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;
reserve q for sequence of LTLStates(v);
reserve U for Choice_Function of BOOL Subformulae v;

theorem
  N is non elementary implies ( chosen_formula(U,N) is Until & w |=
  the_right_argument_of chosen_formula(U,N) implies (the_right_argument_of
  chosen_formula(U,N) in the LTLnew of chosen_succ(w,v,U,N) or
the_right_argument_of chosen_formula(U,N) in the LTLold of N ) & chosen_formula
  (U,N) in the LTLold of chosen_succ(w,v,U,N) )
proof
  set SN = chosen_succ(w,v,U,N);
  set H = chosen_formula(U,N);
  set H2 = the_right_argument_of H;
  set SNO = the LTLold of SN;
  set SNN = the LTLnew of SN;
  set NO = the LTLold of N;
  set NN = the LTLnew of N;
  assume N is non elementary;
  then
A1: H in the LTLnew of N by Th58;
  H is Until & w |= H2 implies (H2 in SNN or H2 in NO) & H in SNO
  proof
    assume that
A2: H is Until and
A3: w |= H2;
A4: SN = SuccNode2(H,N) by A2,A3,Def35;
    LTLNew2 H = {H2} by A2,Def2;
    then
A5: SNN = (NN \ {H}) \/ ({H2} \ NO) by A1,A4,Def5;
A6: now
      per cases;
      suppose
        H2 in NO;
        hence H2 in SNN or H2 in NO;
      end;
      suppose
A7:     not H2 in NO;
        H2 in {H2} by TARSKI:def 1;
        then H2 in {H2} \ NO by A7,XBOOLE_0:def 5;
        hence H2 in SNN or H2 in NO by A5,XBOOLE_0:def 3;
      end;
    end;
A8: H in {H} by TARSKI:def 1;
    SNO =NO \/ {H} by A1,A4,Def5;
    hence thesis by A8,A6,XBOOLE_0:def 3;
  end;
  hence thesis;
end;
