Well-Ordering is not necessarily Usual Ordering

Theorem
Let $S$ be a set of numbers.

According to the Well-Ordering Theorem, $S$ can be well-ordered.

However, the usual ordering on $S$ may not necessarily be a well-ordering.

Proof
From Rational Numbers are Well-Orderable, it is possible to apply a well-ordering to the set of rational numbers $\Q$.

However, the usual ordering on $\Q$ is not a well-ordering.

Indeed:
 * $\set {x \in \Q: x \le 0}$

has no smallest element.