Exchange of Order of Summations over Finite Sets/Cartesian Product/Proof 1

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Theorem

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


Then we have an equality of summations over finite sets:

$\ds \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{t \mathop \in T} \sum_{s \mathop \in S} \map f {s, t}$


Proof