Strict Ordering Preserved under Product with Invertible Element

Theorem
Let $\struct {S, \circ, \preceq}$ 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:


 * $x = \paren {x \circ z} \circ z^{-1} \prec \paren {y \circ z} \circ z^{-1} = y$

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


 * $x = z^{-1} \circ \paren {z \circ x} \prec z^{-1} \circ \paren {z \circ y} = y$