Product of Finite Sets is Finite/Proof 2

Proof
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 minimally inductive set.

Therefore, $\card S \cdot \card T \in \omega$ by Natural Number Multiplication is Closed.

Since $S \times T$ is equinumerous to a member of the minimally inductive set, it follows that $S \times T$ is finite.