reserve n for Nat;

theorem Th22:
  for T being non empty TopSpace, P being Subset of T, s being
  SetSequence of the carrier of T st (for i being Nat holds s.i c= P) holds
  Lim_inf s c= Cl P
proof
  let T be non empty TopSpace, P be Subset of T, s be SetSequence of the
  carrier of T;
  assume
A1: for i being Nat holds s.i c= P;
  let x be object;
  assume
A2: x in Lim_inf s;
  then reconsider p = x as Point of T;
  for G being Subset of T st G is open holds p in G implies P meets G
  proof
    let G be Subset of T;
    assume
A3: G is open;
    assume p in G;
    then reconsider G9 = G as a_neighborhood of p by A3,CONNSP_2:3;
    consider k being Nat such that
A4: for m being Nat st m > k holds s.m meets G9 by A2,Def1;
    set m = k + 1;
    m > k by NAT_1:13;
    then s.m meets G9 by A4;
    hence thesis by A1,XBOOLE_1:63;
  end;
  hence thesis by PRE_TOPC:def 7;
end;
