reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

theorem
  for N1,N2 being strict normal Subgroup of G ex N being strict normal
  Subgroup of G st the carrier of N = carr N1 * carr N2
proof
  let N1,N2 be strict normal Subgroup of G;
  set A = carr N1 * carr N2;
  set B = carr N1;
  set C = carr N2;
  carr N1 * carr N2 = carr N2 * carr N1 by Th125;
  then consider H being strict Subgroup of G such that
A1: the carrier of H = A by GROUP_2:78;
  now
    let a;
    thus a * H = a * N1 * C by A1,GROUP_2:29
      .= N1 * a * C by Th117
      .= B * (a * N2) by GROUP_2:30
      .= B * (N2 * a) by Th117
      .= H * a by A1,GROUP_2:31;
  end;
  then H is normal Subgroup of G by Th117;
  hence thesis by A1;
end;
