Complex Multiplication is Associative

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

Proof
From the definition of complex numbers, we define the following:
 * $z_1 := \left({x_1, y_1}\right)$
 * $z_2 := \left({x_2, y_2}\right)$
 * $z_3 := \left({x_3, y_3}\right)$

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

Thus: