Supremum of Absolute Value of Difference equals Supremum of Difference

Theorem
Let $S$ be a non-empty real set.

Let $\displaystyle \sup_{x, y \mathop \in S} \left\{{x − y}\right\}$ exist.

Then $\displaystyle \sup_{x, y \mathop \in S} \left\{{\left\lvert{x − y}\right\rvert}\right\}$ exists and:


 * $\displaystyle \sup_{x, y \mathop \in S} \left\{{\left\lvert{x − y}\right\rvert}\right\} = \sup_{x, y \mathop \in S} \left\{{x − y}\right\}$.