reserve X for set;

theorem Th13:
  for X be non empty set holds {} in X implies Bottom InclPoset X = {}
proof
  let X be non empty set;
  assume {} in X;
  then reconsider a = {} as Element of InclPoset X;
  for b be Element of InclPoset X st b in X holds a <= b by Th3,XBOOLE_1:2;
  then a is_<=_than X;
  then InclPoset X is lower-bounded by YELLOW_0:def 4;
  then {} is_<=_than a & ex_sup_of {},InclPoset X by YELLOW_0:42;
  then "\/"({},InclPoset X) <= a by YELLOW_0:def 9;
  then
A1: "\/"({},InclPoset X) c= a by Th3;
  thus Bottom InclPoset X = "\/"({},InclPoset X) by YELLOW_0:def 11
    .= {} by A1;
end;
