Cardinal Product Equal to Maximum

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

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

Suppose $S$ is infinite.

Suppose $T > 0$.

Then:


 * $\card {S \times T} = \map \max {\card S, \card T}$

Proof
Let $x$ denote $\map \max {\card S, \card T}$.

Then by Cartesian Product Preserves Cardinality:
 * $S \times T \sim \card S \times \card T$

Let $f: S \times T \to \card S \times \card T$ be a bijection.

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

Hence:

Therefore:
 * $\card {S \times T} \le x$

Conversely:
 * $x = \card S$ if $\card T \le \card S$

and:
 * $x = \card T$ if $\card S \le \card T$

By Relation between Two Ordinals:
 * $x = \card S$ or $x = \card T$

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

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