theorem Th28:
  N2 is_succ_of N1 & not F in the LTLold of N1 & F in the LTLold
  of N2 implies N2 is_succ_of N1,F
proof
  assume that
A1: N2 is_succ_of N1 and
A2: not F in the LTLold of N1 and
A3: F in the LTLold of N2;
  now
    per cases by A1;
    suppose
      N2 is_succ1_of N1;
      then consider H such that
A4:   H in the LTLnew of N1 & N2 = SuccNode1(H,N1);
      the LTLold of N2 =(the LTLold of N1) \/ {H} by A4,Def4;
      then F in (the LTLold of N1) or F in {H} by A3,XBOOLE_0:def 3;
      then F=H by A2,TARSKI:def 1;
      hence thesis by A4;
    end;
    suppose
      N2 is_succ2_of N1;
      then consider H such that
A5:   H in the LTLnew of N1 and
A6:   H is disjunctive or H is Until or H is Release and
A7:   N2=SuccNode2(H,N1);
      the LTLold of N2 =(the LTLold of N1) \/ {H} by A5,A7,Def5;
      then F in (the LTLold of N1) or F in {H} by A3,XBOOLE_0:def 3;
      then F=H by A2,TARSKI:def 1;
      hence thesis by A5,A6,A7;
    end;
  end;
  hence thesis;
end;
