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

Theorem
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Then:
 * $\forall m, n \in \struct {S, \circ, \preceq}: m \preceq n \implies \closedint m {n \circ 1} = \closedint m n \cup \set {n \circ 1}$

where $\closedint m n$ is the closed interval between $m$ and $n$.

Proof
Let $m \preceq n$. Then:

Thus:
 * $\closedint m {n \circ 1} = \closedint m n \cup \set {n \circ 1}$