Complex Cross Product Distributes over Addition

Theorem
Let $z_1, z_2, z_3 \in \C$ be complex numbers.

Then:
 * $z_1 \times \left({z_2 + z_3}\right) = z_1 \times z_2 + z_1 \times z_3$

where $\times$ denotes cross product.

Proof
Let:
 * $z_1 = x_1 + i y_1$
 * $z_2 = x_2 + i y_2$
 * $z_3 = x_3 + i y_3$

Then: