reserve x,X,X2,Y,Y2 for set;
reserve GX for non empty TopSpace;
reserve A2,B2 for Subset of GX;
reserve B for Subset of GX;

theorem
  for V,B being Subset of GX st V c= B holds Cl Down(V,B) c= Cl V
proof
  let V,B be Subset of GX;
  assume V c= B;
  then Cl Down(V,B) = (Cl V) /\ B by Th29;
  hence thesis by XBOOLE_1:17;
end;
