Lexicographically Ordered Pair of Ordered Semigroups with Cancellable Elements

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 $\struct {S_1 \times S_2, \preccurlyeq_l} := \struct {S_1, \preccurlyeq_1} \otimes^l \struct {S_2, \preccurlyeq_2}$ denote the lexicographic order of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$.

Let every element of $\struct {S_1, \circ_1}$ be cancellable.

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