Limit Ordinals Preserved Under Ordinal Multiplication

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

Let $x$ be nonzero.

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, it follows that $\left({ x \times y }\right) \ne 0$.

So by Definition:Limit Ordinal, $\left({ x \times y }\right) \in K_{II} \lor \exists z \in \operatorname{On}: \left({ x \times y }\right) = z^+$

Suppose that $\left({ x \times y }\right) = z^+$ for some ordinal $z$.

It follows that:

But $\left({ x \times w }\right) + 1 \subseteq \left({ x \times w }\right) + x$ by Subset Right Compatible with Ordinal Addition, so:

But $w^+ \in y$ by Successor in Limit Ordinal, so $z^+ \in \left({ x \times y }\right)$, contradicting the fact that $z^+ = \left({ x \times y }\right)$.

Thus, $z^+ \ne \left({ x \times y }\right)$ and $\left({ x \times y }\right) \in K_{II}$