That's a way to state Hensel's lemma for $\mathbb{Q}_{p}$. Proof of the theorem: We inductively construct polynomials $F_{n}, G_{n}$ such that $F_{n}G_{n}\equiv h \mod p^{n}$. For $n=1$, we take $F_{1}=f$ and $G_{1}=g$. For $n\geq 1$, we try to find polynomials $f_{n},g_{n}$ such that we can set $F_{n+1}=F_{n}+p^{n}f_{n}, G_{n+1}=G_{n}+p^{n}g_{n}$. The only thing we need is \[h \equiv F_{n+1}G_{n+1}\equiv F_{n}G_{n}+p^{n}(f_{n}G_{n}+g_{n}F_{n}) \mod p^{n+1},\] thus \[h_{n}\equiv f_{n}G_{n}+g_{n}F_{n}\equiv f_{n}g+g_{n}f \mod p,\] where $h_{n}= \frac{h-F_{n}G_{n}}{p^{n}}\in \Gamma$. By $rf+sg \equiv 1 \mod p$ we get $h_{n}=rfh_{n}+sgh_{n}\mod p$, thus setting $f_{n}=rh_{n}, g_{n}=sh_{n}$ gives our result.