 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem Th13:
  for S be RealUnitarySpace,
      x be Point of S,
      y be Point of MetricSpaceNorm RUSp2RNSp S,
      r be Real
st x=y
holds Ball(x,r) = Ball(y,r)
proof
let S be RealUnitarySpace,
     x be Point of S,
     y be Point of MetricSpaceNorm RUSp2RNSp S,
     r be Real;
assume A1:x=y;
set PM = MetricSpaceNorm RUSp2RNSp S;
A2:Ball (y,r) = { z where z is Element of PM : dist (y,z) < r }
by METRIC_1:def 14;
for p be object holds
   p in Ball (x,r) iff p in Ball (y,r)
proof
let p be object;
 hereby assume p in Ball (x,r); then
 consider z be Point of S such that
   A3: p=z & ||.x - z.|| < r;
   reconsider w=z as Element of PM;
      dist (y,w) < r by A3,Th1,A1;
 hence p in Ball (y,r) by A3,A2;
end;
assume p in Ball (y,r); then
 consider w be Point of PM such that
   A4: p=w & dist(y,w) < r by A2;
   reconsider z=w as Element of S;
      ||.x-z.|| < r by A4,Th1,A1;
 hence p in Ball (x,r) by A4;
end;
hence thesis by TARSKI:2;
end;
