Strict Ordering Preserved under Product with Cancellable Element

Theorem
Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.

Let $x, y, z \in S$ be such that:
 * $(1): \quad z$ is cancellable for $\circ$
 * $(2): \quad x \prec y$

Then:
 * $x \circ z \prec y \circ z$
 * $z \circ x \prec z \circ y$

Proof
Let $z$ be cancellable and $x \prec y$.

Then by the definition of ordered semigroup:
 * $x \circ z \preceq y \circ z$

From the fact that $z$ is cancellable:
 * $x \circ z = y \circ z \iff x = y$

Thus as $x \circ z \ne y \circ z$ it follows from Strictly Precedes is Strict Ordering that:
 * $x \circ z \prec y \circ z$

Similarly, $z \circ x \prec z \circ y$ follows from $z \circ x \preceq z \circ y$.