Completely Additive Function is Additive

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Theorem

Let $f: \N \to \C$ be a completely additive function.


Then $f$ is also additive.


Proof

Let $m, n$ be coprime integers.

Then in particular, $m, n \in \Z$.

Hence, since $f$ is completely additive:

$f \left({m \times n}\right) = f \left({m}\right) + f \left({n}\right)$

and $f$ is additive, as desired.

$\blacksquare$