Complex Multiplication is Commutative

Theorem
The operation of multiplication on the set of complex numbers $\C$ is commutative:
 * $\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$

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

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

Then: