\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{l40\_taylor\_1 (TMN3NKFF8Kf2FaU88V54o61ancmGksB2hoS)}
\maketitle
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\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\_tarski : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_taylor\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_rcomp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_fcont\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k1\_seqfunc : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k5\_taylor\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k2\_rcomp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_seq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k9\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_series\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_taylor\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ r2\_fdiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_fdiff\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k10\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_newton : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_newton : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(k1\_zfmisc\_1~\\
k1\_numbers)){\unicdbdjdcd}((r1\_tarski~X1~(k1\_relset\_1~k1\_numbers~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m2\_subset\_1~X2~k1\_numbers~k5\_numbers){\unicdbdjdcd}((r1\_taylor\_1~X0~X2~X1){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_subset\_1~\\
X4~k1\_numbers){\unicdbdjdcd}(((r1\_tarski~(k1\_rcomp\_1~X3~X4)~X1){\unicdcdcdhd}((v1\_fcont\_1~\\
(k2\_partfun1~k1\_numbers~k1\_numbers~(k1\_seqfunc~k1\_numbers~k1\_numbers~\\
(k5\_taylor\_1~X0~X1)~X2)~(k1\_rcomp\_1~X3~X4))){\unicdcdcdhd}(r1\_taylor\_1~X0~(k2\_nat\_1~\\
X2~np\_\_1)~(k2\_rcomp\_1~X3~X4)))){\unicdbdjdcd}((r1\_xxreal\_0~X4~X3){\unicdcdcdid}({\unicdcdadad} X5 .  \\
((v1\_funct\_1~X5){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers)))){\unicdbdjdcd}(((k1\_relset\_1~k1\_numbers~X5=X1){\unicdcdcdhd}({\unicdcdadad} X6 .  \\
(m1\_subset\_1~X6~k1\_numbers){\unicdbdjdcd}((X6 \in X1){\unicdbdjdcd}(k1\_seq\_1~X5~X6=k9\_real\_1~\\
(k1\_seq\_1~X0~X4)~(k1\_seq\_1~(k3\_series\_1~(k6\_taylor\_1~X0~X1~X6~X4))~\\
X2))))){\unicdbdjdcd}((k1\_seq\_1~X5~X4=k6\_numbers){\unicdcdcdhd}((v1\_fcont\_1~(k2\_partfun1~\\
k1\_numbers~k1\_numbers~X5~(k1\_rcomp\_1~X3~X4))){\unicdcdcdhd}((r2\_fdiff\_1~X5~\\
(k2\_rcomp\_1~X3~X4)){\unicdcdcdhd}({\unicdcdadad} X6 .  (m1\_subset\_1~X6~k1\_numbers){\unicdbdjdcd}\\
((X6 \in k2\_rcomp\_1~X3~X4){\unicdbdjdcd}(k1\_fdiff\_1~X5~X6=k1\_real\_1~(k10\_real\_1~\\
(k8\_real\_1~(k1\_seq\_1~(k1\_seqfunc~k1\_numbers~k1\_numbers~(k5\_taylor\_1~\\
X0~(k2\_rcomp\_1~X3~X4))~(k2\_nat\_1~X2~np\_\_1))~X6)~(k2\_newton~(k9\_real\_1~\\
X4~X6)~X2))~(k3\_newton~X2))))))))))))))))))\end{array}
\label{hyp:mgddjdpfehbgjhmgpgchpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m2\_subset\_1~X0~k1\_numbers~k5\_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 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~k1\_numbers)){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}({\unicdcdaddd} X4 .  ((v1\_funct\_1~\\
X4){\unicdcdcdhd}(m1\_subset\_1~X4~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))){\unicdcdcdhd}\\
((k1\_relset\_1~k1\_numbers~X4=X2){\unicdcdcdhd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~\\
k1\_numbers){\unicdbdjdcd}((X5 \in X2){\unicdbdjdcd}(k1\_seq\_1~X4~X5=k9\_real\_1~(k1\_seq\_1~X1~\\
X3)~(k1\_seq\_1~(k3\_series\_1~(k6\_taylor\_1~X1~X2~X5~X3))~X0)))))))))\end{array}
\label{hyp:mgddhdpfehbgjhmgpgchpfbd}
\end{equation}
\begin{theorem}\label{thm:mgedadpfehbgjhmgpgchpfbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m2\_subset\_1~X0~k1\_numbers~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  \\
((v1\_funct\_1~X1){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~\\
k1\_numbers)))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~k1\_numbers)){\unicdbdjdcd}\\
(((r1\_tarski~X2~(k1\_relset\_1~k1\_numbers~X1)){\unicdcdcdhd}(r1\_taylor\_1~X1~\\
X0~X2)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
(m1\_subset\_1~X4~k1\_numbers){\unicdbdjdcd}(\preoh (\preoh r1\_xxreal\_0~X4~X3){\unicdcdcdhd}((r1\_tarski~\\
(k1\_rcomp\_1~X3~X4)~X2){\unicdcdcdhd}((v1\_fcont\_1~(k2\_partfun1~k1\_numbers~k1\_numbers~\\
(k1\_seqfunc~k1\_numbers~k1\_numbers~(k5\_taylor\_1~X1~X2)~X0)~(k1\_rcomp\_1~\\
X3~X4))){\unicdcdcdhd}((r1\_taylor\_1~X1~(k2\_nat\_1~X0~np\_\_1)~(k2\_rcomp\_1~X3~X4)){\unicdcdcdhd}\\
({\unicdcdadad} X5 .  ((v1\_funct\_1~X5){\unicdcdcdhd}(m1\_subset\_1~X5~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers)))){\unicdbdjdcd}(\preoh (k1\_relset\_1~k1\_numbers~X5=X2){\unicdcdcdhd}\\
(({\unicdcdadad} X6 .  (m1\_subset\_1~X6~k1\_numbers){\unicdbdjdcd}((X6 \in X2){\unicdbdjdcd}(k1\_seq\_1~\\
X5~X6=k9\_real\_1~(k1\_seq\_1~X1~X4)~(k1\_seq\_1~(k3\_series\_1~(k6\_taylor\_1~\\
X1~X2~X6~X4))~X0)))){\unicdcdcdhd}((k1\_seq\_1~X5~X4=k6\_numbers){\unicdcdcdhd}((v1\_fcont\_1~\\
(k2\_partfun1~k1\_numbers~k1\_numbers~X5~(k1\_rcomp\_1~X3~X4))){\unicdcdcdhd}((\\
r2\_fdiff\_1~X5~(k2\_rcomp\_1~X3~X4)){\unicdcdcdhd}({\unicdcdadad} X6 .  (m1\_subset\_1~X6~\\
k1\_numbers){\unicdbdjdcd}((X6 \in k2\_rcomp\_1~X3~X4){\unicdbdjdcd}(k1\_fdiff\_1~X5~X6=k1\_real\_1~\\
(k10\_real\_1~(k8\_real\_1~(k1\_seq\_1~(k1\_seqfunc~k1\_numbers~k1\_numbers~\\
(k5\_taylor\_1~X1~(k2\_rcomp\_1~X3~X4))~(k2\_nat\_1~X0~np\_\_1))~X6)~(k2\_newton~\\
(k9\_real\_1~X4~X6)~X0))~(k3\_newton~X0))))))))))))))))))))\end{array}$$
\end{theorem}
\end{document}
