Power Function Preserves Ordering in Ordered Semigroup/Proof 1

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Theorem

Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.

Let $x, y \in S$ such that $x \preceq y$.

Let $n \in \N_{>0}$ be a strictly positive integer.


Then:

$x^n \preceq y^n$

where $x^n$ is the $n$th power of $x$.


Proof

By definition of ordered semigroup:

$\preceq$ is compatible with $\circ$.

By definition of ordering:

$\preceq$ is transitive.

Thus by Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements:

$x^n \preceq y^n$

$\blacksquare$