Additive Function is Strongly Additive

Theorem
Let $\SS$ be an algebra of sets.

Let $f: \SS \to \overline \R$ be an additive function on $\SS$.

Then $f$ is also strongly additive.

That is:


 * $\forall A, B \in \SS: \map f {A \cup B} + \map f {A \cap B} = \map f A + \map f B$