 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 Th46:
  for G being Group
  for H,K being Subgroup of G st H is 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
proof
  let G be Group;
  let H,K be Subgroup of G;
  assume A1: H is Subgroup of K;
  let N be normal Subgroup of G;
  assume A2: N is Subgroup of K;
  then A3: N is normal Subgroup of K by GROUP_6:8;
  hereby
    assume H,N are_complements_in K;
    then N * H = the carrier of K & H /\ N = (1).K by A1,A3,Th43;
    hence H,((K,N)`*`) are_complements_in K by A1,A2,Th45;
  end;
  assume H,((K,N)`*`) are_complements_in K;
  then N * H = the carrier of K & H /\ N = (1).K by A1,A2,Th45;
  hence H,N are_complements_in K by A1,A3,Th43;
end;
