reserve a for Real;
reserve p,q for Point of TOP-REAL 2;

theorem Th14:
  for n being Element of NAT,p being Point of Euclid n,r being
Real, B being Subset of TOP-REAL n st B=cl_Ball(p,r)
holds B is bounded closed
proof
  let n be Element of NAT,p be Point of Euclid n,
r be Real, B be Subset of
  TOP-REAL n;
  assume
A1: B=cl_Ball(p,r);
  cl_Ball(p,r) c= Ball(p,r+1)
  proof
    let x be object;
A2: r<r+1 by XREAL_1:29;
    assume
A3: x in cl_Ball(p,r);
    then reconsider q=x as Point of Euclid n;
    dist(p,q)<=r by A3,METRIC_1:12;
    then dist(p,q)<r+1 by A2,XXREAL_0:2;
    hence thesis by METRIC_1:11;
  end;
  then cl_Ball(p,r) is bounded by TBSP_1:14;
  hence B is bounded by A1,JORDAN2C:11;
A4: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
  then reconsider BB = B as Subset of TopSpaceMetr Euclid n;
  BB is closed by A1,TOPREAL6:57;
  hence thesis by A4,PRE_TOPC:31;
end;
