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

theorem
  for G being finite commutative Group,H,H1,H2 being Subgroup of G st
  H1 is p-commutative-group & H2 is p-commutative-group &
  the carrier of H = H1 * H2 holds H is p-commutative-group
proof
  let G be finite commutative Group;
  let H,H1,H2 be Subgroup of G;
  assume that
A1: H1 is p-commutative-group &
  H2 is p-commutative-group and
A2: the carrier of H = H1 * H2;
  H is p-group by A1,A2,Th19;
  hence thesis;
end;
