Completely Multiplicative Function of Quotient

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

Then:
 * $\forall x, y \in \R, y \ne 0: f \left({\dfrac x y}\right) = \dfrac {f \left({x}\right)} {f \left({y}\right)}$

whenever $f \left({y}\right) \ne 0$.

Proof
Let $z = \dfrac x y$.

Then: