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 Th38:
  s2 is_next_of s1 & F in the LTLold of s2 implies ex L,m st 1<=
len(L) & L is_Finseq_for v & L.1 = 'X' s1 & L.(len(L)) = s2 & 1 <= m & m <len(L
  ) & CastNode(L.(m+1),v) is_succ_of CastNode(L.m,v),F
proof
  assume that
A1: s2 is_next_of s1 and
A2: F in the LTLold of s2;
  set N1 = 'X' s1;
  consider L such that
A3: 1<=len(L) and
A4: L is_Finseq_for v and
A5: L.1 = 'X' s1 & L.(len(L)) = s2 by A1;
  set n = len(L);
A6: CastNode(L.1,v) = N1 & CastNode(L.n,v) = s2 by A5,Def16;
  1<n by A2,A3,A5,XXREAL_0:1;
  then consider m such that
A7: 1<= m & m<n and
A8: ( not F in the LTLold of CastNode(L.m,v))& F in the LTLold of
  CastNode(L.(m+1) ,v) by A2,A4,A6,Th27;
  set m1 = m+1;
  consider N1,N2 such that
A9: N1 = L.m & N2 = L.(m+1) and
A10: N2 is_succ_of N1 by A4,A7;
  N1 = CastNode(L.m,v) & N2 = CastNode(L.m1,v) by A9,Def16;
  hence thesis by A3,A4,A5,A7,A8,A10,Th28;
end;
