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

Let:

$\left|{S}\right| \ge \omega$

where $\omega$ denotes the minimal infinite successor set.

Then:

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

## Proof

By the hypothesis:

$S \sim \left|{S}\right|$
$S \times S \sim \left|{S}\right| \times \left|{S}\right|$

Therefore:

 $\displaystyle \left\vert{S \times S}\right\vert$ $=$ $\displaystyle \left\vert{\left\vert{S}\right\vert \times \left\vert{S}\right\vert}\right\vert$ Equivalent Sets have Equal Cardinal Numbers $\displaystyle$ $=$ $\displaystyle \left\vert{S}\right\vert$ Non-Finite Cardinal is equal to Cardinal Product

$\blacksquare$