Principle_of_Induction_applied_to_Interval_of_Naturally_Ordered_Semigroup

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $\left[{p \,.\,.\, q}\right]$ be a closed interval of $\left({S, \circ, \preceq}\right)$. Let $T \subseteq \left[{p \,.\,.\, q}\right]$ such that the minimal element of $\left[{p \,.\,.\, q}\right]$ is in $T$. Let $x \in T: x \prec q \implies x \circ 1 \in T$ Then $T = \left[{p \,.\,.\, q}\right]$