Strictly Inductive Semigroup is Inductive Semigroup

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

Then $\struct {S, \circ}$ is an inductive semigroup.

Proof
In accordance with our assertion, let $\struct {S, \circ}$ be a strictly inductive semigroup.

By definition of strictly inductive semigroup:


 * There exists $\beta \in S$ such that the only subset of $S$ containing both $\beta$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.

Let $\alpha = \beta$.

Then we may say:


 * There exist $\alpha, \beta \in S$ such that the only subset of $S$ containing both $\alpha$ and $x \circ \beta$ whenever it contains $x$ is $S$ itself.

Hence, by identifying $\alpha$ with $\beta$, $\struct {S, \circ}$ fulfils the conditions of being an inductive semigroup.