  Let lambda be aleph_0 or a strong limit of cofinality aleph_0. Suppose that
(G_m,p_{m,n}:m =< n<omega) and (H_m,p^t_{m,n}: m=< n < omega) are projective
systems of groups of cardinality less than lambda and suppose that for every
n<omega there is a homomorphism h:H_n-->G_n such that all the diagrams commute.
  If for every mu<lambda there exists (f_i in G_omega:i<mu) such that for
distinct i,j we have: f_i f_j^{-1} notin h_omega(H_omega), then there exists
(f_i in G_omega:i<2^lambda) such that for distinct i,j we have f_i f_j^{-1}
notin h_omega(H_omega).
