 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 Th56:
  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 G ./. N, H are_isomorphic
proof
  let G be Group;
  let H be Subgroup of G;
  let N be normal Subgroup of G;
  assume H,N are_complements_in G;
  then ex phi being Homomorphism of H,G./.N
  st (for h being Element of H
      for g being Element of G st g = h
      holds phi.h = g*N) & phi is bijective by Th55;
  hence G ./. N, H are_isomorphic by GROUP_6:def 11;
end;
