 reserve X for set;
 reserve S for Subset-Family of X;

theorem
  for S be Subset-Family of REAL
  st S = {].a,b.] where a,b is Real: a<=b}
  holds S is cap-closed &
  S is diff-finite-partition-closed with_empty_element &
  S is with_countable_Cover by THS0,THS1,THS2,LemmA1;
