A corollary to the Wedderburn-Artin theorem.

Question

Suppose we proved the Wedderburn-Artin theorem, i.e. we have the fact that if S is a semisimple algebra over a field $F$, then $$ A \cong M_{n_1} (D_1) \times ... \times M_{n_k} (D_k), $$ where $D_1,...,D_k$ are skew fields. How can we prove now using this theorem that if $S$ is a simple algebra, then $S \cong M_n(D)$ for some integer $n$ and skew field $D$? My textbook actually says that it is an obvious corollary, however I cannot see the proof by myself.