Let n = [x1, : : :, xn] be the nth lower central word. Suppose that G is a pro, nite group where the conjugacy classes x n(G) contains less than 2@ 0 elements for any x 2 G. We prove that then n+1(G) has , nite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is , nite. Moreover, it implies that a pro, nite group G is , nite-by-nilpotent if and only if there is a positive integer n such that x n(G) contains less than 2@ 0 elements, for any x 2 G.