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\begin{document}
\title{l85\_sin\_cos (TML12kxrZurZT2nCrt8882iofkzKBFC2V2k)}
\maketitle
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_seq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k24\_sin\_cos : {\iota}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_series\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ 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.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(k4\_series\_1~\\
(k4\_sin\_cos~(k2\_xcmplx\_0~X0~X1))=k8\_real\_1~(k4\_series\_1~(k4\_sin\_cos~\\
X0))~(k4\_series\_1~(k4\_sin\_cos~X1))))\end{array}
\label{hyp:ehedgdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}(v1\_xreal\_0~\\
(k2\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdgfdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k24\_sin\_cos){\unicdcdcdhd}((v1\_funct\_2~k24\_sin\_cos~k1\_numbers~\\
k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~k24\_sin\_cos~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k1\_numbers~k1\_numbers))))\end{array}
\label{hyp:egehpflgcdedpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k1\_numbers~k1\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k1\_numbers~k1\_numbers))))){\unicdbdjdcd}\\
((X0=k24\_sin\_cos){\unicdbdjded}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(k1\_seq\_1~X0~\\
X1=k4\_series\_1~(k4\_sin\_cos~X1))))\end{array}
\label{hyp:egcdcdpfdhjgogpfdgpgdh}
\end{equation}
\begin{theorem}\label{thm:mgidfdpfdhjgogpfdgpgdh}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}(k1\_seq\_1~\\
k24\_sin\_cos~(k2\_xcmplx\_0~X0~X1)=k8\_real\_1~(k1\_seq\_1~k24\_sin\_cos~\\
X0)~(k1\_seq\_1~k24\_sin\_cos~X1)))\end{array}$$
\end{theorem}
\end{document}
