Properties of Ordered Group/OG5

Theorem
Let $\struct {G, \circ, \preceq}$ be an ordered group with identity $e$.

Let $\prec$ be the reflexive reduction of $\preceq$.

Let $x \in G$.

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

Then the following hold:
 * $x \preceq e \implies x^n \preceq e$
 * $e \preceq x \implies e \preceq x^n$
 * $x \prec e \implies x^n \prec e$
 * $e \prec x \implies e \prec x^n$