Subset is Compatible with Ordinal Multiplication

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

Then:


 * $(1): x \le y \implies \left({z \cdot x}\right) \le \left({z \cdot y}\right)$
 * $(2): x \le y \implies \left({x \cdot z}\right) \le \left({y \cdot z}\right)$

Proof
The result follows from Subset is Left Compatible with Ordinal Multiplication and Subset is Right Compatible with Ordinal Multiplication.