Cardinality of Cartesian Product of Finite Sets/Corollary/Proof 2

Proof
Let $f: S \times T \to T \times S$ be the mapping defined as:
 * $\forall \tuple {s, t} \in S \times T: \map f {s, t} = \tuple {t, s}$

which is shown to be bijective as follows:

showing $f$ is an injection.

Let $\tuple {t, s} \in T \times S$.

Then:
 * $\exists \tuple {s, t} \in S \times T: \map f {s, t} = \tuple {t, s}$

showing that $f$ is a surjection.

So we have demonstrated that there exists a bijection from $S \times T$ to $T \times S$.

The result follows by definition of set equivalence.