reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th12:
  for X, Y being TopSpace, A being Subset of [:X,Y:] holds union
  Base-Appr A c= A
proof
  let X, Y be TopSpace, A be Subset of [:X,Y:];
  let e be object;
  assume e in union Base-Appr A;
  then consider B being set such that
A1: e in B and
A2: B in Base-Appr A by TARSKI:def 4;
  ex X1 being Subset of X, Y1 being Subset of Y st B = [:X1,Y1:] & [:X1,Y1
  :] c= A & X1 is open & Y1 is open by A2;
  hence thesis by A1;
end;
