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 B being Subset of GX,V being Subset of GX| B holds Cl V =(Cl Up(V) ) /\ B
proof
  let B be Subset of GX, V be Subset of GX|B;
A1: B=[#](GX|B) by PRE_TOPC:def 5;
  then Cl Down(Up(V),B)=(Cl Up(V))/\ B by Th29;
  hence thesis by A1,XBOOLE_1:28;
end;
