Principle of Induction applied to Interval of Naturally Ordered Semigroup

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

Let $\closedint p q$ be a closed interval of $\struct {S, \circ, \preceq}$.

Let $T \subseteq \closedint p q$ such that the minimal element of $\closedint p q$ is in $T$.

Let:
 * $x \in T: x \prec q \implies x \circ 1 \in T$

Then:
 * $T = \closedint p q$

Proof
Let $T' = T \cup \set {x \in S: q \prec x}$.

Then $T'$ satisfies the conditions of the Principle of Mathematical Induction.

Therefore:
 * $T' = \set {x \in S: p \preceq x}$

Therefore:
 * $T = \closedint p q$