 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 Th49:
  for G being Group
  for H being Subgroup of G
  for N being normal Subgroup of G
  holds H,N are_complements_in G
  iff (H "\/" N = the multMagma of G & H /\ N = (1).G)
proof
  let G be Group;
  let H be Subgroup of G;
  let N be normal Subgroup of G;
  the carrier of G c= the carrier of G; then
  reconsider A = the carrier of G as Subset of G;
  thus H,N are_complements_in G implies
    H "\/" N = the multMagma of G & H /\ N = (1).G
  proof
    assume A1: H,N are_complements_in G;
    then H "\/" N = gr (A) by GROUP_4:50;
    hence H "\/" N = the multMagma of G by Th48;
    thus H /\ N = (1).G by A1;
  end;
  assume A1: H "\/" N = the multMagma of G;
  A2: H * N = carr(H) * N by GROUP_4:43
           .= N * carr(H) by GROUP_3:120
           .= N * H by GROUP_4:43;
  assume H /\ N = (1).G;
  hence H,N are_complements_in G by A1, A2, GROUP_4:51;
end;
