reserve n for Nat;

theorem
  for T being non empty TopSpace, R, S being SetSequence of the carrier
  of T st R is subsequence of S holds Lim_inf S c= Lim_inf R
proof
  let T be non empty TopSpace, R, S be SetSequence of the carrier of T;
  assume R is subsequence of S;
  then consider Nseq being increasing sequence of NAT such that
A1: R = S * Nseq by VALUED_0:def 17;
  let x be object;
  assume
A2: x in Lim_inf S;
  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 R.m meets G
  proof
    let G be a_neighborhood of p;
    consider k being Nat such that
A3: for m being Nat st m > k holds S.m meets G by A2,Def1;
    take k;
    let m1 be Nat;
A4: m1 in NAT by ORDINAL1:def 12;
A5: m1 <= Nseq.m1 by SEQM_3:14;
    assume m1 > k;
    then
A6: Nseq.m1 > k by A5,XXREAL_0:2;
    dom Nseq = NAT by FUNCT_2:def 1;
    then R.m1 = S.(Nseq.m1) by A1,FUNCT_1:13,A4;
    hence thesis by A3,A6;
  end;
  hence thesis by Def1;
end;
