Additive Function is Strongly Additive/Proof 1

Proof
From Set Difference and Intersection form Partition:


 * $A$ is the union of the two disjoint sets $A \setminus B$ and $A \cap B$
 * $B$ is the union of the two disjoint sets $B \setminus A$ and $A \cap B$.

So, by the definition of additive function:
 * $\map f A = \map f {A \setminus B} + \map f {A \cap B}$
 * $\map f B = \map f {B \setminus A} + \map f {A \cap B}$

We also have from Set Difference is Disjoint with Reverse that:
 * $\paren {A \setminus B} \cap \paren {B \setminus A} = \O$

From Sum of Additive Function Values is Defined, it follows that $\map f A + \map f B$ is defined.

Hence:

Hence the result.