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 p in tau1.q & p <> q implies len p < len q
  proof
    defpred P1[Element of l] means
    p in tau1.$1 & p <> $1 implies len p < len $1;
A1: for n holds P1[ prop n]
    proof
      let n;
      tau1.(prop n) = {prop n} by Def4;
      hence thesis by TARSKI:def 1;
    end;
A2: for r, s st P1[r] & P1[s] holds P1[r 'U' s] & P1[r => s]
    proof
      let r,s;
      assume that
A3:   P1[r] and
A4:   P1[s];
      set u = r => s;
      tau1.(r 'U' s) = {r 'U' s} by Def4;
      hence P1[r 'U' s] by TARSKI:def 1;
      thus P1[u]
      proof
        assume that
A5:     p in tau1.u and
A6:     p <> u;
        tau1.u = {u} \/ tau1.r \/ tau1.s by Def4;
        then p in {u} \/ (tau1.r \/ tau1.s) by XBOOLE_1:4,A5;
        then A7: p in {u} or p in tau1.r \/ tau1.s by XBOOLE_0:def 3;
        per cases by A7,TARSKI:def 1,A6,XBOOLE_0:def 3;
        suppose p in tau1.r;
          hence thesis by HILBERT2:16,XXREAL_0:2,A3;
        end;
        suppose p in tau1.s;
          hence thesis by HILBERT2:16,XXREAL_0:2,A4;
        end;
      end;
    end;
    tau1.TFALSUM = {TFALSUM} by Def4;
    then A8: P1[TFALSUM] by TARSKI:def 1;
    for p holds P1[p] from HILBERT2:sch 2(A8,A1,A2);
    hence thesis;
  end;
