Ordinal Exponentiation of Terms

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

Let $n$ be a natural number.

Let $x$ be a limit ordinal.

Let $y$, $z$ and $n$ be greater than $0$.

Then:


 * $\left({ x^y \times n }\right) ^z = x^{y \mathop \times z} \times n$ if $z$ is not a limit ordinal


 * $\left({ x^y \times n }\right) ^z = x^{y \mathop \times z}$ if $z$ is a limit ordinal

Proof
The proof shall proceed by Transfinite Induction on $z$.

Basis for the Induction
The hypothesis requires that $z \ne 0$, so the induction starts at $z = 1$.

This proves the basis for the induction.

Induction Step
This proves the induction step.

Limit Case
Suppose that this statement holds for all $w \in z$ where $z$ is a limit ordinal.

Then:

This proves the limit case.