  In 2006, Kawaguchi proved a lower bound for height of h(f(P)) when f is a
regular affine automorphism of A^2, and he conjectured that a similar estimate
is also true for regular affine automorphisms of A^n for n>2. In this paper we
prove Kawaguchi's conjecture. This implies that Kawaguchi's theory of canonical
heights for regular affine automorphisms of projective space is true in all
dimensions.
