theorem Th20:
  N2 is_succ2_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_succ2_of N1;
  then consider H such that
A1: H in NN1 and
  H is disjunctive or H is Until or H is Release and
A2: N2 = SuccNode2(H,N1);
  set M1 = NN1 \ {H};
  set New2= LTLNew2(H,v);
  set M2 = New2 \ the LTLold of N1;
  reconsider M1 as Subset of Subformulae v;
  reconsider M2 as Subset of Subformulae v;
  New2 = LTLNew2 H by A1,Def28;
  then NN2 = M1 \/ M2 by A1,A2,Def5;
  then
A3: len(NN2)<=len(M1) + len(M2) by Th18;
  reconsider NN1 as Subset of Subformulae v;
A4: len(M2) <= len(New2) by Th15,XBOOLE_1:36;
  len(New2) <= len(H) -1 by A1,Lm28;
  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;
