Finite Ordinal Times Ordinal

Theorem
Let $m$ and $n$ be finite ordinals.

Let $m \ne 0$, where $0$ is the zero ordinal.

Let $x$ be a limit ordinal.

Then:


 * $m \times \left({ x + n }\right) = x + \left({ m \times n }\right)$

Proof
By Ordinal Multiplication is Left Distributive, it follows that:


 * $m \times \left({ x + n }\right) = \left({ m \times x }\right) + \left({ m \times n }\right)$

It remains to prove that $x = \left({ m \times x }\right)$.

Lemma
Suppose $x = \omega$.

Conversely, $\omega \le \left({ m \times \omega }\right)$ by Subset Right Compatible with Ordinal Multiplication.

Since $x$ is a limit ordinal, it follows that: