 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 :: TH44
  for G being Group
  for H,K being Subgroup of G st H is Subgroup of K
  for N being Subgroup of G st N is normal Subgroup of K
  holds H,N are_complements_in K
  iff (H * N = the carrier of K & H /\ N = (1).K)
proof
  let G be Group;
  let H,K be Subgroup of G;
  assume A1: H is Subgroup of K;
  let N be Subgroup of G;
  assume A2: N is normal Subgroup of K;
  then N*H=H*N by A1,Th40;
  hence thesis by A1,A2,Th43;
end;
