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 Th16: X c= tau X
  proof
    let x be object;
    defpred P1[Element of l] means $1 in X implies $1 in tau X;
    assume A1: x in X;
    then reconsider x1 = x as Element of l;
A2: for n holds P1[ prop n]
    proof
      let n;
      set pr = prop n;
      pr in {pr} by TARSKI:def 1;
      then A3: pr in tau1.pr by Def4;
      assume pr in X;
      hence thesis by A3,Def5;
    end;
   A4: for r, s st P1[r] & P1[s] holds P1[r 'U' s] & P1[r => s]
   proof
     let r,s;
     assume that P1[r] and P1[s];
     thus P1[r 'U' s]
     proof
       set f = r 'U' s;
       f in {f} by TARSKI:def 1;
       then A5: f in tau1.f by Def4;
       assume f in X;
       hence thesis by A5,Def5;
     end;
     thus P1[r => s]
     proof
       set f = r => s;
       tau1.f = {f} \/ tau1.r \/ tau1.s by Def4
       .= {f} \/ (tau1.r \/ tau1.s) by XBOOLE_1:4;
       then A6: f in {f} & {f} c= tau1.f by TARSKI:def 1,XBOOLE_1:7;
       assume f in X;
       hence thesis by A6,Def5;
     end;
   end;
A7: P1[TFALSUM]
    proof
      set f = TFALSUM;
      f in {f} by TARSKI:def 1;
      then A8: f in tau1.f by Def4;
      assume f in X;
      hence thesis by A8,Def5;
    end;
    for p holds P1[p] from HILBERT2:sch 2(A7,A2,A4);
    hence x in tau X by A1;
  end;
