reserve a for set;

theorem
  for L being upper-bounded non empty Poset holds uparrow Top L = {Top L}
proof
  let L be upper-bounded non empty Poset;
  thus uparrow Top L c= {Top L}
  proof
    let x be object;
    assume
A1: x in uparrow Top L;
    then reconsider x as Element of L;
A2: x >= Top L by A1,WAYBEL_0:18;
    x <= Top L by YELLOW_0:45;
    then x = Top L by A2,ORDERS_2:2;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {Top L};
  then
A3: x = Top L by TARSKI:def 1;
  Top L <= Top L;
  hence thesis by A3,WAYBEL_0:18;
end;
