  We strengthen Marshall Hall's Theorem to show that free groups are locally
extended residually alternating. Let F be any free group of rank at least two,
let H be a finitely generated subgroup of infinite index in F and let
{g_1,...,g_n} be a finite subset of F-H. Then there is a surjection f from F to
a finite alternating group such that f(g_i) is not in f(H) for any i. The
techniques of this paper can also provide symmetric quotients.
