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


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