Hensel's Lemma/P-adic Integers/Lemma 6

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $\beta \in \Z_p$ be a $p$-adic integer.

Let $\map {F'} X \in \Z_p \sqbrk X$ be a polynomial.

Let $a \in \Z_p$ be a $p$-adic integer:
 * $\map {F'} a \ne 0$

Let $T$ be the set of $p$-adic digits.

Then
 * $\exists b_k \in T : b_k p^k \equiv \dfrac {-\beta p^k} {\map {F'} a} \pmod{p^{k+1}\Z_p}$