Closed Interval of Naturally Ordered Semigroup with Successor equals Union with Successor

Theorem
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 $n$.

Proof
Let $m \preceq n$. Then:

Thus:
 * $\left[{m \, . \, . \, n \circ 1}\right] = \left[{m \, . \, . \, n}\right] \cup \left\{{n \circ 1}\right\}$