  We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is
normal (where the ground field $k$ is algebraically closed of characteristic
zero). This means on the one hand that a Hopf subalgebra is normal when
inducing (then restricting) modules several times as opposed to one time
creates no new simple constituents. This point of view was taken in the paper
http://arxiv.org/abs/math/0409346 which established a normality result in case
H and K are finite group algebras. On the other hand this means that K is
normal in H when H | K is a Galois extension with respect to action of
generalized bialgebras such as bialgebroids, weak Hopf algebras or Hopf
algebroids. The generalized Galois picture of depth two is the point of view we
take here: after showing the centralizer R is separable algebra via Hopf
invariant theory, we compute that the depth two semisimple Hopf algebra pair H
| K is free Frobenius extension with Markov trace satisfying all hypotheses
considered in Kadison and Nikshych, Frobenius Extensions and Weak Hopf
Algebras, J.Algebra 244 (2001). By the main theorem in that paper it is then a
Galois extension with action of semisimple weak Hopf algebra (also regular and
possessing Haar integral). Then the Galois canonical isomorphism (via coring
theory) restricted to integral induces algebra homomorphism from Hopf algebra
into weak Hopf algebra with kernel HK^+ = K^+H.
