diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf
index 512fa95..9be3c50 100644
Binary files a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf and b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf differ
diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex
index 48d5332..f5cb705 100644
--- a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex
+++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex
@@ -11,6 +11,7 @@
 
 \newtheorem*{obs}{Observation}
 \newtheorem*{prop}{Proposition}
+\newtheorem*{lem}{Lemma}
 
 \begin{document}
 \maketitle