Naturally Ordered Semigroup is Unique/Isomorphism is Unique

Theorem
Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be naturally ordered semigroups.

Let:
 * $0'$ be the smallest element of $S'$
 * $1'$ be the smallest element of $S' \setminus \set {0'} = S'^*$.

Then the isomorphism $g: \struct {S, \circ, \preceq} \to \struct {S', \circ', \preceq'}$ defined as:
 * $\forall a \in S: \map g a = \circ'^a 1'$

is unique.

Proof
Let $f: S \to S'$ be another isomorphism different from $g$.

$\map f 1 \ne 1'$.

We show by induction that $1' \notin \Cdm f$.

... Thus $1' \notin \Cdm f$ which is a contradiction.

Thus $\map f 1 = 1$ and it follows

that $f = g$.

Thus the isomorphism $g$ is unique.