Let $(K,v)$ be a valued field where $\Gamma$ is its valued group. Let $\Delta$ be convex subgroup of $\Gamma$ and consider the coarse valuation $\hat{v}:= K \rightarrow \Gamma/\Delta$ which sends each element $x$ to the class $v(x)+\Delta$. A valued field if said to be henselian if given any polynomial $P(x) \in O[x]$ where $O$ is the valuation ring (i.e. $O=\{ x \in K \| \ v(x)\geq 0\}$), if there is some element $\alpha \in K$ such that $v(P(\alpha))>0$ and $v(P'(\alpha))=0$ then we can find an element $\alpha' \in K$ such that $v(\alpha-\alpha')>0$ and $P(\alpha')=0$. (i.e. any non singular zero of the residue field can be lifted to the field.) I would like to know in detail how the following folklore statement is proved:\
**Let $(K,v, \Gamma)$ be a henselian valued field and $(K, \hat{v}, \Gamma/\Delta)$ the valued field with a coarse valuation. Then the valuation $\hat{v}$ is also henselian. **
