reserve n for Nat;

theorem
  for F, G being SetSequence of the carrier of TOP-REAL 2 st for i being
  Nat holds G.i = Cl (F.i) holds Lim_sup G = Lim_sup F
proof
  let F, G be SetSequence of the carrier of TOP-REAL 2;
  assume
A1: for i being Nat holds G.i = Cl (F.i);
  thus Lim_sup G c= Lim_sup F
  proof
    let x be object;
    assume x in Lim_sup G;
    then consider H being subsequence of G such that
A2: x in Lim_inf H by Def2;
    consider NS being increasing sequence of NAT such that
A3: H = G * NS by VALUED_0:def 17;
    set P = F * NS;
    reconsider P as SetSequence of TOP-REAL 2;
    reconsider P as subsequence of F;
    for i being Nat holds H.i = Cl (P.i)
    proof
      let i be Nat;
A4:  i in NAT by ORDINAL1:def 12;
A5:   dom NS = NAT by FUNCT_2:def 1;
      then H.i = G.(NS.i) by A3,FUNCT_1:13,A4
        .= Cl (F.(NS.i)) by A1
        .= Cl (P.i) by A5,FUNCT_1:13,A4;
      hence thesis;
    end;
    then Lim_inf H = Lim_inf P by Th20;
    hence thesis by A2,Def2;
  end;
  for i being Nat holds F.i c= G.i
  proof
    let i be Nat;
    G.i = Cl (F.i) by A1;
    hence thesis by PRE_TOPC:18;
  end;
  hence thesis by Th34;
end;
