Supremum_of_Subset

Let $\left({U, \preceq}\right)$ be an ordered set. Let $S \subseteq U$. Let $T \subseteq S$. Let $S$ admit a supremum (in $U$). If $T$ also admits a supremum (in $U$), then $\sup \left({T}\right) \preceq\sup \left({S}\right)$.