Multiple_of_Supremum_v2

Let $S \subseteq \R: S \ne \varnothing$ be a non-empty subset of the set of real numbers $\R$. Let $S$ be bounded above. Let $z \in \R: z > 0$ be a positive real number. Then $\displaystyle \map {\sup_{x \mathop \in S} } {z x} = z \map {\sup_{x \mathop \in S} } x$