diff --git a/main.tex b/main.tex
index 17c8760..07cc40a 100644
--- a/main.tex
+++ b/main.tex
@@ -147,9 +147,12 @@ What is the first breakpoint on $L(\mu)$?
 We have $-\mu(b-\lambda_c)\leq w(C- F)-\mu(b-c(F))$ for any cut $C$ and $F\subsetneq C$. Thus the first breakpoint is $\mu=\min \frac{w(C- F)}{\lambda_c-c(F)}$, which is the value of normalized mincut.
 \newline
 
-What about other breakpoints? Currently I can only prove the following :(
+What about other breakpoints?
 \begin{lemma}
-$\lambda_i=\min \frac{w(C- F)-w(C_{i-1}- F_{i-1})}{c(F_{i-1})-c(F)}$, where the minimum is taken over all cut $C$ and $F\subset C$ such that both the numerator and denominator are positive.
+$\mu^*\in [\min\limits_{C,F} \frac{w(C\setminus F)}{\lambda_c - c(F)},\min\limits_{C,F} \frac{w(C\setminus F)}{b - c(F)}]$
+\end{lemma}
+\begin{lemma}
+$\mu_i=\min \frac{w(C\setminus F)-w(C_{i-1}\setminus F_{i-1})}{c(F_{i-1})-c(F)}$, where the minimum is taken over all cut $C$ and $F\subset C$ such that both the numerator and denominator are positive.
 \end{lemma}
 \end{frame}