Product of Two Compact Spaces is Compact

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ be topological spaces that are compact, and let $\mathcal{T}$ be the product topology on $X\times Y$, a topology on $X\times Y$ by The Product Topology is a Topology (i).

(i) $(X\times Y,\mathcal{T})$ is compact.