reserve Omega for non empty set,
        Sigma for SigmaField of Omega,
        Prob for Probability of Sigma,
        A for SetSequence of Sigma,
        n,n1,n2 for Nat;

theorem Th17:
 (not Partial_Sums(Prob*A) is convergent &
   A is_all_independent_wrt Prob) implies
  (Prob.lim_inf Complement A = 0 & Prob.lim_sup A = 1)
proof
 assume A1: not Partial_Sums(Prob*A) is convergent;
 assume A2: A is_all_independent_wrt Prob;
 A3: for n being Nat holds (Prob*A).n >= 0
 proof
  let n be Nat;
  dom(Prob*A)=NAT by FUNCT_2:def 1; then
  (Prob*A).n = Prob.(A.n) by FUNCT_1:12,ORDINAL1:def 12;
  hence thesis by PROB_1:def 8;
 end;
 A5: (not (Prob*A) is summable implies
       not Partial_Sums(Prob*A) is bounded_above) &
     not (Prob*A) is summable by A3,A1,SERIES_1:17,def 2;
 Partial_Sums(Prob*A) is divergent_to+infty by A5,A3,LIMFUNC1:29,SERIES_1:16;
 hence thesis by A2,Th16;
end;
