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;
reserve s,s0,s1,s2 for elementary strict LTLnode over v;

theorem Th37:
  s2 is_next_of s1 implies the LTLnext of s1 c= the LTLold of s2
proof
  set N1 = 'X' s1;
A1: the LTLnew of s2 = {} v by Def11;
  assume s2 is_next_of s1;
  then consider L such that
A2: 1<=len(L) and
A3: L is_Finseq_for v and
A4: L.1 = 'X' s1 and
A5: L.(len(L)) = s2;
  set n = len(L);
A6: CastNode(L.n,v) = s2 by A5,Def16;
A7: CastNode(L.1,v) = N1 by A4,Def16;
  the LTLnext of s1 c= the LTLold of s2
  proof
    let x be object;
    assume
A8: x in the LTLnext of s1;
    then x in Subformulae v;
    then reconsider x as LTL-formula by MODELC_2:1;
    1<n by A2,A4,A5,A1,A8,XXREAL_0:1;
    then consider m such that
A9: 1<= m & m<n and
A10: x in the LTLnew of CastNode(L.m,v) & not x in the LTLnew of
    CastNode(L.(m+1) ,v) by A3,A7,A6,A1,A8,Th29;
    set m1 = m+1;
    consider N1,N2 such that
A11: N1 = L.m and
A12: N2 = L.(m+1) and
A13: N2 is_succ_of N1 by A3,A9;
A14: N2 = CastNode(L.m1,v) by A12,Def16;
    1<=m1 & m1<=n by A9,NAT_1:13;
    then
A15: the LTLold of N2 c= the LTLold of CastNode(L.n,v) by A3,A14,Th31;
    N1 = CastNode(L.m,v) by A11,Def16;
    then x in the LTLold of N2 by A10,A13,A14,Th30,Th32;
    hence thesis by A6,A15;
  end;
  hence thesis;
end;
