Lifting The Exponent Lemma for p=2/Corollary

Corollary to Lifting The Exponent Lemma for p=2
Let $x, y \in \Z$ be odd integers.

Let $n \in \Z_{\ge 0}$ be an even positive integer.

Then
 * $\nu_2 \left({x^n - y^n}\right) = \nu_2 \left({x + y}\right) + \nu_2 \left({x - y}\right) + \nu_2 \left({n}\right) - 1$

where $\nu_2$ denotes $2$-adic valuation.

Proof
By Square Modulo 4, $4\mid x^2-y^2$.

By Lifting The Exponent Lemma for p=2: