\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{t21\_anproj\_2 (TMMY9QAued1TSfGQqeEUoZTCWi1U4GBEN7U)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k9\_domain\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ m2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k9\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k1\_funcsdom : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ k5\_funcsdom : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_funcsdom : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}}$ be given.
Let $ k7\_funcsdom : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_domain\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k8\_funcsdom : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k3\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_funct\_2~X1~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_funct\_2~X2~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_funct\_2~X3~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_funct\_2~X4~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~X0){\unicdbdjdcd}\\
({\unicdcdadad} X6 .  (m1\_subset\_1~X6~X0){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~X7~X0){\unicdbdjdcd}\\
({\unicdcdadad} X8 .  (m1\_subset\_1~X8~X0){\unicdbdjdcd}(((X0=k9\_domain\_1~X0~X5~X6~X7~X8){\unicdcdcdhd}\\
((k3\_funct\_2~X0~k1\_numbers~X1~X5=np\_\_1){\unicdcdcdhd}(({\unicdcdadad} X9 .  (X9 \in X0){\unicdbdjdcd}\\
((X9=X5){\unicdcdcdid}(k1\_funct\_1~X1~X9=k6\_numbers))){\unicdcdcdhd}((k3\_funct\_2~X0~k1\_numbers~\\
X2~X6=np\_\_1){\unicdcdcdhd}(({\unicdcdadad} X9 .  (X9 \in X0){\unicdbdjdcd}((X9=X6){\unicdcdcdid}(k1\_funct\_1~X2~X9=\\
k6\_numbers))){\unicdcdcdhd}((k3\_funct\_2~X0~k1\_numbers~X3~X7=np\_\_1){\unicdcdcdhd}(({\unicdcdadad} X9 .  \\
(X9 \in X0){\unicdbdjdcd}((X9=X7){\unicdcdcdid}(k1\_funct\_1~X3~X9=k6\_numbers))){\unicdcdcdhd}((k3\_funct\_2~\\
X0~k1\_numbers~X4~X8=np\_\_1){\unicdcdcdhd}({\unicdcdadad} X9 .  (X9 \in X0){\unicdbdjdcd}((X9=X8){\unicdcdcdid}(k1\_funct\_1~\\
X4~X9=k6\_numbers))))))))))){\unicdbdjdcd}((X5=X6){\unicdcdcdid}((X5=X7){\unicdcdcdid}((X5=X8){\unicdcdcdid}(\\
(X6=X7){\unicdcdcdid}((X6=X8){\unicdcdcdid}((X7=X8){\unicdcdcdid}({\unicdcdadad} X9 .  (m2\_funct\_2~X9~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdaddd} X10 .  (m1\_subset\_1~X10~k1\_numbers){\unicdcdcdhd}\\
({\unicdcdaddd} X11 .  (m1\_subset\_1~X11~k1\_numbers){\unicdcdcdhd}({\unicdcdaddd} X12 .  (m1\_subset\_1~\\
X12~k1\_numbers){\unicdcdcdhd}({\unicdcdaddd} X13 .  (m1\_subset\_1~X13~k1\_numbers){\unicdcdcdhd}(\\
r2\_funct\_2~X0~k1\_numbers~X9~(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~\\
X0)~(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~X0)~(k1\_funcsdom~X0~\\
k1\_numbers~(k5\_funcsdom~X0)~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~X10~X1))~(k2\_funcsdom~X0~k1\_numbers~\\
k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~\\
k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~X11~X2)))~(k2\_funcsdom~X0~\\
k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~(k7\_funcsdom~\\
X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~X12~X3)))~\\
(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
X13~X4))))))))))))))))))))))))\end{array}
\label{hyp:ehbdjdpfbgogahchpgkgpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_funct\_2~X1~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_funct\_2~X2~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_funct\_2~X3~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_funct\_2~X4~X0~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~X0){\unicdbdjdcd}\\
({\unicdcdadad} X6 .  (m1\_subset\_1~X6~X0){\unicdbdjdcd}({\unicdcdadad} X7 .  (m1\_subset\_1~X7~X0){\unicdbdjdcd}\\
({\unicdcdadad} X8 .  (m1\_subset\_1~X8~X0){\unicdbdjdcd}(((k3\_funct\_2~X0~k1\_numbers~X1~\\
X5=np\_\_1){\unicdcdcdhd}(({\unicdcdadad} X9 .  (X9 \in X0){\unicdbdjdcd}((X9=X5){\unicdcdcdid}(k1\_funct\_1~X1~X9=\\
k6\_numbers))){\unicdcdcdhd}((k3\_funct\_2~X0~k1\_numbers~X2~X6=np\_\_1){\unicdcdcdhd}(({\unicdcdadad} X9 .  \\
(X9 \in X0){\unicdbdjdcd}((X9=X6){\unicdcdcdid}(k1\_funct\_1~X2~X9=k6\_numbers))){\unicdcdcdhd}((k3\_funct\_2~\\
X0~k1\_numbers~X3~X7=np\_\_1){\unicdcdcdhd}(({\unicdcdadad} X9 .  (X9 \in X0){\unicdbdjdcd}((X9=X7){\unicdcdcdid}(k1\_funct\_1~\\
X3~X9=k6\_numbers))){\unicdcdcdhd}((k3\_funct\_2~X0~k1\_numbers~X4~X8=np\_\_1){\unicdcdcdhd}\\
({\unicdcdadad} X9 .  (X9 \in X0){\unicdbdjdcd}((X9=X8){\unicdcdcdid}(k1\_funct\_1~X4~X9=k6\_numbers)))))))))){\unicdbdjdcd}\\
((X5=X6){\unicdcdcdid}((X5=X7){\unicdcdcdid}((X5=X8){\unicdcdcdid}((X6=X7){\unicdcdcdid}((X6=X8){\unicdcdcdid}((X7=X8){\unicdcdcdid}(\\
{\unicdcdadad} X9 .  (m1\_subset\_1~X9~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X10 .  (m1\_subset\_1~\\
X10~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X11 .  (m1\_subset\_1~X11~k1\_numbers){\unicdbdjdcd}(\\
{\unicdcdadad} X12 .  (m1\_subset\_1~X12~k1\_numbers){\unicdbdjdcd}((r2\_funct\_2~X0~k1\_numbers~\\
(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~X0)~(k1\_funcsdom~X0~k1\_numbers~\\
(k5\_funcsdom~X0)~(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~X0)~(\\
k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
X9~X1))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
X10~X2)))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~\\
k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~\\
X0~k1\_numbers)~X11~X3)))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~X12~X4)))~(k8\_funcsdom~X0)){\unicdbdjdcd}((X9=k6\_numbers){\unicdcdcdhd}\\
((X10=k6\_numbers){\unicdcdcdhd}((X11=k6\_numbers){\unicdcdcdhd}(X12=k6\_numbers))))))))))))))))))))))))\end{array}
\label{hyp:ehbdhdpfbgogahchpgkgpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
({\unicdcdaddd} X2 .  (m2\_funct\_2~X2~X0~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)){\unicdcdcdhd}\\
((k3\_funct\_2~X0~k1\_numbers~X2~X1=np\_\_1){\unicdcdcdhd}({\unicdcdadad} X3 .  (X3 \in X0){\unicdbdjdcd}\\
((X3=X1){\unicdcdcdid}(k1\_funct\_1~X2~X3=k6\_numbers))))))\end{array}
\label{hyp:ehbdadpfbgogahchpgkgpfcd}
\end{equation}
\begin{theorem}\label{thm:ehcdbdpfbgogahchpgkgpfcd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~X0){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~X0){\unicdbdjdcd}\\
({\unicdcdadad} X4 .  (m1\_subset\_1~X4~X0){\unicdbdjdcd}(\preoh (X0=k9\_domain\_1~X0~X1~X2~X3~X4){\unicdcdcdhd}\\
((X1{\unicdcdgdad}X2){\unicdcdcdhd}((X1{\unicdcdgdad}X3){\unicdcdcdhd}((X1{\unicdcdgdad}X4){\unicdcdcdhd}((X2{\unicdcdgdad}X3){\unicdcdcdhd}((X2{\unicdcdgdad}X4){\unicdcdcdhd}((X3{\unicdcdgdad}X4){\unicdcdcdhd}(\\
{\unicdcdadad} X5 .  (m2\_funct\_2~X5~X0~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X6 .  (m2\_funct\_2~X6~X0~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X7 .  (m2\_funct\_2~X7~X0~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X8 .  (m2\_funct\_2~X8~X0~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}\\
(\preoh ({\unicdcdadad} X9 .  (m1\_subset\_1~X9~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X10 .  (m1\_subset\_1~\\
X10~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X11 .  (m1\_subset\_1~X11~k1\_numbers){\unicdbdjdcd}(\\
{\unicdcdadad} X12 .  (m1\_subset\_1~X12~k1\_numbers){\unicdbdjdcd}((r2\_funct\_2~X0~k1\_numbers~\\
(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~X0)~(k1\_funcsdom~X0~k1\_numbers~\\
(k5\_funcsdom~X0)~(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~X0)~(\\
k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
X9~X5))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
X10~X6)))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~\\
k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~\\
X0~k1\_numbers)~X11~X7)))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~X12~X8)))~(k8\_funcsdom~X0)){\unicdbdjdcd}((X9=k6\_numbers){\unicdcdcdhd}\\
((X10=k6\_numbers){\unicdcdcdhd}((X11=k6\_numbers){\unicdcdcdhd}(X12=k6\_numbers))))))))){\unicdcdcdhd}\\
({\unicdcdadad} X9 .  (m2\_funct\_2~X9~X0~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)){\unicdbdjdcd}\\
({\unicdcdaddd} X10 .  (m1\_subset\_1~X10~k1\_numbers){\unicdcdcdhd}({\unicdcdaddd} X11 .  (m1\_subset\_1~\\
X11~k1\_numbers){\unicdcdcdhd}({\unicdcdaddd} X12 .  (m1\_subset\_1~X12~k1\_numbers){\unicdcdcdhd}(\\
{\unicdcdaddd} X13 .  (m1\_subset\_1~X13~k1\_numbers){\unicdcdcdhd}(r2\_funct\_2~X0~k1\_numbers~\\
X9~(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~X0)~(k1\_funcsdom~X0~\\
k1\_numbers~(k5\_funcsdom~X0)~(k1\_funcsdom~X0~k1\_numbers~(k5\_funcsdom~\\
X0)~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
X10~X5))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~X0~k1\_numbers)~\\
X11~X6)))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~(k9\_funct\_2~X0~\\
k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~(k9\_funct\_2~\\
X0~k1\_numbers)~X12~X7)))~(k2\_funcsdom~X0~k1\_numbers~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~(k7\_funcsdom~X0)~(k1\_domain\_1~k1\_numbers~\\
(k9\_funct\_2~X0~k1\_numbers)~X13~X8)))))))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
