
theorem Th5:
  for X being set, F being SetSequence of X, x being object holds x in
lim_sup F iff for n being Nat ex k being Nat st 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_sup F = meet f and
A2: for n being Nat holds f.n = Union (F ^\ n) by Def2;
  hereby
    assume
A3: x in lim_sup F;
    let n be Nat;
    set G = F ^\ n;
    consider f being SetSequence of X such that
A4: lim_sup F = meet f and
A5: for n being Nat holds f.n = Union (F ^\ n) by Def2;
    f.n = Union G by A5;
    then x in Union G by A3,A4,Th3;
    then consider k being Nat such that
A6: x in G.k by PROB_1:12;
     reconsider k as Nat;
    take k;
    thus x in F.(n+k) by A6,NAT_1:def 3;
  end;
  assume
A7: for n being Nat ex k being Nat st x in F.(n+k);
  for z being Nat holds x in f.z
  proof
    let z be Nat;
    set G = F ^\ z;
    consider k being Nat such that
A8: x in F.(z+k) by A7;
    f.z = Union G & G.k = F.(z + k) by A2,NAT_1:def 3;
    hence thesis by A8,PROB_1:12;
  end;
  hence thesis by A1,Th3;
end;
