reserve n for Nat;

theorem
  for N being Nat, F being sequence of TOP-REAL N, x being
  Point of TOP-REAL N, x9 being Point of Euclid N st x = x9 holds
  x is_a_cluster_point_of F iff for r being Real, n being Nat st
  r > 0 holds ex m being Nat st n <= m & F.m in Ball (x9, r)
proof
  let N be Nat, F be sequence of TOP-REAL N, x be Point of TOP-REAL
  N, x9 be Point of Euclid N;
  assume
A1: x = x9;
  hereby
    assume
A2: x is_a_cluster_point_of F;
    let r be Real, n be Nat;
    reconsider O = Ball (x9, r) as open Subset of TOP-REAL N by Th1;
    assume r > 0;
    then x in O by A1,GOBOARD6:1;
    then consider m being Element of NAT such that
A3: n <= m & F.m in O by A2,FRECHET2:def 3;
    reconsider m as Nat;
    take m;
    thus n <= m & F.m in Ball (x9, r) by A3;
  end;
  assume
A4: for r being Real, n being Nat st r > 0 holds ex m
  being Nat st n <= m & F.m in Ball (x9, r);
  for O being Subset of TOP-REAL N, n being Nat  st O is open &
  x in O ex m being Element of NAT st n <= m & F.m in O
  proof
    let O be Subset of TOP-REAL N, n be Nat;
    assume that
A5: O is open and
A6: x in O;
    reconsider n9=n as Nat;
A7: the TopStruct of TOP-REAL N = TopSpaceMetr Euclid N by EUCLID:def 8;
    then reconsider G9 = O as Subset of TopSpaceMetr Euclid N;
    G9 is open by A5,A7,PRE_TOPC:30;
    then consider r being Real such that
A8: r > 0 and
A9: Ball (x9, r) c= G9 by A1,A6,TOPMETR:15;
    consider m being Nat such that
A10: n9 <= m & F.m in Ball (x9, r) by A4,A8;
     reconsider m as Element of NAT by ORDINAL1:def 12;
    take m;
    thus thesis by A9,A10;
  end;
  hence thesis by FRECHET2:def 3;
end;
