reserve a for set;

theorem Th23:
  for L being lower-bounded non empty Poset holds
  downarrow Bottom L = {Bottom L}
proof
  let L be lower-bounded non empty Poset;
  thus downarrow Bottom L c= {Bottom L}
  proof
    let a be object;
    assume a in downarrow Bottom L;
    then a in {x where x is Element of L : ex y being Element of L st x <= y &
    y in {Bottom L}} by WAYBEL_0:14;
    then consider a9 be Element of L such that
A1: a9 = a and
A2: ex y being Element of L st a9 <= y & y in {Bottom L};
    consider y being Element of L such that
A3: a9 <= y and
A4: y in {Bottom L} by A2;
A5: Bottom L <= a9 by YELLOW_0:44;
    y = Bottom L by A4,TARSKI:def 1;
    hence thesis by A1,A3,A4,A5,ORDERS_2:2;
  end;
  let a be object;
  assume a in {Bottom L};
  then
A6: a = Bottom L by TARSKI:def 1;
  Bottom L <= Bottom L;
  hence thesis by A6,WAYBEL_0:17;
end;
