Ordinal Multiplication by Zero

Theorem
Let $x$ be an ordinal.

Proof
For $\left({\varnothing \cdot x}\right) = \varnothing$, the proof shall proceed by Transfinite Induction on $x$.

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.