theorem Th32:
  N2 is_succ_of N1,F implies F in the LTLold of N2
proof
  assume
A1: N2 is_succ_of N1,F;
  now
    per cases by A1;
    suppose
      F in the LTLnew of N1 & N2 = SuccNode1(F,N1);
      then the LTLold of N2 =(the LTLold of N1) \/ {F} by Def4;
      then
A2:   {F} c= the LTLold of N2 by XBOOLE_1:7;
      F in {F} by TARSKI:def 1;
      hence thesis by A2;
    end;
    suppose
      F in the LTLnew of N1 & (F is disjunctive or F is Until or F is
      Release) & N2=SuccNode2(F,N1);
      then the LTLold of N2 =(the LTLold of N1) \/ {F} by Def5;
      then
A3:   {F} c= the LTLold of N2 by XBOOLE_1:7;
      F in {F} by TARSKI:def 1;
      hence thesis by A3;
    end;
  end;
  hence thesis;
end;
