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 Th48:
  for f being Function of LTLNodes(v),LTLNodes(v) st f
is_succ_homomorphism v,w holds for x st x in LTLNodes(v) & CastNode(x,v) is non
elementary & w |= *CastNode(x,v) holds for k st (for i st i<=k holds CastNode((
  f|**i).x,v) is non elementary) holds CastNode((f|**(k+1)).x,v) is_succ_of
  CastNode((f|**k).x,v) & w |= *CastNode((f|**k).x,v)
proof
  set LN = LTLNodes(v);
  let f be Function of LN,LN;
  assume
A1: f is_succ_homomorphism v,w;
  then
A2: f is_homomorphism v,w;
  for x st x in LN & CastNode(x,v) is non elementary & w |= *CastNode(x,v)
  holds for k st (for i st i<=k holds CastNode((f|**i).x,v) is non elementary)
  holds CastNode((f|**(k+1)).x,v) is_succ_of CastNode((f|**k).x,v) & w |= *
  CastNode((f|**k).x,v)
  proof
    let x such that
A3: x in LN and
A4: CastNode(x,v) is non elementary & w |= *CastNode(x,v);
    for k st (for i st i<=k holds CastNode((f|**i).x,v) is non elementary)
    holds CastNode((f|**(k+1)).x,v) is_succ_of CastNode((f|**k).x,v) & w |= *
    CastNode((f|**k).x,v)
    proof
      let k such that
A5:   for i st i<=k holds CastNode((f|**i).x,v) is non elementary;
      set y = (f|**k).x;
A6:   y in LN by A3,FUNCT_2:5;
A7:   (f|**(k+1)).x = (f*(f|**k)).x by FUNCT_7:71
        .= f.y by A3,FUNCT_2:15;
      CastNode(y,v) is non elementary & w |= *CastNode(y,v) by A2,A3,A4,A5,Th47
;
      hence thesis by A1,A6,A7;
    end;
    hence thesis;
  end;
  hence thesis;
end;
