diff --git a/main.tex b/main.tex
index 533f032..0159724 100644
--- a/main.tex
+++ b/main.tex
@@ -30,7 +30,7 @@ for $e\in E-F^*$:\\
 
 Consider this process on the lattice of flats. The set of flats of $M$ forms a geometric lattice. Take two flats $A,B$ that $A\subset B$ in the lattice and consider the sublattice between $A$ and $B$. This sublattice is exactly the lattice of flats of matroid $(M/A)\setminus (E-B)$.
 \begin{lemma}
-    Consider the ideal utilizations $u^*(e)$.
+Consider the ideal utilizations $u^*(e)$ assigned by the above algorithm.
 \begin{enumerate}
 \item $\sigma(M)\geq \sigma(M|F^*)$
 \item $u^*(e)$ is unique even though the $F^*$ may not be unique.
@@ -54,6 +54,7 @@ the last inequality follows from $\frac{c(E-F')}{r(E)-r(F')}\geq \frac{c(E-F^*)}
 \frac{c(E)-c(F_1)-c(F_2)}{r(E)-r(F_1)-r(F_2)}
 \leq \sigma
 \]
+...
 \end{enumerate}
 \end{proof}