Ordinal Multiplication by One

Theorem
Let $x$ be an ordinal.

Let $1$ denote the successor of $\varnothing$.

Proof
The proof of the other equality shall proceed by Transfinite Induction.

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

Induction Step
This proves the induction step.

Limit Case
This proves the limit case.