  In this paper we shall give sufficient conditions for local CR
diffeomorphisms between two real analytic submanifolds of $\Bbb C^N$ to be
determined by finitely many derivatives at finitely many points. These
conditions will also be shown to be necessary in model cases. We shall also
show that under the same conditions, the Lie algebra of the infinitesimal CR
automorphisms at a point is finite dimensional.
