reserve X for set;

theorem Th14:
  for X be non empty set holds union X in X implies Top InclPoset X = union X
proof
  let X be non empty set;
  assume union X in X;
  then reconsider a = union X as Element of InclPoset X;
  for b be Element of InclPoset X st b in X holds b <= a by Th3,ZFMISC_1:74;
  then a is_>=_than X;
  then InclPoset X is upper-bounded by YELLOW_0:def 5;
  then {} is_>=_than a & ex_inf_of {},InclPoset X by YELLOW_0:43;
  then a <= "/\"({},InclPoset X) by YELLOW_0:def 10;
  then
A1: "/\"({},InclPoset X) c= a & a c= "/\"({},InclPoset X) by Th3,ZFMISC_1:74;
  thus Top InclPoset X = "/\"({},InclPoset X) by YELLOW_0:def 12
    .= union X by A1;
end;
