Inductive Semigroup whose Inductive Elements Commute is Commutative Semigroup

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let there exist $\alpha, \beta \in S$ which fulfil the condition for $\struct {S, \circ}$ to be an inductive semigroup:
 * the only subset of $S$ containing both $\alpha$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.

Let $\alpha$ and $\beta$ commute.

Then $\struct {S, \circ}$ is a commutative semigroup.