  The Multivariate Hensel Lemma for local rings is usually proved as a
consequence of the Grothendieck version of Zariski's Main Theorem. This version
deals with a more general situation that is a priori much more difficult. In
this paper, we give a direct proof of the Multivariate Hensel Lemma for
ultrametric fields, in the framework of constructive mathematics and without
using~ZMT. In the framework of classical mathematics, our result entails the
Lemma for rank-one valued fields.
