
theorem Th1: :: cf. WAYBEL13:9
  for L being upper-bounded Semilattice, F being non empty directed
  Subset of InclPoset Filt L holds sup F = union F
proof
  let L be upper-bounded Semilattice, F be non empty directed Subset of
  InclPoset Filt L;
  Filt L = Ids (L opp) by WAYBEL16:7;
  hence thesis by WAYBEL13:9;
end;
