\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{l30\_scmpds\_i (TMJBpg6PFwzM1USB11nUAkASrY1b2Eb5rC4)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_scmpds\_i : {\iota}$ be given.
Let $ k2\_compos\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_9 : {\iota}$ be given.
Let $ np\_\_10 : {\iota}$ be given.
Let $ np\_\_11 : {\iota}$ be given.
Let $ np\_\_12 : {\iota}$ be given.
Let $ np\_\_13 : {\iota}$ be given.
Let $ k5\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_xboole\_0 : {\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 $ k4\_xtuple\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_numbers : {\iota}$ be given.
Let $ np\_\_14 : {\iota}$ be given.
Let $ k12\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_scm\_inst : {\iota}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ k7\_card\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_15 : {\iota}$ be given.
Let $ k2\_scmpds\_i : {{\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 $ k7\_finseq\_4 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k4\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_scmpds\_i){\unicdbdjdcd}(\preoh (\preoh (X0 \in k1\_tarski~(\\
k3\_xtuple\_0~k6\_numbers~k1\_xboole\_0~k1\_xboole\_0)){\unicdcdcdhd}(k4\_xtuple\_0~\\
X0=k6\_numbers)){\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~(k12\_finseq\_1~k4\_numbers~X1))){\unicdcdcdhd}(k4\_xtuple\_0~X0=\\
np\_\_14)){\unicdcdcdhd}((\preoh (X0 \in ReplSep~(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~k2\_scm\_inst))~\\
({\uniddcgcg} X1 : \iota .  True)~({\uniddcgcg} X1 : \iota .  k3\_xtuple\_0~np\_\_1~k1\_xboole\_0~\\
(k12\_finseq\_1~k2\_scm\_inst~X1))){\unicdcdcdhd}(k4\_xtuple\_0~X0=np\_\_1)){\unicdcdcdhd}((\\
\preoh (X0 \in ReplSep3~(toset~({\uniddcgcg} X1 : \iota .  m2\_subset\_1~X1~k4\_ordinal1~\\
(k7\_card\_1~np\_\_15)))~({\uniddcgcg} X1 : \iota .  toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~\\
X2~k2\_scm\_inst))~({\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~(k2\_scmpds\_i~X2~X3))){\unicdcdcdhd}((k4\_xtuple\_0~\\
X0=np\_\_2){\unicdcdcdid}(k4\_xtuple\_0~X0=np\_\_3))){\unicdcdcdhd}((\preoh (X0 \in ReplSep4~(toset~\\
({\uniddcgcg} X1 : \iota .  m2\_subset\_1~X1~k4\_ordinal1~(k7\_card\_1~np\_\_15)))~\\
({\uniddcgcg} X1 : \iota .  toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~X2~k2\_scm\_inst))~\\
({\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}(\preoh (k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_4){\unicdcdcdhd}((k4\_xtuple\_0~\\
X0{\unicdcdgdad}np\_\_5){\unicdcdcdhd}((k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_6){\unicdcdcdhd}((k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_7){\unicdcdcdhd}\\
(k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_8)))))){\unicdcdcdhd}(\preoh (X0 \in ReplSep5~(toset~({\uniddcgcg} X1 : \\
\iota .  m2\_subset\_1~X1~k4\_ordinal1~(k7\_card\_1~np\_\_15)))~({\uniddcgcg} X1 : \\
\iota .  toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~X2~k2\_scm\_inst))~({\uniddcgcg} X1 : \\
\iota .  {\uniddcgcg} X2 : \iota .  toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~k2\_scm\_inst))~\\
({\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 .  toset~({\uniddcgcg} X5 : \iota .  m1\_subset\_1~X5~k4\_numbers))~({\uniddcgcg} X1 : \iota .  \\
{\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \iota .  {\uniddcgcg} X4 : \iota .  {\uniddcgcg} X5 : \iota .  X1 \in k3\_enumset1~\\
np\_\_9~np\_\_10~np\_\_11~np\_\_12~np\_\_13)~({\uniddcgcg} X1 : \iota .  {\uniddcgcg} X2 : \iota .  {\uniddcgcg} X3 : \\
\iota .  {\uniddcgcg} X4 : \iota .  {\uniddcgcg} X5 : \iota .  k3\_xtuple\_0~X1~k1\_xboole\_0~(k7\_finseq\_4~\\
X2~X3~X4~X5))){\unicdcdcdhd}(\preoh (k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_9){\unicdcdcdhd}((k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_10){\unicdcdcdhd}\\
((k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_11){\unicdcdcdhd}((k4\_xtuple\_0~X0{\unicdcdgdad}np\_\_12){\unicdcdcdhd}(k4\_xtuple\_0~\\
X0{\unicdcdgdad}np\_\_13)))))))))))\end{array}
\label{hyp:ehidpfdhdgngahegdhpfjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (k4\_xboole\_0~(k1\_tarski~X0)~X1=k1\_xboole\_0){\unicdbdjded}\\
(X0 \in X1)\end{array}
\label{hyp:ehgdadpfkhggngjgdhdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (k4\_xboole\_0~(k1\_tarski~X0)~(k1\_tarski~\\
X1)=k1\_xboole\_0){\unicdbdjdcd}(X0=X1)\end{array}
\label{hyp:ehbdfdpfkhggngjgdhdgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k6\_numbers}={k1\_xboole\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflggdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  k5\_xtuple\_0~(k3\_xtuple\_0~X0~\\
X1~X2)=X1\end{array}
\label{hyp:chegfdpfihehfhahmgfgpfad}
\end{equation}
\begin{theorem}\label{thm:mgddadpfdhdgngahegdhpfjg}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_scmpds\_i){\unicdbdjdcd}((\preoh (k2\_compos\_0~k1\_scmpds\_i~\\
X0{\unicdcdgdad}np\_\_9){\unicdcdcdhd}((k2\_compos\_0~k1\_scmpds\_i~X0{\unicdcdgdad}np\_\_10){\unicdcdcdhd}((k2\_compos\_0~\\
k1\_scmpds\_i~X0{\unicdcdgdad}np\_\_11){\unicdcdcdhd}((k2\_compos\_0~k1\_scmpds\_i~X0{\unicdcdgdad}np\_\_12){\unicdcdcdhd}\\
(k2\_compos\_0~k1\_scmpds\_i~X0{\unicdcdgdad}np\_\_13))))){\unicdbdjdcd}(k5\_xtuple\_0~X0=k1\_xboole\_0))\end{array}$$
\end{theorem}
\end{document}
