Well-Ordered Subset
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Theorem
Let $R$ strictly well-order $A$ and $B \subseteq A$. Then $R$ strictly well-orders $B$.
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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, $R$ well-orders $B$. $\blacksquare$
