Sum over Disjoint Union of Finite Sets

Theorem
Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$.

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

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

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

Then we have the equality of summations over finite sets:


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

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