reserve X for set;

theorem
  for T being non empty TopSpace, X being Subset of InclPoset the
  topology of T holds sup X = union X
proof
  let T be non empty TopSpace;
  set L = InclPoset the topology of T;
  let X be Subset of L;
  reconsider X as Subset-Family of T by XBOOLE_1:1;
  reconsider Un = union X as Element of L by PRE_TOPC:def 1;
A1: now
    let b be Element of L;
    assume b is_>=_than X;
    then for Z being set st Z in X holds Z c= b by Th3;
    then Un c= b by ZFMISC_1:76;
    hence Un <= b by Th3;
  end;
  for b being Element of L st b in X holds b <= Un by Th3,ZFMISC_1:74;
  then Un is_>=_than X;
  hence thesis by A1,YELLOW_0:30;
end;
