Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup

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

If either:
 * $x \circ z \prec y \circ z$

or
 * $z \circ x \prec z \circ y$

then $x \prec y$.

Proof
Let $\preceq$ be a total ordering.

Let $x \circ z \prec y \circ z$.

But we have, by hypothesis:
 * $x \succeq y \implies x \circ z \succeq y \circ z$

which contradicts $x \circ z \prec y \circ z$.

So $x \prec y$.

Similarly for $z \circ x \prec z \circ y$.