
theorem
  for L being non empty Poset, x being Element of L holds
  ex_sup_of downarrow x, L & sup downarrow x = x
proof
  let L be non empty Poset, x be Element of L;
  ex_sup_of {x}, L by YELLOW_0:38;
  hence ex_sup_of downarrow x, L by Th32;
  thus sup downarrow x = sup {x} by Th33,YELLOW_0:38
    .= x by YELLOW_0:39;
end;
