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

theorem
  for G being finite Group, N being normal Subgroup of G
  st N is p-group & G./.N is p-group holds
  G is p-group
proof
  let G be finite Group;
  let N be normal Subgroup of G;
  assume that
A1: N is p-group and
A2: G./.N is p-group;
  consider r1 be Nat such that
A3: card N = p |^ r1 by A1;
  consider r2 be Nat such that
A4: card (G./.N) = p |^ r2 by A2;
 card(G./.N) = index N by GROUP_6:27;
  then card G = p |^ r1 * p |^ r2 by A3,A4,GROUP_2:147
        .= p |^ (r1 + r2) by NEWTON:8;
  hence thesis;
end;
