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 Th33:
  L is_Finseq_for v & 1<= len(L) & the LTLnew of CastNode(L.(len(L
)),v) = {} v implies the LTLnew of CastNode(L.1,v) c= the LTLold of CastNode(L.
  (len(L)),v)
proof
  assume that
A1: L is_Finseq_for v and
A2: 1<=len(L) and
A3: the LTLnew of CastNode(L.(len(L)),v) = {} v;
  set n= len(L);
  the LTLnew of CastNode(L.1,v) c= the LTLold of CastNode(L.n,v)
  proof
    let x be object;
    assume
A4: x in the LTLnew of CastNode(L.1,v);
    then x in Subformulae v;
    then reconsider x as LTL-formula by MODELC_2:1;
    1<n by A2,A3,A4,XXREAL_0:1;
    then consider m such that
A5: 1<= m & m<n and
A6: x in the LTLnew of CastNode(L.m,v) & not x in the LTLnew of
    CastNode(L.(m+1) ,v) by A1,A3,A4,Th29;
    set m1 = m+1;
    1<=m1 & m1<=n by A5,NAT_1:13;
    then
A7: the LTLold of CastNode(L.m1,v) c= the LTLold of CastNode(L.n,v) by A1,Th31;
    consider N1,N2 such that
A8: N1 = L.m and
A9: N2 = L.(m+1) and
A10: N2 is_succ_of N1 by A1,A5;
A11: N2 = CastNode(L.m1,v) by A9,Def16;
    N1 = CastNode(L.m,v) by A8,Def16;
    then x in the LTLold of N2 by A6,A10,A11,Th30,Th32;
    hence thesis by A11,A7;
  end;
  hence thesis;
end;
