 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 H,K being normal Subgroup of G
  st H /\ K = (1).G
  for h,k being Element of G st h in H & k in K
  holds h*k = k*h
proof
  let H,K be normal Subgroup of G;
  assume A1: H /\ K = (1).G;
  let h,k be Element of G;
  assume A2: h in H;
  assume A3: k in K;
  [. h, k .] in H /\ K
  proof
    A4: [. h, k .] in H
    proof
      h" in H & h |^ k in H by A2, ThNorm, GROUP_2:51;
      then (h") * (h |^ k) in H by GROUP_2:50;
      hence [. h, k .] in H by GROUP_5:18;
    end;
    [. h, k .] in K
    proof
      k" in K by A3, GROUP_2:51;
      then (k") |^ h in K by ThNorm;
      then ((k") |^ h) * k in K by A3, GROUP_2:50;
      hence [. h, k .] in K by GROUP_5:18;
    end;
    hence thesis by A4, GROUP_2:82;
  end;
  then 1_G = [. h, k .] by A1, GROUP_5:1;
  hence thesis by GROUP_5:36;
end;
