reserve k,n,n1,m,m1,m0,h,i,j for Nat,
  a,x,y,X,X1,X2,X3,X4,Y for set;
reserve L,L1,L2 for FinSequence;
reserve F,F1,G,G1,H for LTL-formula;
reserve W,W1,W2 for Subset of Subformulae H;
reserve v for LTL-formula;
reserve N,N1,N2,N10,N20,M for strict LTLnode over v;
reserve w for Element of Inf_seq(AtomicFamily);
reserve R1,R2 for Real_Sequence;

theorem Th8:
  rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae
  H & L2 is one-to-one implies len(L1,W) = len(L2,W)
proof
  defpred P[Nat] means for W1 st card W1 <=$1 holds (rng L1 = Subformulae H &
L1 is one-to-one) & (rng L2 = Subformulae H & L2 is one-to-one) implies len(L1,
  W1) = len(L2,W1);
  set k = card W;
A1: for k being Nat st P[k] holds P[k + 1]
  proof
    let k be Nat such that
A2: P[k];
    P[k+1]
    proof
      let W1 such that
A3:   card W1 <=k+1;
      rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H &
      L2 is one-to-one implies len(L1,W1) = len(L2,W1)
      proof
        assume that
A4:     rng L1 = Subformulae H & L1 is one-to-one and
A5:     rng L2 = Subformulae H & L2 is one-to-one;
        now
          per cases by A3,NAT_1:8;
          suppose
            card W1 <=k;
            hence thesis by A2,A4,A5;
          end;
          suppose
A6:         card W1 = k+1;
            then W1 <> {};
            then consider F being object such that
A7:         F in W1 by XBOOLE_0:def 1;
            F in Subformulae H by A7;
            then reconsider F as LTL-formula by MODELC_2:1;
            {F} c= W1 by A7,ZFMISC_1:31;
            then
A8:         card (W1 \ {F}) = card W1 - card {F} by CARD_2:44
              .= card W1 - 1 by CARD_1:30
              .= k by A6;
A9:         len(L1,W1) = (len(L1,W1) - len F) + len F
              .= len(L1,W1\{F}) + len F by A4,A7,Th6;
            len(L2,W1) = (len(L2,W1) - len F) + len F
              .= len(L2,W1\{F}) + len F by A5,A7,Th6
              .= len(L1,W1\{F}) + len F by A2,A4,A5,A8;
            hence thesis by A9;
          end;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
A10: P[0]
  proof
    let W1;
    assume card W1 <=0;
    then
A11: W1 = {} H;
    then len(L1,W1) = 0 by Th4;
    hence thesis by A11,Th4;
  end;
  for k being Nat holds P[k] from NAT_1:sch 2 (A10,A1);
  then P[k];
  hence thesis;
end;
