Restriction of Strict Well-Ordering is Strict Well-Ordering

Theorem
Let $R$ strictly well-order $A$ and $B \subseteq A$. Then $R$ strictly well-orders $B$.

Proof
By Foundational Relation Subset, $R$ is a foundational relation on $B$.

By Totally Ordered Subset, $R$ totally orders $B$.

By the above two statements, $R$ strictly well-orders $B$.