Strict Ordering Preserved under Product with Invertible Element

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

If:
 * $(1): \quad z$ is invertible
 * $(2): \quad$ Either $x \circ z \prec y \circ z$ or $z \circ x \prec z \circ y$

then $x \prec y$.

Proof
Let $z$ be invertible.

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

Because $S$ is a semigroup, $z$ is cancellable from Invertible Element of Monoid is Cancellable.

Then from Cancellability in Ordered Semigroup:
 * $x = \left({x \circ z}\right) \circ z^{-1} \prec \left({y \circ z}\right) \circ z^{-1} = y$