Strict Ordering Preserved under Product with Invertible Element

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

Let $z \in S$ be invertible.

Suppose that either $x \circ z \prec y \circ z$ or $z \circ x \prec z \circ y$.

Then $x \prec y$.

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

By Invertible Element of Monoid is Cancellable, $z^{-1}$ is cancellable.

Then from Strict Ordering Preserved under Product with Cancellable Element:


 * $x = \left({x \circ z}\right) \circ z^{-1} \prec \left({y \circ z}\right) \circ z^{-1} = y$

Likewise, if $z \circ x \prec z \circ y$:


 * $x = z^{-1} \circ \left({z \circ x}\right) \prec z^{-1} \circ \left({z \circ y}\right) = y$