\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\_arytm\_2 (TMK4bM8Lm6r1XZ6nXJaNDYsWi4yi1rHJGZU)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k2\_arytm\_2 : {\iota}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k3\_arytm\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_arytm\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k11\_arytm\_3 : {\iota}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ k5\_arytm\_3 : {\iota}$ be given.
Let $ r3\_arytm\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_arytm\_2 : {\iota}$ be given.
Let $ k2\_xboole\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ k4\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_arytm\_3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k1\_tarski : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_subset\_1~X2~(k1\_zfmisc\_1~X0)){\unicdbdjdcd}(({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~X0){\unicdbdjdcd}((X3 \in X1){\unicdbdjdcd}(X3 \in X2))){\unicdbdjdcd}(r1\_tarski~X1~X2)))\end{array}
\label{hyp:ehcdpfdhfhcgdhfgehpfbd}
\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 .  {\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}\\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~\\
X2~X0~X1){\unicdbdjded}(m1\_subset\_1~X2~X1))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngcdpfdhfhcgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k11\_arytm\_3}={k1\_xboole\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdbdpfbgchjhehngpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k2\_arytm\_2){\unicdbdjdcd}((k3\_arytm\_2~X0 \in ReplSep~\\
(toset~({\uniddcgcg} X1 : \iota .  m1\_subset\_1~X1~k5\_arytm\_3))~({\uniddcgcg} X1 : \iota .  \\
X1{\unicdcdgdad}k11\_arytm\_3)~({\uniddcgcg} X1 : \iota .  ReplSep~(toset~({\uniddcgcg} X2 : \iota .  m1\_subset\_1~\\
X2~k5\_arytm\_3))~({\uniddcgcg} X2 : \iota .  \preoh r3\_arytm\_3~X1~X2)~({\uniddcgcg} X2 : \iota .  X2))){\unicdbdjdcd}\\
(X0 \in k5\_arytm\_3))\end{array}
\label{hyp:mgcdjdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_arytm\_3){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k5\_arytm\_3){\unicdbdjdcd}({\unicdcdadad} X2 .  ((X2 \in k1\_arytm\_2){\unicdcdcdhd}((X0 \in X2){\unicdcdcdhd}(r3\_arytm\_3~\\
X1~X0))){\unicdbdjdcd}(X1 \in X2)))\end{array}
\label{hyp:mgcdgdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k2\_arytm\_2){\unicdbdjdcd}((k3\_arytm\_2~X0=k11\_arytm\_3){\unicdbdjded}\\
(X0=k11\_arytm\_3))\end{array}
\label{hyp:mgbdjdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{v1\_xboole\_0}~{k5\_arytm\_3}}}}\label{hyp:ggdgddpfbgchjhehngpfdd}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{v1\_xboole\_0}~{k1\_arytm\_2}}}}\label{hyp:ggdgbdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((\preoh v1\_xboole\_0~X0){\unicdcdcdhd}((\preoh v1\_xboole\_0~X1){\unicdcdcdhd}\\
(m1\_subset\_1~X1~(k1\_zfmisc\_1~X0)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~\\
X2~X0~X1){\unicdbdjdcd}(m1\_subset\_1~X2~X0))\end{array}
\label{hyp:egehpfngcdpfdhfhcgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k2\_arytm\_2){\unicdbdjdcd}(m2\_subset\_1~(k3\_arytm\_2~\\
X0)~(k1\_zfmisc\_1~k5\_arytm\_3)~k1\_arytm\_2)\end{array}
\label{hyp:egehpflgddpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{{m1\_subset\_1}~{k1\_arytm\_2}}~{({{k1\_zfmisc\_1}~{({{k1\_zfmisc\_1}~{k5\_arytm\_3}})}})}}\label{hyp:egehpflgbdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{{m1\_subset\_1}~{k11\_arytm\_3}}~{k5\_arytm\_3}}\label{hyp:egehpflgbdbdpfbgchjhehngpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}k5\_arytm\_3=k2\_xboole\_0~(k6\_subset\_1~(ReplSep2~(toset~({\uniddcgcg} X0 : \\
\iota .  m1\_subset\_1~X0~k4\_ordinal1))~({\uniddcgcg} X0 : \iota .  toset~({\uniddcgcg} X1 : \\
\iota .  m1\_subset\_1~X1~k4\_ordinal1))~({\uniddcgcg} X0 : \iota .  {\uniddcgcg} X1 : \iota .  (r1\_arytm\_3~\\
X0~X1){\unicdcdcdhd}(X1{\unicdcdgdad}k1\_xboole\_0))~({\uniddcgcg} X0 : \iota .  {\uniddcgcg} X1 : \iota .  k4\_tarski~\\
X0~X1))~(ReplSep~(toset~({\uniddcgcg} X0 : \iota .  m1\_subset\_1~X0~k4\_ordinal1))~\\
({\uniddcgcg} X0 : \iota .  True)~({\uniddcgcg} X0 : \iota .  k4\_tarski~X0~np\_\_1)))~k4\_ordinal1\end{array}
\label{hyp:eghdpfbgchjhehngpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k2\_arytm\_2){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k2\_arytm\_2){\unicdbdjdcd}((((X0 \in k5\_arytm\_3){\unicdcdcdhd}(X1 \in k5\_arytm\_3)){\unicdbdjdcd}((r1\_arytm\_2~\\
X0~X1){\unicdbdjded}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~k5\_arytm\_3){\unicdcdcdhd}({\unicdcdaddd} X3 .  (\\
m1\_subset\_1~X3~k5\_arytm\_3){\unicdcdcdhd}((X0=X2){\unicdcdcdhd}((X1=X3){\unicdcdcdhd}(r3\_arytm\_3~X2~\\
X3))))))){\unicdcdcdhd}(((X0 \in k5\_arytm\_3){\unicdbdjdcd}((X1 \in k5\_arytm\_3){\unicdcdcdid}((r1\_arytm\_2~\\
X0~X1){\unicdbdjded}(X0 \in X1)))){\unicdcdcdhd}(((X1 \in k5\_arytm\_3){\unicdbdjdcd}((X0 \in k5\_arytm\_3){\unicdcdcdid}((r1\_arytm\_2~\\
X0~X1){\unicdbdjded}(\preoh X1 \in X0)))){\unicdcdcdhd}(\preoh (\preoh (X0 \in k5\_arytm\_3){\unicdcdcdhd}(X1 \in k5\_arytm\_3)){\unicdcdcdhd}\\
((\preoh (X0 \in k5\_arytm\_3){\unicdcdcdhd}(\preoh X1 \in k5\_arytm\_3)){\unicdcdcdhd}((\preoh (\preoh X0 \in k5\_arytm\_3){\unicdcdcdhd}\\
(X1 \in k5\_arytm\_3)){\unicdcdcdhd}(\preoh (r1\_arytm\_2~X0~X1){\unicdbdjded}(r1\_tarski~X0~X1)))))))))\end{array}
\label{hyp:egfdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k2\_arytm\_2){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_subset\_1~\\
X1~(k1\_zfmisc\_1~k5\_arytm\_3)~k1\_arytm\_2){\unicdbdjdcd}(((X0 \in k5\_arytm\_3){\unicdbdjdcd}\\
((X1=k3\_arytm\_2~X0){\unicdbdjded}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~k5\_arytm\_3){\unicdcdcdhd}\\
((X0=X2){\unicdcdcdhd}(X1=ReplSep~(toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~k5\_arytm\_3))~\\
({\uniddcgcg} X3 : \iota .  \preoh r3\_arytm\_3~X2~X3)~({\uniddcgcg} X3 : \iota .  X3)))))){\unicdcdcdhd}((\preoh X0 \in \\
k5\_arytm\_3){\unicdbdjdcd}((X1=k3\_arytm\_2~X0){\unicdbdjded}(X1=X0)))))\end{array}
\label{hyp:egddpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}k1\_arytm\_2=k6\_subset\_1~(ReplSep~(toset~({\uniddcgcg} X0 : \iota .  m1\_subset\_1~\\
X0~(k1\_zfmisc\_1~k5\_arytm\_3)))~({\uniddcgcg} X0 : \iota .  {\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k5\_arytm\_3){\unicdbdjdcd}((X1 \in X0){\unicdbdjdcd}(({\unicdcdadad} X2 .  (m1\_subset\_1~X2~k5\_arytm\_3){\unicdbdjdcd}\\
((r3\_arytm\_3~X2~X1){\unicdbdjdcd}(X2 \in X0))){\unicdcdcdhd}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~k5\_arytm\_3){\unicdcdcdhd}\\
((X2 \in X0){\unicdcdcdhd}(\preoh r3\_arytm\_3~X2~X1))))))~({\uniddcgcg} X0 : \iota .  X0))~(k1\_tarski~\\
k5\_arytm\_3)\end{array}
\label{hyp:egbdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{X0}={X1}})}{\unicdbdjded}{({{({{{r1\_tarski}~{X0}}~{X1}})}{\unicdcdcdhd}{({{{r1\_tarski}~{X1}}~{X0}})}})}}}}\label{hyp:egbdadpfihcgpgpgmgfgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k5\_arytm\_3){\unicdcdcdhd}(m1\_subset\_1~\\
X1~k5\_arytm\_3)){\unicdbdjdcd}((r3\_arytm\_3~X0~X1){\unicdcdcdid}(r3\_arytm\_3~X1~X0))\end{array}
\label{hyp:dgpgogogfgdgehfgegogfgdhdhpfchddpfbgchjhehngpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k2\_arytm\_2){\unicdcdcdhd}(m1\_subset\_1~\\
X1~k2\_arytm\_2)){\unicdbdjdcd}((r1\_arytm\_2~X0~X1){\unicdcdcdid}(r1\_arytm\_2~X1~X0))\end{array}
\label{hyp:dgpgogogfgdgehfgegogfgdhdhpfchbdpfbgchjhehngpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~\\
X0)){\unicdbdjdcd}(v1\_xboole\_0~X1))\end{array}
\label{hyp:dgdgbdpfdhfhcgdhfgehpfbd}
\end{equation}
\begin{theorem}\label{thm:mgddadpfbgchjhehngpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k2\_arytm\_2){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k2\_arytm\_2){\unicdbdjdcd}((r1\_tarski~(k3\_arytm\_2~X0)~(k3\_arytm\_2~X1)){\unicdbdjdcd}\\
(r1\_arytm\_2~X0~X1)))\end{array}$$
\end{theorem}
\end{document}
