Cardinal Product Equal to Maximum

Theorem
Let $S$ and $T$ be sets that are equinumerous to their cardinal number.

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

Suppose $S$ is infinite.

Suppose $T > 0$.

Then:


 * $\left|{S \times T}\right| = \max\left({\left|{S}\right|, \left|{T}\right| }\right)$

Proof
Let $x$ denote $\max \left({ \left|{ S }\right|, \left|{ T }\right| }\right)$.

Then by Cartesian Product Preserves Cardinality:
 * $S \times T \sim \left|{ S }\right| \times \left|{ T }\right|$

Let $f : S \times T \to \left|{ S }\right| \times \left|{ T }\right|$ be a bijection.

It follows that $f : S \times T \to x \times x$ is an injection.

Hence:

Therefore:
 * $\left|{ S \times T }\right| \le x$

Conversely:
 * $x = \left|{ S }\right|$ if $\left|{ T }\right| \le \left|{ S }\right|$

and:
 * $x = \left|{ T }\right|$ if $\left|{ S }\right| \le \left|{ T }\right|$

By Relation between Two Ordinals:
 * $x = \left|{ S }\right|$ or $x = \left|{ T }\right|$

It follows by Set Less than Cardinal Product that:
 * $x \le \left|{ S \times T }\right|$

Combining the two lemmas, it follows that:
 * $x = \left|{ S \times T }\right|$