reserve n for Nat;

theorem Th15:
  for T being non empty TopSpace, S being SetSequence of the
  carrier of T holds Cl Lim_inf S = Lim_inf S
proof
  let T be non empty TopSpace;
  let S be SetSequence of the carrier of T;
  thus Cl Lim_inf S c= Lim_inf S
  proof
    let x be object;
    assume
A1: x in Cl Lim_inf S;
    then reconsider x9 = x as Point of T;
    now
      let G be a_neighborhood of x9;
      set H = Int G;
      x9 in H by CONNSP_2:def 1;
      then Lim_inf S meets H by A1,PRE_TOPC:24;
      then consider z being object such that
A2:   z in Lim_inf S and
A3:   z in H by XBOOLE_0:3;
      reconsider z as Point of T by A2;
      z in Int H by A3;
      then H is a_neighborhood of z by CONNSP_2:def 1;
      then consider k being Nat such that
A4:   for m being Nat st m > k holds S.m meets H by A2,Def1;
      take k;
      let m be Nat;
      assume m > k;
      then S.m meets H by A4;
      hence S.m meets G by TOPS_1:16,XBOOLE_1:63;
    end;
    hence thesis by Def1;
  end;
  thus thesis by PRE_TOPC:18;
end;
