First Element of Geometric Sequence not dividing Second/Proof 2

Proof
From Form of Geometric Progression of Integers, the terms of $P$ are in the form:
 * $(1): \quad: a_j = k q^j p^{n - j}$

where $p \perp q$.

Let $a_0 \nmid a_1$.

That is:

Aiming for a contradiction, suppose that:
 * $\exists i, j \in \left\{{0, 1, \ldots, n}\right\}, i \ne j: a_i \mathop \backslash a_j$

let $i < j$.

But we have that $p \perp q$.

From Powers of Coprime Numbers are Coprime:
 * $q^{j - i} \perp p^{j - i}$

This can happen only when $q = 1$.

This is incompatible with $q \nmid p$.

From this contradiction, the result follows.