# Exchange of Order of Summations over Finite Sets/Cartesian Product/Proof 3

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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

Let $n$ be the cardinality of $T$.

The proof goes by induction on $n$.

### Basis for the Induction

Let $n = 0$.

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### Induction Step

Let $n > 0$.

Let $t \in T$.

Use Cardinality of Set minus Singleton

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