Complex Multiplication is Associative

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

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: