Commensurability and elementary equivalence of polycyclic groups

It is shown that two polycyclic-by-finite groups G and H, satisfying the same sentences with one alternation of quantifiers, are commensurable. In fact we show something stronger: given n > 1 there is a subgroup H-n of H and a subgroup G(n) of G such that H-n similar or equal to G, G(n) similar or equal to H and the indices [G:G(n)] and [H:H-n] are finite and prime to n.