Non-Finite Cardinal is equal to Cardinal Product/Corollary

Corollary to Non-Finite Cardinal is equal to Cardinal Product
Let $S$ be a set that is equinumerous to its cardinal number.

Let $\card S$ denote the cardinal number of $S$.

Let:
 * $\card S \ge \omega$

where $\omega$ denotes the minimally inductive set.

Then:
 * $\card {S \times S} = \card S$

Proof
By hypothesis:
 * $S \sim \card S$

By Cartesian Product Preserves Cardinality:
 * $S \times S \sim \card S \times \card S$

Therefore: