Subset is Compatible with Ordinal Multiplication

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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.

$\blacksquare$