 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem
  for H1,H2 being Subgroup of G st [. H1, H2 .] = (1).G
  for a,b being Element of G st a in H1 & b in H2
  holds a*b = b*a
proof
  let H1,H2 be Subgroup of G;
  assume A1: [. H1, H2 .] = (1).G;
  let a,b be Element of G;
  assume A2: a in H1;
  assume A3: b in H2;
  [.a,b.] = 1_G by A1, A2, A3, GROUP_5:1, GROUP_5:65;
  hence thesis by GROUP_5:36;
end;
