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

Theorem
Let $$\left({S, \circ; \preceq}\right)$$ be a naturally ordered semigroup.

$$\forall m, n \in \left({S, \circ; \preceq}\right): m \preceq n \Longrightarrow \left[{m \,. \, . \, n \circ 1}\right] = \left[{m \,. \, . \, n}\right] \cup \left\{{n \circ 1}\right\}$$

... where $$\left[{m \,. \, . \, n}\right]$$ is the closed interval between $m$ and $n$.

Proof
Let $$m \preceq n$$. Then:

$$ $$ $$

$$ $$ $$ $$

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