Let $ np\_\_5 : {\iota}$ be given.
Let $ np\_\_6 : {\iota}$ be given.
Let $ np\_\_7 : {\iota}$ be given.
Let $ np\_\_8 : {\iota}$ be given.
Let $ k2\_compos\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_compos\_1~k1\_scmpds\_2)){\unicdbdjdcd}((X0 \in \\
k1\_tarski~(k3\_xtuple\_0~k6\_numbers~k1\_xboole\_0~k1\_xboole\_0)){\unicdbdjdcd}\\
(k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0=k6\_numbers))\end{array}
\label{hyp:mgedjdpfdhdgngahegdhpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_compos\_1~k1\_scmpds\_2)){\unicdbdjdcd}(\preoh (X0 \in \\
ReplSep4~(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~(k7\_card\_1~np\_\_15)))~\\
({\uniddcgcg} X1 : \iota .  toset~({\uniddcgcg} X2 : \iota .  m2\_subset\_1~X2~k1\_ami\_2~k2\_ami\_2))~\\
({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~k4\_numbers))~\\
({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  toset~({\uniddcgcg} X4 : \iota .  m1\_subset\_1~\\
X4~k4\_numbers))~({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  {\uniddcgcg} X4 : \\
\iota .  X1 \in k3\_enumset1~np\_\_4~np\_\_5~np\_\_6~np\_\_7~np\_\_8)~({\uniddcgcg} X1 : \iota .  \\
{\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  {\uniddcgcg} X4 : \iota .  k3\_xtuple\_0~X1~k1\_xboole\_0~\\
(k11\_finseq\_1~X2~X3~X4))){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~\\
X0{\unicdcdgdad}np\_\_4){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_5){\unicdcdcdhd}\\
((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_6){\unicdcdcdhd}((k2\_compos\_0~\\
(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_7){\unicdcdcdhd}(k2\_compos\_0~(u1\_compos\_1~\\
k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_8))))))\end{array}
\label{hyp:mgedhdpfdhdgngahegdhpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_compos\_1~k1\_scmpds\_2)){\unicdbdjdcd}(\preoh (X0 \in \\
ReplSep3~(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~(k7\_card\_1~np\_\_15)))~\\
({\uniddcgcg} X1 : \iota .  toset~({\uniddcgcg} X2 : \iota .  m2\_subset\_1~X2~k1\_ami\_2~k2\_ami\_2))~\\
({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~k4\_numbers))~\\
({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  X1 \in k2\_tarski~np\_\_2~np\_\_3)~\\
({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  k3\_xtuple\_0~X1~k1\_xboole\_0~\\
(k3\_scmpds\_1~X2~X3))){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~\\
X0{\unicdcdgdad}np\_\_2){\unicdcdcdhd}(k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_3)))\end{array}
\label{hyp:mgedgdpfdhdgngahegdhpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_compos\_1~k1\_scmpds\_2)){\unicdbdjdcd}((X0 \in \\
ReplSep~(toset~({\uniddcgcg} X1 : \iota .  m2\_subset\_1~X1~k1\_ami\_2~k2\_ami\_2))~\\
({\uniddcgcg} X1 : \iota .  True)~({\uniddcgcg} X1 : \iota .  k3\_xtuple\_0~np\_\_1~k1\_xboole\_0~\\
(k12\_finseq\_1~k2\_ami\_2~X1))){\unicdbdjdcd}(k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~\\
X0=np\_\_1))\end{array}
\label{hyp:mgedfdpfdhdgngahegdhpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_compos\_1~k1\_scmpds\_2)){\unicdbdjdcd}((X0 \in \\
ReplSep~(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~k4\_numbers))~({\uniddcgcg} X1 : \\
\iota .  True)~({\uniddcgcg} X1 : \iota .  k3\_xtuple\_0~np\_\_14~k1\_xboole\_0~(k9\_finseq\_1~\\
X1))){\unicdbdjdcd}(k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0=np\_\_14))\end{array}
\label{hyp:mgededpfdhdgngahegdhpfcd}
\end{equation}
\begin{theorem}\label{thm:mggdadpfdhdgngahegdhpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~(u1\_compos\_1~k1\_scmpds\_2)){\unicdbdjdcd}(\preoh (\preoh (\\
\preoh X0 \in k1\_tarski~(k3\_xtuple\_0~k6\_numbers~k1\_xboole\_0~k1\_xboole\_0)){\unicdcdcdhd}\\
((\preoh X0 \in ReplSep~(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~k4\_numbers))~\\
({\uniddcgcg} X1 : \iota .  True)~({\uniddcgcg} X1 : \iota .  k3\_xtuple\_0~np\_\_14~k1\_xboole\_0~\\
(k9\_finseq\_1~X1))){\unicdcdcdhd}((\preoh X0 \in ReplSep~(toset~({\uniddcgcg} X1 : \iota .  m2\_subset\_1~\\
X1~k1\_ami\_2~k2\_ami\_2))~({\uniddcgcg} X1 : \iota .  True)~({\uniddcgcg} X1 : \iota .  k3\_xtuple\_0~\\
np\_\_1~k1\_xboole\_0~(k12\_finseq\_1~k2\_ami\_2~X1))){\unicdcdcdhd}((\preoh X0 \in ReplSep3~\\
(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~(k7\_card\_1~np\_\_15)))~({\uniddcgcg} X1 : \\
\iota .  toset~({\uniddcgcg} X2 : \iota .  m2\_subset\_1~X2~k1\_ami\_2~k2\_ami\_2))~({\uniddcgcg} X1 : \\
\iota .  {\uniddcgcg} X2 : \iota .  toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~k4\_numbers))~\\
({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  X1 \in k2\_tarski~np\_\_2~np\_\_3)~\\
({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  k3\_xtuple\_0~X1~k1\_xboole\_0~\\
(k3\_scmpds\_1~X2~X3))){\unicdcdcdhd}(\preoh X0 \in ReplSep4~(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~\\
X1~(k7\_card\_1~np\_\_15)))~({\uniddcgcg} X1 : \iota .  toset~({\uniddcgcg} X2 : \iota .  m2\_subset\_1~\\
X2~k1\_ami\_2~k2\_ami\_2))~({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  toset~({\uniddcgcg} X3 : \\
\iota .  m1\_subset\_1~X3~k4\_numbers))~({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \\
\iota .  toset~({\uniddcgcg} X4 : \iota .  m1\_subset\_1~X4~k4\_numbers))~({\uniddcgcg} X1 : \iota .  \\
{\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  {\uniddcgcg} X4 : \iota .  X1 \in k3\_enumset1~np\_\_4~np\_\_5~\\
np\_\_6~np\_\_7~np\_\_8)~({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  {\uniddcgcg} X4 : \\
\iota .  k3\_xtuple\_0~X1~k1\_xboole\_0~(k11\_finseq\_1~X2~X3~X4))))))){\unicdcdcdhd}\\
((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}k6\_numbers){\unicdcdcdhd}((k2\_compos\_0~\\
(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_14){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~\\
k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_1){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~\\
X0{\unicdcdgdad}np\_\_2){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_3){\unicdcdcdhd}\\
((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_4){\unicdcdcdhd}((k2\_compos\_0~\\
(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_5){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~\\
k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_6){\unicdcdcdhd}((k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~\\
X0{\unicdcdgdad}np\_\_7){\unicdcdcdhd}(k2\_compos\_0~(u1\_compos\_1~k1\_scmpds\_2)~X0{\unicdcdgdad}np\_\_8)))))))))))\end{array}$$
\end{theorem}
\end{document}
