De Moivre's Formula/Proof 1

Theorem
Let $z \in \C$ be a complex number expressed in complex form:
 * $z = r \paren {\cos x + i \sin x}$

Then:
 * $\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \, \map \sin {\omega x}$