reserve n for Nat;

theorem Th23:
  for T being non empty TopSpace, F being SetSequence of the
  carrier of T, A being Subset of T st for i being Nat holds F.i = A holds
  Lim_inf F = Cl A
proof
  let T be non empty TopSpace, F be SetSequence of the carrier of T, A be
  Subset of T;
  assume
A1: for i being Nat holds F.i = A;
  then for i being Nat holds F.i c= A;
  hence Lim_inf F c= Cl A by Th22;
  thus Cl A c= Lim_inf F
  proof
    let x be object;
    assume
A2: x in Cl A;
    then reconsider p = x as Point of T;
    for G being a_neighborhood of p ex k being Nat st for m
    being Nat st m > k holds F.m meets G
    proof
      let G being a_neighborhood of p;
      take k = 1;
      let m be Nat;
      assume m > k;
      F.m = A by A1;
      hence thesis by A2,CONNSP_2:27;
    end;
    hence thesis by Def1;
  end;
end;
