Non-Finite Cardinal is equal to Cardinal Product/Corollary

Theorem
Let $S$ be a set that is equinumerous to its cardinal number.

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

Suppose that $\left|{ S }\right| \ge \omega$, where $\omega$ denotes the minimal infinite successor set.

Then:


 * $\left|{ S \times S }\right| = \left|{ S }\right|$

Proof
By the hypothesis, $S \sim \left|{ S }\right|$.

By Cartesian Product Preserves Cardinality, $S \times S \sim \left|{ S }\right| \times \left|{ S }\right|$.

Therefore: