  For an arrangement of $n$ lines in the real projective plane, we denote by
$f$ the number of regions into which the real projective plane is divided by
the lines. Using Bojanowski's inequality, we establish a new lower bound for
$f$. In particular, we show that if no more than $\frac{2}{3}n$ lines intersect
at any point, then $f \ge \frac{1}{6}n^{2}$
