 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem Th63:
  for G being Group
  for H1,H2 being Subgroup of G
  st H1 * H2 = the carrier of (H1 "\/" H2)
  holds H1 * H2 = H2 * H1
proof
  let G be Group;
  let H1,H2 be Subgroup of G;
  assume A1: H1 * H2 = the carrier of (H1 "\/" H2);

  then A2: H2 * H1 c= H1 * H2 by Th62;
  for x being Element of G st x in H1 * H2 holds x in H2 * H1
  proof
    let x be Element of G;
    assume x in H1 * H2;
    then x in H1 "\/" H2 by A1;
    then x" in H1 "\/" H2 by GROUP_2:51;
    then consider h1,h2 being Element of G such that
    B2: x" = h1*h2 & h1 in H1 & h2 in H2 by A1, GROUP_5:4;
    B3: x = (x")"
         .= (h2") * (h1") by B2, GROUP_1:17;
    h2" in H2 & h1" in H1 by B2,GROUP_2:51;
    hence x in H2 * H1 by B3,GROUP_5:4;
  end;

  then H1 * H2 c= H2 * H1;
  hence thesis by A2,XBOOLE_0:def 10;
end;
