Principle of Mathematical Induction/Naturally Ordered Semigroup/General Result

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

Let $p \in S$.

Let $T \subseteq S$ such that:
 * $x \in T \implies p \preceq x \land \paren {x \in T \implies x \circ 1 \in T}$

Then:
 * $S \setminus S_p \subseteq T$

where:
 * $\setminus$ denotes set difference
 * $S_p$ denotes the set of all elements of $S$ preceding $p$.

Proof
Let $S_p$ be the set of all elements of $S$ preceding $p$:
 * $S_p = \set {x \in S: x \prec p}$

Let $T' = T \cup S_p$.

Then the set $T'$ satisfies the conditions of the Principle of Mathematical Induction for a Naturally Ordered Semigroup.

From that result:
 * $T' = S$

By Set Difference with Union is Set Difference:
 * $S \setminus S_p = T \setminus S_p$

By Set Difference is Subset:
 * $T \setminus S_p \subseteq T$

completing the proof.