  If G is a finitely generated powerful pro-p group satisfying a certain law
v=1, and if G can be generated by a normal subset T of finite width which
satisfies a positive law, we prove that G is nilpotent. Furthermore, the
nilpotency class of G can be bounded in terms of the prime p, the number of
generators of G, the law v=1, the width of T, and the degree of the positive
law. The main interest of this result is the application to verbal subgroups:
if G is a p-adic analytic pro-p group in which all values of a word w satisfy
positive law, and if the verbal subgroup w(G) is powerful, then w(G) is
nilpotent.
