Geometric Sequence in Lowest Terms has Coprime Extremes/Proof 2

Theorem
A geometric progression of integers in lowest terms has extremes which are coprime.

Proof
Let $P$ be a geometric progression of natural numbers of length $n$.

Let the common ratio of $P$ be expressed in canonical form as $\dfrac p q$.

From Construction of Geometric Progression in Lowest Terms:
 * $P = \left({q^{n-1}, p q^{n-2}, p^2 q^{n-3}, \ldots, p^{n - 2} q, p^{n - 1} }\right)$

By definition of canonical form:
 * $p \perp q$

It follows from Powers of Coprime Numbers are Coprime that:
 * $p^{n - 1} \perp q^{n - 1}$

Hence the result.