Subset of Well-Ordered Set is Well-Ordered/Proof 2

Proof
By definition of well-ordered set, $\struct {S, \preceq}$ is:
 * a totally ordered set

and:
 * a well-founded set.

By Subset of Toset is Toset, $\struct {T, \preceq'}$ is a totally ordered set.

By Subset of Well-Founded Relation is Well-Founded, $\preceq'$ is a well-founded relation.

Hence the result.