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

theorem Th64:
  for G being Group
  for H being Subgroup of G
  for N being normal Subgroup of G
  st H,N are_complements_in G
  holds H * N = the carrier of G & N * H = the carrier of G
proof
  let G be Group;
  let H be Subgroup of G;
  let N be normal Subgroup of G;
  assume A1: H,N are_complements_in G;
  then H "\/" N = the multMagma of G by Th49;
  hence thesis by A1, Th63;
end;
