Exchange of Order of Summations over Finite Sets/Cartesian Product

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

Let $S, T$ be finite sets.

Let $S \times T$ be their cartesian product. Let $f: S \times T \to \mathbb A$ be a mapping.

Then we have an equality of summations over finite sets:
 * $\ds \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{t \mathop \in T} \sum_{s \mathop \in S} \map f {s, t}$