Principle of Induction applied to Interval of Naturally Ordered Semigroup

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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:

$T = \closedint p q$

$\blacksquare$


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