 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem Th40:
  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 N * H = H * N
proof
  let G be Group;
  let H,K be Subgroup of G;
  assume H is Subgroup of K;
  then reconsider H0 = H as Subgroup of K;
  let N be Subgroup of G;
  assume N is normal Subgroup of K;
  then reconsider N0 = N as normal Subgroup of K;
  A1: the multMagma of N0 = the multMagma of N;
  A2: the multMagma of H0 = the multMagma of H;
  hence N * H = N0 * H0 by ThProdLemma,A1
            .= H0 * N0 by GROUP_5:8
            .= H * N by ThProdLemma,A1,A2;
end;
