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.
$\blacksquare$