Product of Finite Sets is Finite/Proof 2

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Theorem

Let $S$ and $T$ be finite sets.


Then $S \times T$ is a finite set.


Proof

Let $\left|{ S }\right|$ denote the cardinal number of $S$.

Let $\cdot$ denote ordinal multiplication.


By Cardinal Product Equinumerous to Ordinal Product, it follows that $S \times T \sim \left|{ S }\right| \cdot \left|{ T }\right|$.

But then $\left|{ S }\right|$ and $\left|{ T }\right|$ are members of the minimal infinite successor set.

Therefore, $\left|{ S }\right| \cdot \left|{ T }\right| \in \omega$ by Natural Number Multiplication is Closed.


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

$\blacksquare$


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