reserve n for Nat;

theorem Th20:
  for T being non empty TopSpace, F, G being SetSequence of the
carrier of T st for i being Nat holds G.i = Cl (F.i) holds Lim_inf G
  = Lim_inf F
proof
  let T be non empty TopSpace, F, G be SetSequence of the carrier of T;
  assume
A1: for i being Nat holds G.i = Cl (F.i);
  thus Lim_inf G c= Lim_inf F
  proof
    let x be object;
    assume
A2: x in Lim_inf G;
    then reconsider p = x as Point of T;
    for H being a_neighborhood of p ex k being Nat st for m
    being Nat st m > k holds F.m meets H
    proof
      let H be a_neighborhood of p;
      consider H1 being non empty Subset of T such that
A3:   H1 is a_neighborhood of p and
A4:   H1 is open and
A5:   H1 c= H by CONNSP_2:5;
      consider k being Nat such that
A6:   for m being Nat st m > k holds G.m meets H1 by A2,A3,Def1;
      take k;
      let m be Nat;
      assume m > k;
      then G.m meets H1 by A6;
      then consider y being object such that
A7:   y in G.m and
A8:   y in H1 by XBOOLE_0:3;
      reconsider y as Point of T by A7;
      H1 is a_neighborhood of y by A4,A8,CONNSP_2:3;
      then consider H9 being non empty Subset of T such that
A9:   H9 is a_neighborhood of y and
      H9 is open and
A10:  H9 c= H1 by CONNSP_2:5;
      y in Cl (F.m) by A1,A7;
      then H9 meets F.m by A9,CONNSP_2:27;
      then H1 meets F.m by A10,XBOOLE_1:63;
      hence thesis by A5,XBOOLE_1:63;
    end;
    hence thesis by Def1;
  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 Lim_inf F c= Lim_inf G by Th17;
end;
