Product of Finite Sets is Finite/Proof 2
Let $S$ and $T$ be finite sets.
Then $S \times T$ is a finite set.
Let $\card S$ denote the cardinal number of $S$.
Let $\cdot$ denote ordinal multiplication.
By Cardinal Product Equinumerous to Ordinal Product, it follows that $S \times T \sim \card S \cdot \card T$.
But then $\card S$ and $\card T$ are members of the minimal infinite successor set.
Therefore, $\card S \cdot \card T \in \omega$ by Natural Number Multiplication is Closed.