Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum

Theorem
Let $f$ be a real function.

Let $S$ be a subset of the domain of $f$.

Let $\displaystyle \sup_{x \mathop \in S} \left\{{f \left({x}\right)}\right\}$ and $\displaystyle \inf_{x \mathop \in S} \left\{{f \left({x}\right)}\right\}$ exist.

Then $\displaystyle \sup_{x, y \mathop \in S} \left\{{\left\vert{f \left({x}\right) - f \left({y}\right)}\right\vert}\right\}$ exists and:


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