theorem
  for G being finite addGroup, I, H being Subgroup of G, J being Subgroup
  of H holds I = J implies index I = index J * index H
proof
  let G be finite addGroup, I, H be Subgroup of G, J be Subgroup of H;
  assume
A1: I = J;
  card G = card H * index H & card H = card J * index J by Th147;
  then card I * (index J * index H) = card I * index I by A1,Th147;
  hence thesis by XCMPLX_1:5;
end;
