Prove Lemma (7.13).
Let's remind ourselves of Lemma (7.13).
Let $p$ be an odd prime and $a$ be an integer such that $p \not \mid a$. Then
$$ a, 2a, 3a, ⋯ , (\frac{p-1}{2})a \not \equiv 0 \pmod p $$
Consider the number $ka$ where $k$ is an integer such that $1 \le k \le (\frac{p-1}{2})$.
Since $p$ does not divide $a$, then we consider whether $p$ divides $k$. We can see that $p$ does not divide $k$ because $k < p$.
And so $p$ does not divide $ka$. Equivalently, $ka \not \equiv 0 \pmod p$.
That is,
$$ a, 2a, 3a, ⋯ , (\frac{p-1}{2})a \not \equiv 0 \pmod p $$