 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 Th43:
  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 (N * H = the carrier of K & H /\ N = (1).K)
proof
  let G be Group;
  let H,K be Subgroup of G;
  assume H is Subgroup of K;
  then reconsider H1=the multMagma of H as strict Subgroup of K
  by Th1;
  let N be Subgroup of G;
  assume N is normal Subgroup of K;
  then reconsider N1=the multMagma of N as strict normal Subgroup of K
  by Th2;
  A1: N * H = N1 * H1 by ThProdLemma;
  hence H,N are_complements_in K implies
    ((N * H = the carrier of K) & (H /\ N = (1).K))
      by ThCapLemma, GROUP_5:8;

  assume A2: N * H = the carrier of K;
  assume H /\ N = (1).K;
  then H1, N1 are_complements_in K by A1, A2, ThCapLemma, GROUP_5:8;
  hence H,N are_complements_in K;
end;
