Limit Ordinals Preserved Under Ordinal Multiplication

Theorem
Let $x$ and $y$ be ordinals.

Let $x$ be non-empty.

Let $y$ be a limit ordinal.

It follows that the ordinal product $\left({x \times y}\right)$ is a limit ordinal.

Proof
$y$ is a limit ordinal and thus is nonzero, by definition.

$x$ and $y$ are both nonzero.

So by Ordinals have No Zero Divisors:
 * $x \times y \ne 0$

So by definition of limit ordinal:
 * $x \times y \in K_{II} \lor \exists z \in \On: x \times y = z^+$

Suppose that $x \times y = z^+$ for some ordinal $z$.

It follows that:

But by Subset is Right Compatible with Ordinal Addition:
 * $\paren {x \times w} + 1 \subseteq \paren {x \times w} + x$

Therefore:

But by Successor in Limit Ordinal:
 * $w^+ \in y$

Therefore:
 * $z^+ \in x \times y$

contradicting the fact that $z^+ = x \times y$.

Thus:
 * $z^+ \ne x \times y$

and:
 * $x \times y \in K_{II}$