\begin{frame}{Basic Properties of Context-Free Languages}
  \begin{block}{Theorem}
    If $L_1$ is \alert{context-free} and $L_2$ \alert{regular}, then $L_1 \backslash L_2$ is \alert{context-free}.
  \end{block}
  \pause\medskip
  
  \begin{proof}
    $\overline{L_2}$ is regular,
    thus $L_1 \backslash L_2 = L_1 \cap \overline{L_2}$ is context-free.
  \end{proof}
  \pause\bigskip
  
  \begin{alertblock}{}
    $L_2 \backslash L_1$ is \emph{not} always context-free. Namely
    \begin{talign}
      \overline{L_1}=\Sigma^*\backslash L_1
    \end{talign} 
  \end{alertblock}
\end{frame}