Restriction of Strict Well-Ordering is Strict Well-Ordering
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Theorem
Let $R$ be a strict well-ordering of $A$.
Let $B \subseteq A$.
Then $R$ is a strict well-ordering of $B$.
Proof
By Restriction of Strictly Well-Founded Relation is Strictly Well-Founded, $R$ is a strictly well-founded relation on $B$.
By Restriction of Total Ordering is Total Ordering, $R$ is a total ordering on $B$.
By the above two statements, $R$ is a strict well-ordering of $B$.
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$\blacksquare$