
theorem Th32:
  for L being non empty transitive reflexive RelStr, X be Subset of L holds
  ex_sup_of X,L iff ex_sup_of downarrow X,L
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_0:46;
end;
