Complex Multiplication Distributes over Addition

Theorem
The operation of multiplication on the set of complex numbers $\C$ is distributive over the operation of addition.
 * $\forall z_1, z_2, z_3 \in \C:$
 * $z_1 \paren {z_2 + z_3} = z_1 z_2 + z_1 z_3$
 * $\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$

Proof
From the definition of complex numbers, we define the following:

where $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.

Thus:

The result $\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$ follows directly from the above, and the fact that Complex Multiplication is Commutative.