Unity Function is Completely Multiplicative

Theorem
Let $$f_1: \Z^*_+ \to \Z^*_+$$ be the constant function:
 * $$\forall n \in \Z^*_+: f_1 \left({n}\right) = 1$$.

Then $$f_1$$ is completely multiplicative.

Proof
$$\forall m, n \in \Z^*_+: f_1 \left({m n}\right) = 1 = f_1 \left({m}\right) f_1 \left({n}\right)$$