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 Lm5;
  then consider H being strict Subgroup of G such that
A1: the carrier of H = A by Th78;
  now
    let a;
    thus a + H = a + N1 + C by A1,ThB29
      .= N1 + a + C by Th117
      .= B + (a + N2) by ThA30
      .= B + (N2 + a) by Th117
      .= H + a by A1,ThB31;
  end;
  then H is normal Subgroup of G by Th117;
  hence thesis by A1;
end;
