reserve
  x, y for object,
  i, n for Nat,
  r, s for Real,
  f1, f2 for n-element real-valued FinSequence;
reserve e, e1 for Point of Euclid n;

theorem Th19:
  e1 in Ball(e,r) implies
  ex m being non zero Element of NAT st OpenHypercube(e1,1/m) c= Ball(e,r)
  proof
    reconsider B = Ball(e,r) as Subset of TopSpaceMetr Euclid n;
    assume
A1: e1 in Ball(e,r);
    B is open by TOPMETR:14;
    then consider s being Real such that
A2: s > 0 and
A3: Ball(e1,s) c= B by A1,TOPMETR:15;
    per cases;
    suppose
A4:   n <> 0;
      then consider m being Nat such that
A5:   1/m < s/sqrt(n) and
A6:   m > 0 by A2,FRECHET:36;
      reconsider m as non zero Element of NAT by A6,ORDINAL1:def 12;
A7:   OpenHypercube(e1,s/sqrt(n)) c= Ball(e1,s) by A4,Th17;
      OpenHypercube(e1,1/m) c= OpenHypercube(e1,s/sqrt(n)) by A5,Th13;
      then OpenHypercube(e1,1/m) c= Ball(e1,s) by A7;
      hence thesis by A3,XBOOLE_1:1;
    end;
    suppose n = 0;
      then (OpenHypercube(e1,1/1) = {} or OpenHypercube(e1,1/1) = {{}}) &
      Ball(e,r) = {{}} by A1,EUCLID:77,ZFMISC_1:33;
      hence thesis;
    end;
end;
