Closed_Interval_of_Naturally_Ordered_Semigroup_with_Successor_equals_Union_with_Successor

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Then $\forall m, n \in \left({S, \circ, \preceq}\right): m \preceq n \implies \left[{m \,.\,.\, n \circ 1}\right] = \left[{m \,.\,.\, n}\right] \cup \left\{{n \circ 1}\right\}$ where $\left[{m \,.\,.\, n}\right]$ is the closed interval between $m$ and $n$.