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:

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