reserve A,B,p,q,r,s for Element of LTLB_WFF,
  n for Element of NAT,
  X for Subset of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y for set;

theorem tau tau X = tau X
  proof
    thus tau tau X c= tau X
    proof
      let x be object;
      assume x in tau tau X;
      then consider p such that
A1:   p in tau X and
A2:   x in tau1.p by Def5;
      consider q such that
A3:   q in X and
A4:   p in tau1.q by A1,Def5;
      tau1.p c= tau1.q by A4,Th8;
      hence x in tau X by A2,A3,Def5;
    end;
    thus tau X c= tau tau X by Th16;
  end;
