--- a/slides/slides02.tex	Wed Sep 29 21:04:41 2021 +0100
+++ b/slides/slides02.tex	Wed Oct 06 20:43:35 2021 +0100
@@ -1194,6 +1194,20 @@
 \begin{frame}<1-8>[c]
 \end{frame}
 
+\begin{frame}[c]
+\end{frame}
+
+\begin{frame}[c]
+\end{frame}
+
+\begin{frame}[c]
+\end{frame}
+
+\begin{frame}<1-10>[c]
+\end{frame}
+
+
+
 \end{document}
 % below are slides for proving.