  In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n
pseudolines, there are at least $\frac{6}{13}n$ simple points of intersection.
Since pseudoline arrangements are the topological representations of
reorientation classes of oriented matroids of rank $3$, in this paper, we will
use this result to prove by induction that an oriented paving matroid of rank
$r \ge 3$ on $n$ elements, where $n \geq 5+ r$, has at least
$\frac{12}{13(r-1)} \binom{n}{r-2}$ independent hyperplanes, yielding a new
necessary condition for a paving matroid to be orientable.
