reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem Th21:
  for G being finite Group,N1,N2 being normal Subgroup of G
  holds N1 is p-group &
  N2 is p-group implies ex N being strict normal Subgroup of G
  st the carrier of N = N1 * N2 & N is p-group
proof
  let G be finite Group;
  let N1,N2 be normal Subgroup of G;
  assume N1 is p-group & N2 is p-group; then
  consider N be strict Subgroup of G such that
A1: the carrier of N = N1 * N2 & N is p-group by Th20;
  for x,y be Element of G st y in N holds x * y * x" in N
  proof
    let x,y be Element of G;
    assume y in N;
    then y in the carrier of N;
    then consider a,b be Element of G such that
A2: y = a * b & a in N1 & b in N2 by A1,GROUP_11:6;
A3: x * y * x" =((x * a) * b) * x" by A2,GROUP_1:def 3
              .=(x * a) * (b * x") by GROUP_1:def 3
              .=(x * a) * 1_G * (b * x") by GROUP_1:def 4
              .=(x * a) * (x" * x) * (b * x") by GROUP_1:def 5
              .=(x * a) * x" * x * (b * x") by GROUP_1:def 3
              .=((x * a) * x") * (x * (b * x")) by GROUP_1:def 3
              .=(x * a * x") * (x * b * x") by GROUP_1:def 3;
    x * a * x" in N1 & x * b * x" in N2 by A2,GROUP_11:4;
    then x * y * x" in N1 * N2 by A3,GROUP_11:6;
    hence thesis by A1;
  end;
  then N is normal Subgroup of G by GROUP_11:5;
  hence thesis by A1;
end;
