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frozen 3.x
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EthanDeng committed Apr 11, 2020
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4 changes: 1 addition & 3 deletions elegantbook-en.tex
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\logo{logo-blue.png}
\cover{cover.jpg}

\providecommand\qed{}
\renewcommand{\qed}{\hfill\ensuremath{\square}}

\begin{document}

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\end{proposition}

\begin{proof}
Let $z$ be some element of $xH \cap yH$. Then $z = xa$ for some $a \in H$, and $z = yb$ for some $b \in H$. If $h$ is any element of $H$ then $ah \in H$ and $a^{-1}h \in H$, since $H$ is a subgroup of $G$. But $zh = x(ah)$ and $xh = z(a^{-1}h)$ for all $h \in H$. Therefore $zH \subset xH$ and $xH \subset zH$, and thus $xH = zH$. Similarly $yH = zH$, and thus $xH = yH$, as required. \qed
Let $z$ be some element of $xH \cap yH$. Then $z = xa$ for some $a \in H$, and $z = yb$ for some $b \in H$. If $h$ is any element of $H$ then $ah \in H$ and $a^{-1}h \in H$, since $H$ is a subgroup of $G$. But $zh = x(ah)$ and $xh = z(a^{-1}h)$ for all $h \in H$. Therefore $zH \subset xH$ and $xH \subset zH$, and thus $xH = zH$. Similarly $yH = zH$, and thus $xH = yH$, as required.
\end{proof}

\begin{figure}[htbp]
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2 changes: 1 addition & 1 deletion elegantbook.cls
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%%%%%%%%%%%%%%%%%%%%%
% % !Mode:: "TeX:UTF-8"
\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{elegantbook}[2020/02/10 v3.10 ElegantBook document class]
\ProvidesClass{elegantbook}[2020/04/11 v4.0.0 ElegantBook document class]

\RequirePackage{kvoptions}
\RequirePackage{etoolbox}
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