Exponent Base of One

Theorem
Let $x$ be an ordinal.

Then:


 * $1^x = 1$

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

Basis for the Induction
This proves the basis for the induction.

Induction Step
The inductive hypothesis supposes that $1^x = 1$ for some $x$.

This proves the induction step.

Limit Case
The inductive hypothesis supposes that $\forall y \in x: 1^y = 1$.

It follows that:

This proves the limit case.