reserve n for Nat;

theorem
  for S being SetSequence of TOP-REAL 2 holds lim_inf S c= Lim_inf S
proof
  let S be SetSequence of TOP-REAL 2;
  let x be object;
  assume
A1: x in lim_inf S;
  then reconsider p = x as Point of Euclid 2 by TOPREAL3:8;
  reconsider y = x as Point of TOP-REAL 2 by A1;
  consider k being Nat such that
A2: for n being Nat holds x in S.(k+n) by A1,KURATO_0:4;
  for r being Real st r > 0 ex k being Nat st for m
  being Nat st m > k holds S.m meets Ball (p, r)
  proof
    let r be Real;
    assume r > 0;
    then
A3: x in Ball (p, r) by GOBOARD6:1;
     reconsider k as Nat;
    take k;
    let m be Nat;
    assume m > k;
    then consider h being Nat such that
A4: m = k + h by NAT_1:10;
    x in S.m by A2,A4;
    hence thesis by A3,XBOOLE_0:3;
  end;
  then y in Lim_inf S by Th14;
  hence thesis;
end;
