\documentclass{article}
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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
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\begin{document}
\title{t8\_integra8 (TMaKWQMZgFmVpi9wXWLTvdKEgSVzbJejUCC)}
\maketitle
Let $ k18\_sin\_cos : {{\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 $ k32\_sin\_cos : {\iota}$ be given.
Let $ np\_\_4 : {\iota}$ be given.
Let $ k7\_square\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ k1\_seq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k16\_sin\_cos : {\iota}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k19\_sin\_cos : {\iota}$ be given.
Let $ k8\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_3 : {\iota}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
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 $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k17\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}k18\_sin\_cos~(k10\_real\_1~k32\_sin\_cos~np\_\_4)=k10\_real\_1~(k7\_square\_1~\\
np\_\_2)~np\_\_2\end{array}
\label{hyp:ehhdpfjgogehfghgchbgid}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k1\_seq\_1~k16\_sin\_cos~(k1\_real\_1~(k10\_real\_1~k32\_sin\_cos~np\_\_4))=\\
k1\_real\_1~(k10\_real\_1~np\_\_1~(k7\_square\_1~np\_\_2))){\unicdcdcdhd}((k1\_seq\_1~\\
k19\_sin\_cos~(k1\_real\_1~(k10\_real\_1~k32\_sin\_cos~np\_\_4))=k10\_real\_1~\\
np\_\_1~(k7\_square\_1~np\_\_2)){\unicdcdcdhd}((k1\_seq\_1~k16\_sin\_cos~(k8\_real\_1~\\
(k10\_real\_1~np\_\_3~np\_\_4)~k32\_sin\_cos)=k10\_real\_1~np\_\_1~(k7\_square\_1~\\
np\_\_2)){\unicdcdcdhd}(k1\_seq\_1~k19\_sin\_cos~(k8\_real\_1~(k10\_real\_1~np\_\_3~np\_\_4)~\\
k32\_sin\_cos)=k1\_real\_1~(k10\_real\_1~np\_\_1~(k7\_square\_1~np\_\_2)))))\end{array}
\label{hyp:ehddadpfdhjgogdgpgdhbdad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k1\_seq\_1~k16\_sin\_cos~(k10\_real\_1~k32\_sin\_cos~np\_\_4)=k10\_real\_1~\\
np\_\_1~(k7\_square\_1~np\_\_2)){\unicdcdcdhd}(k1\_seq\_1~k19\_sin\_cos~(k10\_real\_1~\\
k32\_sin\_cos~np\_\_4)=k10\_real\_1~np\_\_1~(k7\_square\_1~np\_\_2))\end{array}
\label{hyp:ehcdjdpfdhjgogdgpgdhbdad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_4){\unicdcdcdhd}(m2\_subset\_1~np\_\_4~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_4~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_4~k1\_numbers))\end{array}
\label{hyp:dhahdgedpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(k18\_sin\_cos~X0=k17\_sin\_cos~\\
X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdidpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
{{{m1\_subset\_1}~{k32\_sin\_cos}}~{k1\_numbers}}\label{hyp:egehpflgddcdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(m1\_subset\_1~(k1\_real\_1~\\
X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k1\_numbers){\unicdcdcdhd}(v1\_xreal\_0~\\
X1)){\unicdbdjdcd}(m1\_subset\_1~(k10\_real\_1~X0~X1)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdadpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{{k17\_sin\_cos}~{X0}}={{{k1\_seq\_1}~{k16\_sin\_cos}}~{X0}}})}}}\label{hyp:egbdhdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfihchfgbgmgpfad}
\end{equation}
\begin{theorem}\label{thm:ehidpfjgogehfghgchbgid}
$$\begin{array}{c}k18\_sin\_cos~(k1\_real\_1~(k10\_real\_1~k32\_sin\_cos~np\_\_4))=k1\_real\_1~\\
(k10\_real\_1~(k7\_square\_1~np\_\_2)~np\_\_2)\end{array}$$
\end{theorem}
\end{document}
