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

theorem Th19:
  for G being finite commutative Group,H,H1,H2 being Subgroup of G st
  H1 is p-group & H2 is p-group &
  the carrier of H = H1 * H2 holds H is p-group
proof
  let G be finite commutative Group;
  let H,H1,H2 be Subgroup of G;
  assume that
A1: H1 is p-group & H2 is p-group and
A2: the carrier of H = H1 * H2;
for a being Element of H holds a is p-power
  proof
    let a be Element of H;
    a in H1 * H2 by A2;
    then consider b1,b2 be Element of G such that
A3: a = b1 * b2 & b1 in H1 & b2 in H2 by GROUP_11:6;
    b1 is p-power & b2 is p-power by A1,A3,Th15;
    then consider r be Nat such that
A4: ord (b1 * b2) = p |^ r by Def1;
    ord a = p |^ r by A3,A4,GROUP_8:5;
    hence thesis;
  end;
  hence thesis by Th17;
end;
