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

theorem Th42:
  for G being Group
  for H1,N1,H2,N2 being Subgroup of G
  st the multMagma of H1 = the multMagma of H2
  & the multMagma of N1 = the multMagma of N2
  holds H1 * N1 = H2 * N2 & H1 /\ N1 = H2 /\ N2
proof
  let G be Group;
  let H1,N1,H2,N2 be Subgroup of G;
  assume A1: the multMagma of H1 = the multMagma of H2;
  assume A2: the multMagma of N1 = the multMagma of N2;
  reconsider K=G as Subgroup of G by GROUP_2:54;
  reconsider HK=the multMagma of H1,NK=the multMagma of N1 as Subgroup of K
  by Th1;
  thus H1 * N1 = HK * NK by ThProdLemma
              .= H2 * N2 by A1,A2,ThProdLemma;
  thus H1 /\ N1 = HK /\ NK by ThCapLemma
               .= H2 /\ N2 by A1,A2,ThCapLemma;
end;
