theorem Th54:
  H is Until & H in the LTLold of CastNode(q.1,v) & (for i holds
CastNode(q.(i+1),v) is_next_of CastNode(q.i,v)) implies (for i st i>=1 holds (H
  in the LTLold of CastNode(q.i,v)) & (the_left_argument_of H in the LTLold of
CastNode(q.i,v)) & not (the_right_argument_of H in the LTLold of CastNode(q.i,v
))) or ex j st j>=1 & the_right_argument_of H in the LTLold of CastNode(q.j,v)
  & for i st 1<=i & i<j holds H in the LTLold of CastNode(q.i,v) &
  the_left_argument_of H in the LTLold of CastNode(q.i,v)
proof
  deffunc Node(Nat) = CastNode(q.$1,v);
  assume that
A1: H is Until and
A2: H in the LTLold of Node(1) & for i holds Node(i+1) is_next_of Node(i );
  set G = the_right_argument_of H;
  set F = the_left_argument_of H;
  H = F 'U' G by A1,MODELC_2:8;
  hence thesis by A2,Lm32;
end;
