Subset is Compatible with Ordinal Multiplication

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

Then:


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

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