theorem Th19:
  N2 is_succ1_of N1 implies len(the LTLnew of N2) <= len(the LTLnew of N1) - 1
proof
  set NN1 = the LTLnew of N1;
  set NN2 = the LTLnew of N2;
  assume N2 is_succ1_of N1;
  then consider H such that
A1: H in NN1 and
A2: N2 = SuccNode1(H,N1);
  set M1 = NN1 \ {H};
  set New1= LTLNew1(H,v);
  set M2 = New1 \ the LTLold of N1;
  reconsider M1 as Subset of Subformulae v;
  reconsider M2 as Subset of Subformulae v;
  New1 = LTLNew1 H by A1,Def27;
  then NN2 = M1 \/ M2 by A1,A2,Def4;
  then
A3: len(NN2)<=len(M1) + len(M2) by Th18;
  reconsider NN1 as Subset of Subformulae v;
A4: len(M2) <= len(New1) by Th15,XBOOLE_1:36;
  len(New1) <= len(H) -1 by A1,Lm27;
  then len(M2) <= len(H) -1 by A4,XXREAL_0:2;
  then
A5: len(M1) + len(M2) <= len(M1) + (len(H) -1) by XREAL_1:6;
  len(M1) = len(NN1)-len(H) by A1,Th10;
  hence thesis by A5,A3,XXREAL_0:2;
end;
