
theorem Th33:
  for L being non empty transitive reflexive RelStr, X be Subset of L
  st ex_sup_of X,L holds sup X = sup downarrow X
proof
  let L be non empty transitive reflexive RelStr, X be Subset of L;
  for x being Element of L holds x is_>=_than X iff x is_>=_than downarrow X
  by Th31;
  hence thesis by YELLOW_0:47;
end;
