Cancellability in Naturally Ordered Semigroup

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

Then:
 * $\forall m, n, p \in S: m \preceq n \iff m \circ p \preceq n \circ p$

It follows that:
 * $\forall m, n, p \in S: m \prec n \iff m \circ p \prec n \circ p$

Proof
From naturally ordered semigroup: NO 1, $\left({S, \circ, \preceq}\right)$ is a well-ordering, and therefore $\preceq$ is a total ordering.

From naturally ordered semigroup: NO 2, we have that all elements of $S$ are cancellable.

Thus from Strict Ordering Preserved under Product with Cancellable Element: