Cancellability in Ordered Semigroup

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Theorem

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

Let:

$(1): \quad z$ be cancellable for $\circ$

$(2): \quad x \prec y$.

Then:

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 a Strict Ordering that:

$x \circ z \prec y \circ z$

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

$\blacksquare$