Exchange of Order of Summations over Finite Sets/Cartesian Product/Proof 2

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

Let $\N_{< m}$ denote an initial segment of the natural numbers.

Let $\sigma: \N_{< m} \to S$ and $\tau : \N_{< n} \to T$ be bijections.

We have: