The number of points in a matroid with no n-point line as a minor

For any positive integer l we prove that if M is a simple matroid with no ( l + 2 ) -point line as a minor and with sufficiently large rank, then | E ( M ) | ⩽ q r ( M ) − 1 q − 1 , where q is the largest prime power less than or equal to l. Equality is attained by projective geometries over GF ( q ) .