reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;

theorem Th5:
  for A, B being Subset of Y st B c= A holds A is anti-discrete
  implies B is anti-discrete
proof
  let A, B be Subset of Y;
  assume
A1: B c= A;
  assume
A2: A is anti-discrete;
  now
    let G be Subset of Y;
    assume
A3: G is open;
    assume B meets G;
    then B /\ G <> {};
    then A /\ G <> {} by A1,XBOOLE_1:3,26;
    then A meets G;
    then A c= G by A2,A3;
    hence B c= G by A1;
  end;
  hence thesis;
end;
