Completely Multiplicative Function of Quotient

Theorem
Let $f: \R \to \R$ be a completely multiplicative function.

Then:
 * $\forall x, y \in \R, y \ne 0: \map f {\dfrac x y} = \dfrac {\map f x} {\map f y}$

whenever $\map f y \ne 0$.

Proof
Let $z = \dfrac x y$.

Then: