P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 4

Theorem
Let $p \ge 3$ be a prime number.

Let $x_1 \in \Z_{\gt 0}: \dfrac {p + 1} 2 \le x_1 \lt p$

Let $a = x_1^2 + p$

Let $f \paren{X} \in \Z [X]$ be the polynomial $X^2 - a$

Let $f' \paren{X} \in \Z [X]$ be the formal derivative of $f \paren{X}$.

Then:
 * $\map {f’} {x_1} \not \equiv 0 \pmod p$

Proof
By Corollary to Absolute Value of Integer is not less than Divisors then:
 * $p \nmid x_1$
 * $p \nmid 2$

By Euclid's Lemma for Prime Divisors then:
 * $p \nmid 2x_1$

Hence:
 * $2x_1 \not \equiv 0 \mod p$

The formal derivative $f' \paren{X} \in \Z [X]$ of $f \paren{X}$ is by definition:
 * $2X$

Then:
 * $\map {f’} {x_1} = 2x_1 \not \equiv 0 \pmod p$