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 Cancellability in Ordered Semigroup:


 * $$m \preceq n \iff m \circ p \preceq n \circ p$$;
 * $$m \prec n \iff m \circ p \prec n \circ p$$.