theorem
A is_all_independent_wrt Prob implies
  (Prob.lim_inf Complement A = 0 or
   Prob.lim_inf Complement A = 1) &
   (Prob.lim_sup A = 0 or Prob.lim_sup A = 1)
proof
 assume A1: A is_all_independent_wrt Prob;
 per cases;
  suppose Partial_Sums(Prob*A) is convergent;
   hence thesis by Th16;
  end;
  suppose not Partial_Sums(Prob*A) is convergent;
   hence thesis by A1,Th17;
  end;
end;
