Power Function Preserves Ordering in Ordered Group/Proof 2

Proof
An ordered group is an ordered structure which is also a group.

Hence an ordered group is also an ordered semigroup.

From Power Function Preserves Ordering in Ordered Semigroup:
 * $\forall x, y \in S: x \preceq y \implies x^n \preceq y^n$

From the Cancellation Laws, every element of a group is cancellable.

Hence from Power Function with Cancellable Element Preserves Strict Ordering in Ordered Semigroup:
 * $\forall x, y \in S: x \prec y \implies x^n \prec y^n$