Restriction of Well-Founded Ordering
Let $\preceq'$ be the restriction of $\preceq$ to $T$.
Then $\preceq'$ is a well-founded ordering of $T$.
Let $A$ be a non-empty subset of $T$.
Then $A$ is a non-empty subset of $S$.
Let $x \in A$ and suppose $x \preceq' m$.
Then by the definition of restriction, $x \preceq m$.
Thus by the definition of a minimal element, $x = m$.
As this holds for all $x \in A$, $m$ is minimal in $A$ with respect to $\preceq'$.
As this holds for all subsets $A$ of $T$, $\preceq'$ is a well-founded ordering of $T$.