Sum over Disjoint Union of Finite Sets

Theorem
Let $G$ be an abelian group.

Let $S$ and $T$ be finite disjoint sets.

Let $f : S \to G$ and $g : T \to G$ be mappings.

Let $h = f\cup g : S\cup T \to G$ be their union.

Then we have the equality of summations:


 * $\displaystyle \sum_{s \in S}f(s) + \sum_{t\in T}g(t) = \sum_{u \in S \cup T} h(u)$

Proof
Note that by Union of Finite Sets is Finite, $S\cup T$ is finite.