Equivalence of Definitions of Strictly Inductive Semigroup

$(1)$ implies $(2)$
Let $\struct {S, \circ}$ be a strictly inductive semigroup by definition $1$.

Then by definition $1$:

Thus $\struct {S, \circ}$ is a strictly inductive semigroup by definition $2$.

$(2)$ implies $(1)$
Let $\struct {S, \circ}$ be a strictly inductive semigroup by definition $2$.

Then by definition $2$:

Thus $\struct {S, \circ}$ is a strictly inductive semigroup by definition $1$.