 Coprime-factor Hensel lifting

A monic polynomial over a complete discrete valuation ring factors with prescribed coprime monic residue factors. Given monic lifts $g_r,h_r$ correct modulo ${\pi }^r$, the degree-bounded map $(u,v)\mapsto u{\bar{h}}_r+v{\bar{g}}_r$ is a bijection over the residue field, by coprimality and equality of dimensions. Solve it for the error $(f-g_r{h}_r)/{\pi }^r$, and replace the lifts by $g_r+{\pi }^{r}u,h_r+{\pi }^{r}v$. Their coefficients converge by completeness and the limiting product is $f$. The completeness assumption cannot be replaced just by localization.