Simple Order Product of Pair of Ordered Semigroups is Ordered Semigroup

Theorem
Let $\struct {S_1, \circ_1, \preccurlyeq_1}$ and $\struct {S_2, \circ_2, \preccurlyeq_2}$ be ordered semigroups.

Let $\struct {S_1 \times S_2, \odot} := \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}$ denote the external direct product of $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$.

Let $\struct {S_1 \times S_2, \preccurlyeq_s} := \struct {S_1, \preccurlyeq_1} \otimes^s \struct {S_2, \preccurlyeq_2}$ denote the simple (order) product of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$.

Then $\struct {S_1 \times S_2, \odot, \preccurlyeq_s}$ is also an ordered semigroup.

Proof
From Simple Order Product of Pair of Ordered Sets is Ordered Set, $\struct {S_1 \times S_2, \otimes^s}$ is an ordered set.

From External Direct Product of Semigroups, $\struct {S_1 \times S_2, \odot}$ is a semigroup.

It remains to be shown that $\preccurlyeq_s$ is compatible with $\odot$.

Let $\tuple {x_1, x_2}, \tuple {y_1, y_2} \in S_1 \times S_2$ be arbitrary such that $\tuple {x_1, x_2} \preccurlyeq_s \tuple {y_1, y_2}$.

and the proof is complete.