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:


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

where $i = \sqrt {-1}$ and $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.

Thus: