Equivalence of Definitions of Strictly Well-Ordered Set

Proof
By definition, a strictly totally ordered set is a relational structure such that $\RR$ is a strict total ordering.

By definition, a strict well-ordering $\RR$ is a strict total ordering such that $\RR$ is a strictly well-founded relation.

So $\struct {S, \RR}$ is a relational structure such that $\RR$ is a strict well-ordering on $S$.

Hence both definitions specify:


 * a relational structure $\struct {S, \RR}$

such that:
 * $\RR$ is a strict total ordering
 * $\RR$ is a strictly well-founded relation.