Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $x \in \Q_p$.

Let the canonical expansion of $x$ be eventually periodic.

Then:
 * $\exists r \in \Q, n \in \Z, y \in \Q_p$:
 * $(1) \quad x = r + p^n y$
 * $(2) \quad$the canonical expansion of $y$ is periodic