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:


 * $m \times \paren {x + n} = x + \paren {m \times n}$

Lemma
By Ordinal Multiplication is Left Distributive:


 * $m \times \paren {x + n} = \paren {m \times x} + \paren {m \times n}$

It remains to prove that $x = m \times x$.

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