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 GX being TopSpace for V,B being Subset of GX st V is open holds
  Down(V,B) is open
proof
  let GX be TopSpace;
  let V,B be Subset of GX;
  assume V is open;
  then
A1: V in the topology of GX by PRE_TOPC:def 2;
  Down(V,B)=V /\ [#](GX|B) by PRE_TOPC:def 5;
  then Down(V,B) in the topology of GX|B by A1,PRE_TOPC:def 4;
  hence thesis by PRE_TOPC:def 2;
end;
