reserve X for RealNormSpace;

theorem Th3:
  for X be RealNormSpace, z be Element of MetricSpaceNorm X, r be
  Real holds ex x be Point of X st x = z & cl_Ball(z,r) = {y where y is
  Point of X: ||.x-y.|| <= r}
proof
  let X be RealNormSpace, z be Element of MetricSpaceNorm X, r be Real;
  reconsider x=z as Point of X;
  set M = MetricSpaceNorm X;
A1: cl_Ball(z,r) = {q where q is Element of M : dist(z,q) <= r} by
METRIC_1:def 15;
  now
    let a be object;
    assume a in {y where y is Point of X: ||.x-y.|| <= r};
    then consider y be Point of X such that
A2: a=y & ||.x-y.|| <= r;
    reconsider t=y as Element of M;
    ||.x-y.|| = dist(z,t) by Def1;
    hence a in {q where q is Element of M : dist(z,q) <= r} by A2;
  end;
  then
A3: {y where y is Point of X: ||.x-y.|| <= r}
     c= {q where q is Element of M: dist(z,q) <= r};
  now
    let a be object;
    assume a in {q where q is Element of M : dist(z,q) <= r};
    then consider q be Element of M such that
A4: a=q & dist(z,q) <= r;
    reconsider t=q as Point of X;
    ||.x-t.|| = dist(z,q) by Def1;
    hence a in {y where y is Point of X: ||.x-y.|| <= r} by A4;
  end;
  then
  {q where q is Element of M : dist(z,q) <= r} c= {y where y is Point of
X: ||.x-y.|| <= r};
  then
  {q where q is Element of M : dist(z,q) <= r} = {y where y is Point of
X: ||.x-y.|| <= r} by A3,XBOOLE_0:def 10;
  hence thesis by A1;
end;
