Vector Cross Product is Anticommutative/Complex

Theorem
The vector cross product is anticommutative:
 * $\forall z_1, z_2 \in \C: z_1 \times z_2 = -\left({z_2 \times z_1}\right)$

Proof
Let:
 * $z_1 := x_1 + i y_1, z_2 = x_2 + i y_2$

Then: