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 Th29:
  for V,B being Subset of GX st V c= B holds Cl Down(V,B) =(Cl V) /\ B
proof
  let V,B be Subset of GX;
  assume V c= B;
  then Down(V,B)=V by XBOOLE_1:28;
  then Cl Down(V,B) =(Cl V) /\ ([#](GX|B)) by PRE_TOPC:17;
  hence thesis by PRE_TOPC:def 5;
end;
