Let $K$ be a complete field with nonarchimedian valuation $|\cdot|$ and $\mathcal{O}_K := \{x \in K: \; |x| \le 1\}$, $\mathfrak{m} := \{x \in K: \; |x| < 1\}$.
I have seen two statements that do not seem to be equivalent called Hensel's Lemma:
1: Let $f(X) \in \mathcal{O}_K[X]$, $f(X) \notin \mathfrak{m}[X]$ such that $[f] = \overline{g} \cdot \overline{h}$ in $\mathcal{O}_K[X]/\mathfrak{m}$. Then there exist polynomials $g,h \in \mathcal{O}_K[X]$ with $f = gh$ and $\mathrm{deg}(g) = \mathrm{deg}(\overline{g})$ and $[g] = \overline{g}, [h] = \overline{h}$ in $\mathcal{O}_K[X]/\mathfrak{m}$.
2: Let $f(X) \in \mathcal{O}_K[X]$ and $\alpha \in \mathcal{O}_K$ such that $|f(\alpha)| < |f'(\alpha)|^2$. Then there exists some $\beta \in \mathcal{O}_K$ such that $f(\beta) = 0$ and $|\alpha - \beta| < |f'(\alpha)|$.
Does the first somehow imply the second?