 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 :: TH47
  for G being Group
  for K being Subgroup of G
  for H being Subgroup of K
  for N being normal Subgroup of G st N is Subgroup of K
  holds H,N are_complements_in K
  iff H,((K,N)`*`) are_complements_in K by Th46;
