Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup

Theorem
Let $\struct {S, \circ, \preceq}$ 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 $x \circ z \prec y \circ z$.

$x \succeq y$.

As $\struct {S, \circ, \preceq}$ is an ordered semigroup, $\preceq$ is compatible with $\circ$.

Hence we have:
 * $x \succeq y \implies x \circ z \succeq y \circ z$

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

We have that $\preceq$ is a total ordering, and that it is not the case that $x \succeq y$.

Hence by the Trichotomy Law:
 * $x \prec y$

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