 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 :: TH57
  for G being Group
  for H1, H2 being Subgroup of G
  for N being normal Subgroup of G
  st H1,N are_complements_in G & H2,N are_complements_in G
  holds H1,H2 are_isomorphic
proof
  let G be Group;
  let H1, H2 be Subgroup of G;
  let N be normal Subgroup of G;
  assume H1,N are_complements_in G; then
  A3: G ./. N, H1 are_isomorphic by Th56;
  assume H2,N are_complements_in G; then
  G ./. N, H2 are_isomorphic by Th56;
  hence H1,H2 are_isomorphic by A3,GROUP_6:67;
end;
