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;
reserve v for neg-inner-most LTL-formula;
reserve U for Choice_Function of BOOL Subformulae v;
reserve N,N1,N2,M1 for strict LTLnode over v;
reserve s,s1 for elementary strict LTLnode over v;

theorem Th73:
  w |= v implies for n holds CastNode(chosen_run(w,v,U).(n+1),v)
  is_next_of CastNode(chosen_run(w,v,U).n,v) & Shift(w,n) |= * ('X' CastNode(
  chosen_run(w,v,U).n,v))
proof
  set s = init v;
  deffunc R(Nat) = CastNode(chosen_run(w,v,U).$1,v);
  defpred P[Nat] means R($1+1) is_next_of R($1) & Shift(w,$1) |= *('X' R($1));
  assume w |= v;
  then
A1: w |= *('X' s) by Th72;
A2: CastNode(chosen_run(w,v,U).0,v) = CastNode(s,v) by Def50
    .= s by Def16;
A3: for n st P[n] holds P[n+1]
  proof
    let n;
    set s1=R(n);
    s1 is elementary strict LTLnode over v
    proof
      now
        per cases;
        suppose
          n = 0;
          then s1 = CastNode(s,v) by Def50
            .= s by Def16;
          hence thesis;
        end;
        suppose
A4:       0 <n;
          set m = n-1;
          reconsider m as Nat by A4,NAT_1:20;
          n = m+1;
          then
          s1 = CastNode(chosen_next(Shift(w,m),v,U, CastNode(chosen_run(w
          ,v,U).m,v)),v) by Def50
            .= chosen_next(Shift(w,m),v,U, CastNode(chosen_run(w,v,U).m,v))
          by Def16;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    then reconsider s1 as elementary strict LTLnode over v;
    set n1 =n+1;
    set w1 = Shift(w,n);
    set w2 = Shift(w,n1);
    set s2=R(n1);
    set s3=R(n1+1);
A5: s2 = CastNode( chosen_next (w1,v,U, CastNode(chosen_run(w,v,U).n,v) ),
    v) by Def50
      .= chosen_next(w1,v,U,s1) by Def16;
    then reconsider s2 as elementary strict LTLnode over v;
A6: s3 = CastNode( chosen_next (w2,v,U, CastNode(chosen_run(w,v,U).n1,v) )
    ,v) by Def50
      .= chosen_next(w2,v,U,s2) by Def16;
    assume P[n];
    then w2 = Shift(w1,1) & w1 |= *chosen_next(w1,v,U,s1) by Th69,MODELC_2:80;
    then w2 |= * ('X' s2) by A5,Th70;
    hence thesis by A6,Th69;
  end;
  R(0+1) = CastNode( chosen_next (Shift(w,0),v,U, CastNode(chosen_run(w,v,
  U).0,v) ),v) by Def50
    .= CastNode(chosen_next(w,v,U,s),v) by A2,MODELC_2:79
    .= chosen_next(w,v,U,s) by Def16;
  then
A7: P[0] by A1,A2,Th69,MODELC_2:79;
  for n holds P[n] from NAT_1:sch 2(A7,A3);
  hence thesis;
end;
