  We prove that if M is a vertically 4-connected matroid with a modular flat X
of rank at least three, then every representation of M | X over a finite field
F extends to a unique F-representation of M. A corollary is that when F has
order q, any vertically 4-connected matroid with a PG(2, F)-restriction is
either F-representable or has a U_{2, q^2+1}-minor. We also show that no
excluded minor for the class of F-representable matroids has a PG(2,
F)-restriction.
