
theorem Th4:
  for X being set, F being SetSequence of X, x being object holds x in
  lim_inf F iff ex n being Nat st for k being Nat holds x
  in F.(n+k)
proof
  let X be set, F be SetSequence of X, x be object;
  consider f being SetSequence of X such that
A1: lim_inf F = Union f and
A2: for n being Nat holds f.n = meet (F ^\ n) by Def1;
  hereby
    consider f being SetSequence of X such that
A3: lim_inf F = Union f and
A4: for n being Nat holds f.n = meet (F ^\ n) by Def1;
    assume x in lim_inf F;
    then consider n being Nat such that
A5: x in f.n by A3,PROB_1:12;
    set G = F ^\ n;
     reconsider n as Nat;
    take n;
    let k be Nat;
A6: G.k = F.(n + k) by NAT_1:def 3;
    x in meet (F ^\ n) by A4,A5;
    hence x in F.(n+k) by A6,Th3;
  end;
  given n being Nat such that
A7: for k being Nat holds x in F.(n+k);
  set G = F ^\ n;
  for z being Nat holds x in G.z
  proof
    let z be Nat;
    G.z = F.(n + z) by NAT_1:def 3;
    hence thesis by A7;
  end;
  then x in meet G by Th3;
  then x in f.n by A2;
  hence thesis by A1,PROB_1:12;
end;
