
theorem Th3:
  for R being non empty reflexive transitive RelStr, D being non
  empty directed Subset of InclPoset Ids R holds sup D = union D
proof
  let R be non empty reflexive transitive RelStr, D be non empty directed
  Subset of InclPoset Ids R;
  reconsider UD = union D as Ideal of R by Th2;
A1: ex_sup_of D,InclPoset Ids R by WAYBEL_0:75;
  UD in Ids R;
  then reconsider UD as Element of InclPoset Ids R by YELLOW_1:1;
A2: for b being Element of InclPoset Ids R st b is_>=_than D holds UD <= b
  by Lm2;
  D is_<=_than UD by Lm1;
  hence thesis by A2,A1,YELLOW_0:def 9;
end;
