\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{t12\_fdiff\_2 (TMGfer8o5Dx47wC4x3SmENs1r9S6n1VFs5t)}
\maketitle
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ v1\_funct\_1 : {{\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 $ r1\_fdiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_fdiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m1\_rcomp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r1\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v2\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ v1\_fdiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ v3\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_tarski : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_valued\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ v2\_comseq\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k20\_valued\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k37\_valued\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k47\_valued\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k8\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k2\_seq\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
(\preoh ({\unicdcdaddd} X2 .  (m1\_rcomp\_1~X2~X0){\unicdcdcdhd}(r1\_tarski~X2~(k1\_relset\_1~k1\_numbers~\\
X1))){\unicdcdcdhd}({\unicdcdadad} X2 .  ((v2\_relat\_1~X2){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~k5\_numbers~k1\_numbers){\unicdcdcdhd}((v1\_fdiff\_1~X2~k6\_numbers){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))))){\unicdbdjdcd}(\\
{\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v3\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~\\
k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k1\_numbers)))))){\unicdbdjdcd}(\preoh (k2\_relset\_1~k1\_numbers~X3=k1\_tarski~\\
X0){\unicdcdcdhd}((r1\_tarski~(k2\_relset\_1~k1\_numbers~(k3\_valued\_1~k5\_numbers~\\
k1\_numbers~k1\_numbers~X2~X3))~(k1\_relset\_1~k1\_numbers~X1)){\unicdcdcdhd}(r1\_tarski~\\
(k1\_tarski~X0)~(k1\_relset\_1~k1\_numbers~X1))))))))\end{array}
\label{hyp:ehidpfggegjgggggpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
((r1\_fdiff\_1~X1~X0){\unicdbdjded}(({\unicdcdaddd} X2 .  (m1\_rcomp\_1~X2~X0){\unicdcdcdhd}(r1\_tarski~\\
X2~(k1\_relset\_1~k1\_numbers~X1))){\unicdcdcdhd}({\unicdcdadad} X2 .  ((v2\_relat\_1~X2){\unicdcdcdhd}\\
((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~X2~k5\_numbers~k1\_numbers){\unicdcdcdhd}((v1\_fdiff\_1~\\
X2~k6\_numbers){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~\\
k1\_numbers))))))){\unicdbdjdcd}({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v3\_funct\_1~\\
X3){\unicdcdcdhd}((v1\_funct\_2~X3~k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~X3~(\\
k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers)))))){\unicdbdjdcd}(((k2\_relset\_1~\\
k1\_numbers~X3=k1\_tarski~X0){\unicdcdcdhd}(r1\_tarski~(k2\_relset\_1~k1\_numbers~\\
(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~X2~X3))~(k1\_relset\_1~\\
k1\_numbers~X1))){\unicdbdjdcd}(v2\_comseq\_2~(k20\_valued\_1~k5\_numbers~k1\_numbers~\\
k1\_numbers~(k37\_valued\_1~k5\_numbers~k1\_numbers~X2)~(k47\_valued\_1~\\
k5\_numbers~k1\_numbers~k1\_numbers~(k8\_funct\_2~k5\_numbers~k1\_numbers~\\
k1\_numbers~(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~X2~X3)~\\
X1)~(k8\_funct\_2~k5\_numbers~k1\_numbers~k1\_numbers~X3~X1))))))))))\end{array}
\label{hyp:ehbdbdpfggegjgggggpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  ((v1\_funct\_1~\\
X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}\\
(({\unicdcdadad} X2 .  ((v2\_relat\_1~X2){\unicdcdcdhd}((v1\_funct\_1~X2){\unicdcdcdhd}((v1\_funct\_2~\\
X2~k5\_numbers~k1\_numbers){\unicdcdcdhd}((v1\_fdiff\_1~X2~k6\_numbers){\unicdcdcdhd}(m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))))){\unicdbdjdcd}(\\
{\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v3\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~\\
k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k1\_numbers)))))){\unicdbdjdcd}(((k2\_relset\_1~k1\_numbers~X3=k1\_tarski~\\
X0){\unicdcdcdhd}(r1\_tarski~(k2\_relset\_1~k1\_numbers~(k3\_valued\_1~k5\_numbers~\\
k1\_numbers~k1\_numbers~X2~X3))~(k1\_relset\_1~k1\_numbers~X1))){\unicdbdjdcd}(\\
v2\_comseq\_2~(k20\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~(\\
k37\_valued\_1~k5\_numbers~k1\_numbers~X2)~(k47\_valued\_1~k5\_numbers~\\
k1\_numbers~k1\_numbers~(k8\_funct\_2~k5\_numbers~k1\_numbers~k1\_numbers~\\
(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~X2~X3)~X1)~(k8\_funct\_2~\\
k5\_numbers~k1\_numbers~k1\_numbers~X3~X1))))))){\unicdbdjdcd}(({\unicdcdadad} X2 .  (\\
m1\_rcomp\_1~X2~X0){\unicdbdjdcd}(\preoh r1\_tarski~X2~(k1\_relset\_1~k1\_numbers~X1))){\unicdcdcdid}\\
((r1\_fdiff\_1~X1~X0){\unicdcdcdhd}({\unicdcdadad} X2 .  ((v2\_relat\_1~X2){\unicdcdcdhd}((v1\_funct\_1~\\
X2){\unicdcdcdhd}((v1\_funct\_2~X2~k5\_numbers~k1\_numbers){\unicdcdcdhd}((v1\_fdiff\_1~X2~k6\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))))){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  ((v1\_funct\_1~X3){\unicdcdcdhd}((v3\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~\\
X3~k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k1\_numbers)))))){\unicdbdjdcd}(((k2\_relset\_1~k1\_numbers~X3=k1\_tarski~\\
X0){\unicdcdcdhd}(r1\_tarski~(k2\_relset\_1~k1\_numbers~(k3\_valued\_1~k5\_numbers~\\
k1\_numbers~k1\_numbers~X2~X3))~(k1\_relset\_1~k1\_numbers~X1))){\unicdbdjdcd}(\\
k1\_fdiff\_1~X1~X0=k2\_seq\_2~(k20\_valued\_1~k5\_numbers~k1\_numbers~\\
k1\_numbers~(k37\_valued\_1~k5\_numbers~k1\_numbers~X2)~(k47\_valued\_1~\\
k5\_numbers~k1\_numbers~k1\_numbers~(k8\_funct\_2~k5\_numbers~k1\_numbers~\\
k1\_numbers~(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~X2~X3)~\\
X1)~(k8\_funct\_2~k5\_numbers~k1\_numbers~k1\_numbers~X3~X1)))))))))))\end{array}
\label{hyp:mgbdbdpfggegjgggggpfcd}
\end{equation}
\begin{theorem}\label{thm:ehbdcdpfggegjgggggpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X2 .  ((v1\_funct\_1~X2){\unicdcdcdhd}(m1\_subset\_1~X2~\\
(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdbdjdcd}((((r1\_fdiff\_1~\\
X2~X0){\unicdcdcdhd}(k1\_fdiff\_1~X2~X0=X1)){\unicdbdjdcd}(({\unicdcdaddd} X3 .  (m1\_rcomp\_1~X3~X0){\unicdcdcdhd}\\
(r1\_tarski~X3~(k1\_relset\_1~k1\_numbers~X2))){\unicdcdcdhd}({\unicdcdadad} X3 .  ((v2\_relat\_1~\\
X3){\unicdcdcdhd}((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~k5\_numbers~k1\_numbers){\unicdcdcdhd}\\
((v1\_fdiff\_1~X3~k6\_numbers){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k1\_numbers))))))){\unicdbdjdcd}({\unicdcdadad} X4 .  ((v1\_funct\_1~X4){\unicdcdcdhd}\\
((v3\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~X4~k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers)))))){\unicdbdjdcd}((\\
(k2\_relset\_1~k1\_numbers~X4=k1\_tarski~X0){\unicdcdcdhd}(r1\_tarski~(k2\_relset\_1~\\
k1\_numbers~(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~X3~X4))~\\
(k1\_relset\_1~k1\_numbers~X2))){\unicdbdjdcd}((v2\_comseq\_2~(k20\_valued\_1~k5\_numbers~\\
k1\_numbers~k1\_numbers~(k37\_valued\_1~k5\_numbers~k1\_numbers~X3)~\\
(k47\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~(k8\_funct\_2~k5\_numbers~\\
k1\_numbers~k1\_numbers~(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~\\
X3~X4)~X2)~(k8\_funct\_2~k5\_numbers~k1\_numbers~k1\_numbers~X4~X2)))){\unicdcdcdhd}\\
(k2\_seq\_2~(k20\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~(k37\_valued\_1~\\
k5\_numbers~k1\_numbers~X3)~(k47\_valued\_1~k5\_numbers~k1\_numbers~\\
k1\_numbers~(k8\_funct\_2~k5\_numbers~k1\_numbers~k1\_numbers~(k3\_valued\_1~\\
k5\_numbers~k1\_numbers~k1\_numbers~X3~X4)~X2)~(k8\_funct\_2~k5\_numbers~\\
k1\_numbers~k1\_numbers~X4~X2)))=X1))))))){\unicdcdcdhd}(({\unicdcdadad} X3 .  ((v2\_relat\_1~\\
X3){\unicdcdcdhd}((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~k5\_numbers~k1\_numbers){\unicdcdcdhd}\\
((v1\_fdiff\_1~X3~k6\_numbers){\unicdcdcdhd}(m1\_subset\_1~X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k1\_numbers))))))){\unicdbdjdcd}({\unicdcdadad} X4 .  ((v1\_funct\_1~X4){\unicdcdcdhd}\\
((v3\_funct\_1~X4){\unicdcdcdhd}((v1\_funct\_2~X4~k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~\\
X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers)))))){\unicdbdjdcd}((\\
(k2\_relset\_1~k1\_numbers~X4=k1\_tarski~X0){\unicdcdcdhd}(r1\_tarski~(k2\_relset\_1~\\
k1\_numbers~(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~X3~X4))~\\
(k1\_relset\_1~k1\_numbers~X2))){\unicdbdjdcd}((v2\_comseq\_2~(k20\_valued\_1~k5\_numbers~\\
k1\_numbers~k1\_numbers~(k37\_valued\_1~k5\_numbers~k1\_numbers~X3)~\\
(k47\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~(k8\_funct\_2~k5\_numbers~\\
k1\_numbers~k1\_numbers~(k3\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~\\
X3~X4)~X2)~(k8\_funct\_2~k5\_numbers~k1\_numbers~k1\_numbers~X4~X2)))){\unicdcdcdhd}\\
(k2\_seq\_2~(k20\_valued\_1~k5\_numbers~k1\_numbers~k1\_numbers~(k37\_valued\_1~\\
k5\_numbers~k1\_numbers~X3)~(k47\_valued\_1~k5\_numbers~k1\_numbers~\\
k1\_numbers~(k8\_funct\_2~k5\_numbers~k1\_numbers~k1\_numbers~(k3\_valued\_1~\\
k5\_numbers~k1\_numbers~k1\_numbers~X3~X4)~X2)~(k8\_funct\_2~k5\_numbers~\\
k1\_numbers~k1\_numbers~X4~X2)))=X1))))){\unicdbdjdcd}(({\unicdcdadad} X3 .  (m1\_rcomp\_1~\\
X3~X0){\unicdbdjdcd}(\preoh r1\_tarski~X3~(k1\_relset\_1~k1\_numbers~X2))){\unicdcdcdid}((r1\_fdiff\_1~\\
X2~X0){\unicdcdcdhd}(k1\_fdiff\_1~X2~X0=X1)))))))\end{array}$$
\end{theorem}
\end{document}
