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:

$$ $$ $$ $$ $$ $$ $$ $$