reserve G for strict Group,
  a,b,x,y,z for Element of G,
  H,K for strict Subgroup of G,
  p for Element of NAT,
  A for Subset of G;
reserve G for Group;
reserve H, B, A for Subgroup of G,
  D for Subgroup of A;

theorem Th20:
  for G being finite Group, H be Subgroup of G holds index (G,H) > 0
proof
  let G be finite Group, H be Subgroup of G;
  card G = card H * index H by GROUP_2:147;
  hence thesis;
end;
