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 Th29:
  L is_Finseq_for v & F in the LTLnew of CastNode(L.1,v) & 1< n &
n <= len(L) & not F in the LTLnew of CastNode(L.n,v) implies ex m st 1<= m & m<
n & F in the LTLnew of CastNode(L.m,v) & not F in the LTLnew of CastNode(L.(m+1
  ),v)
proof
  assume
A1: L is_Finseq_for v & F in the LTLnew of CastNode(L.1,v) & 1< n & n <=
  len (L) & not F in the LTLnew of CastNode(L.n,v);
  defpred P[Nat] means for F1,n1,L1 st len(L1) <= $1 holds L1 is_Finseq_for v
  & F1 in the LTLnew of CastNode(L1.1,v) & (1< n1 & n1 <= len(L1) & not (F1 in
  the LTLnew of CastNode(L1.n1,v)) ) implies ex m st 1<= m & m<n1 & F1 in the
  LTLnew of CastNode(L1.m,v) & not (F1 in the LTLnew of CastNode(L1.(m+1),v));
A2: for k being Nat st P[k] holds P[k + 1]
  proof
    let k such that
A3: P[k];
    P[k + 1]
    proof
      let F1,n1,L1 such that
A4:   len(L1) <= k+1;
      now
        per cases by A4,NAT_1:8;
        suppose
          len(L1)<=k;
          hence thesis by A3;
        end;
        suppose
A5:       len(L1)=k+1;
          L1 is_Finseq_for v & F1 in the LTLnew of CastNode(L1.1,v) & 1<
n1 & n1 <= len(L1) & not F1 in the LTLnew of CastNode(L1.n1,v) implies ex m st
1<= m & m<n1 & F1 in the LTLnew of CastNode(L1.m,v) & not F1 in the LTLnew of
          CastNode(L1.(m+1),v)
          proof
            assume that
A6:         L1 is_Finseq_for v and
A7:         F1 in the LTLnew of CastNode(L1.1,v) and
A8:         1< n1 and
A9:         n1 <= len(L1) and
A10:        not F1 in the LTLnew of CastNode(L1.n1,v);
            now
              per cases by A5,A9,NAT_1:8;
              suppose
A11:            n1<=k;
                set L2=L1|Seg k;
                reconsider L2 as FinSequence by FINSEQ_1:15;
A12:            k+0 <= k+1 by XREAL_1:7;
                then
A13:            dom L2 = Seg k by A5,FINSEQ_1:17;
                then n1 in dom L2 by A8,A11,FINSEQ_1:1;
                then
A14:            L2.n1 = L1.n1 by FUNCT_1:47;
                1< k by A8,A11,XXREAL_0:2;
                then 1 in dom L2 by A13,FINSEQ_1:1;
                then
A15:            F1 in the LTLnew of CastNode(L2.1,v) by A7,FUNCT_1:47;
                len(L2) = k & L2 is_Finseq_for v by A5,A6,A12,Th26,FINSEQ_1:17;
                then consider m such that
A16:            1<= m and
A17:            m<n1 and
A18:            F1 in the LTLnew of CastNode(L2.m,v) & not F1 in the
                LTLnew of CastNode(L2.(m+ 1),v) by A3,A8,A10,A11,A15,A14;
                m+1<=n1 by A17,NAT_1:13;
                then
A19:            m+1 <=k by A11,XXREAL_0:2;
                1<= m+1 by A16,NAT_1:13;
                then m+1 in dom L2 by A13,A19,FINSEQ_1:1;
                then
A20:            L2.(m+1) = L1.(m+1) by FUNCT_1:47;
                m <=k by A11,A17,XXREAL_0:2;
                then m in dom L2 by A13,A16,FINSEQ_1:1;
                then L2.m = L1.m by FUNCT_1:47;
                hence thesis by A16,A17,A18,A20;
              end;
              suppose
A21:            n1=k+1;
                then
A22:            1<= k by A8,NAT_1:13;
A23:            k+0 < k+1 by XREAL_1:8;
                now
                  per cases;
                  suppose
                    F1 in the LTLnew of CastNode(L1.k,v);
                    hence thesis by A10,A21,A23,A22;
                  end;
                  suppose
A24:                not F1 in the LTLnew of CastNode(L1.k,v);
A25:                1<k
                    proof
                      set b=1-k;
                      set a=k-1;
A26:                  a+b=0 & 1-1<=k-1 by A22,XREAL_1:9;
                      now
                        assume k<=1;
                        then 1-1<=1-k by XREAL_1:10;
                        then a=0 by A26;
                        hence contradiction by A7,A24;
                      end;
                      hence thesis;
                    end;
                    set L2=L1|Seg k;
                    reconsider L2 as FinSequence by FINSEQ_1:15;
A27:                k+0 <= k+1 by XREAL_1:7;
                    then
A28:                dom L2 = Seg k by A5,FINSEQ_1:17;
                    then k in dom L2 by A22,FINSEQ_1:1;
                    then
A29:                not F1 in the LTLnew of CastNode(L2.k,v) by A24,FUNCT_1:47;
                    1 in dom L2 by A22,A28,FINSEQ_1:1;
                    then
A30:                F1 in the LTLnew of CastNode(L2.1,v) by A7,FUNCT_1:47;
                    len(L2) = k & L2 is_Finseq_for v by A5,A6,A27,Th26,
FINSEQ_1:17;
                    then consider m such that
A31:                1<= m and
A32:                m<k and
A33:                F1 in the LTLnew of CastNode(L2.m,v) and
A34:                not F1 in the LTLnew of CastNode(L2.(m+1),v) by A3,A30,A29
,A25;
                    m in dom L2 by A28,A31,A32,FINSEQ_1:1;
                    then
A35:                F1 in the LTLnew of CastNode(L1.m,v) by A33,FUNCT_1:47;
                    1<= m+1 & m+1<=k by A31,A32,NAT_1:13;
                    then m+1 in dom L2 by A28,FINSEQ_1:1;
                    then
A36:                L2.(m+1) = L1.(m+1) by FUNCT_1:47;
                    m<n1 by A21,A23,A32,XXREAL_0:2;
                    hence thesis by A31,A34,A35,A36;
                  end;
                end;
                hence thesis;
              end;
            end;
            hence thesis;
          end;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
A37: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2 (A37,A2);
  hence thesis by A1;
end;
