Lemma 5.4. Assuming that:

$(K, | ∙ |)$ is a complete discretely valued field
such that $k : = O_{K} ∕ m$ is a perfect field of characterist $p$
$π \in O_{K}$ a fixed uniformiser
$x, y \in O_{K}$ such that $x \equiv y mod π^{k}$ ( $k \geq 1$ )

Then

x^{p} \equiv y^{p} mod π^{k + 1}

.