 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 Th45:
  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,((K,N)`*`) are_complements_in K
  iff (N * H = 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;
  then reconsider H1=the multMagma of H as strict Subgroup of K by Th1;
  let N be normal Subgroup of G;
  assume N is Subgroup of K;
  then A2: N is normal Subgroup of K by GROUP_6:8;
  then the multMagma of (K,N)`*` = the multMagma of N by Th41;
  then H1,((K,N)`*`) are_complements_in K iff H,N are_complements_in K;
  hence thesis by A1,A2,Th43;
end;
