Relation Compatibility in Totally Ordered Semigroup

Theorem

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

$(1): \quad$ All the elements of $\left({S, \circ, \preceq}\right)$ are cancellable for $\circ$

$(2): \quad \preceq$ is a total ordering.

Then:

$\forall x, y, z \in S: x \circ z \preceq y \circ z \iff x \preceq y$

Proof

From Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup:

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

From the definition of cancellable element:

$x \circ z = y \circ z \implies x = y$

$\blacksquare$