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:
 * $\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in T} f(s, t) = \sum_{t \mathop \in T} \sum_{s \mathop \in S} f(s, t)$

Outline of Proof
Using the definition of summations over finite sets, we can reduce this to Exchange of Order of Indexed Summations over Rectangular Domain.

Proof
Let $m$ be the cardinality of $S$ and $n$ be the cardinality of $T$.

Let $\N_{ 0$.

Let $t\in T$.

Use Cardinality of Set minus Singleton