
theorem
  for X being set, F being SetSequence of X
   holds lim_inf F = (lim_sup Complement F)`
proof
  let X be set, F be SetSequence of X;
  set G = Complement F;
  thus lim_inf F c= (lim_sup Complement F)`
  proof
    let x be object;
    assume
A1: x in lim_inf F;
    then consider n being Nat such that
A2: for k being Nat holds x in F.(n+k) by Th4;
    for k being Nat holds not x in G.(n+k)
    proof
      let k be Nat;
       reconsider nk = n+k as Element of NAT by ORDINAL1:def 12;
A3:   G.(n+k) = (F.(nk))` by PROB_1:def 2;
      assume x in G.(n+k);
      then not x in F.(n+k) by A3,XBOOLE_0:def 5;
      hence thesis by A2;
    end;
    then not x in lim_sup G by Th5;
    hence thesis by A1,XBOOLE_0:def 5;
  end;
  thus (lim_sup Complement F)` c= lim_inf F
  proof
    let x be object;
    assume
A4: x in (lim_sup Complement F)`;
    then not x in lim_sup Complement F by XBOOLE_0:def 5;
    then consider n being Nat such that
A5: for k being Nat holds not x in G.(n+k) by Th5;
    for k being Nat holds x in F.(n+k)
    proof
      let k be Nat;
       reconsider nk = n+k as Element of NAT by ORDINAL1:def 12;
      assume not x in F.(n+k);
      then x in (F.(nk))` by A4,XBOOLE_0:def 5;
      then x in G.(n+k) by PROB_1:def 2;
      hence thesis by A5;
    end;
    hence thesis by Th4;
  end;
end;
