Congruence Relation/Examples/Equal Fourth Powers over Complex Numbers for Multiplication

Example of Congruence Relation
Let $\C$ denote the set of complex numbers.

Let $\RR$ denote the equivalence relation on $\C$ defined as:
 * $\forall w, z \in \C: z \mathrel \RR w \iff z^4 = w^4$

Then $\RR$ is a congruence relation for multiplication on $\C$.

Proof
Note that by Equivalence Relation Examples: Equal Fourth Powers over Complex Numbers, $\RR$ is an equivalence relation.

It remains to be shown that it is a congruence.

Let $w_1, w_2, z_1, z_2 \in \C$ such that:


 * $\paren {w_1 \mathrel \RR z_1} \land \paren {z_1 \mathrel \RR z_2}$

Then: