reserve n for Nat;

theorem Th14:
  for S being SetSequence of the carrier of TOP-REAL n, p being
Point of TOP-REAL n, p9 being Point of Euclid n st p = p9 holds p in Lim_inf S
  iff for r being Real st r > 0 ex k being Nat st for m being
  Nat st m > k holds S.m meets Ball (p9, r)
proof
  let S be SetSequence of the carrier of TOP-REAL n, p be Point of TOP-REAL n,
  p9 be Point of Euclid n;
  assume
A1: p = p9;
  hereby
    assume
A2: p in Lim_inf S;
    let r be Real;
    assume r > 0;
    then reconsider G = Ball (p9, r) as a_neighborhood of p by A1,GOBOARD6:2;
    ex k being Nat st for m being Nat st m > k holds
    S.m meets G by A2,Def1;
    hence ex k being Nat st for m being Nat st m > k
    holds S.m meets Ball (p9, r);
  end;
  assume
A3: for r being Real st r > 0 ex k being Nat st for m
  being Nat st m > k holds S.m meets Ball (p9, r);
  now
    let G be a_neighborhood of p;
A4: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
    then reconsider G9 = Int G as Subset of TopSpaceMetr Euclid n;
A5: p9 in G9 by A1,CONNSP_2:def 1;
    G9 is open by A4,PRE_TOPC:30;
    then consider r being Real such that
A6: r > 0 and
A7: Ball (p9, r) c= G9 by A5,TOPMETR:15;
    consider k being Nat such that
A8: for m being Nat st m > k holds S.m meets Ball (p9, r) by A3,A6;
    take k;
    let m be Nat;
    assume m > k;
    then G9 c= G & S.m meets Ball (p9, r) by A8,TOPS_1:16;
    hence S.m meets G by A7,XBOOLE_1:1,63;
  end;
  hence thesis by Def1;
end;
