Summation over Cartesian Product as Double Summation

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.

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

Outline of proof
We use induction on the cardinality of $T$. In the induction step, we use Sum over Disjoint Union of Finite Sets and Summation of Sum of Mappings on Finite Set.

Also see

 * Exchange of Order of Summation over Cartesian Product of Finite Sets