reserve X, Y for RealNormSpace;

theorem Th2:
  for y1 be Point of X, r be Real holds Ball(y1,r) = y1 + Ball(0.X,r)
proof
  let y1 be Point of X, r be Real;
  thus Ball(y1,r) c=y1 + Ball(0.X,r)
  proof
    let t be object;
    assume
A1: t in Ball(y1,r);
    then reconsider z1=t as Point of X;
    set z0=z1-y1;
    ex zz1 be Point of X st z1=zz1 & ||.y1-zz1.|| < r by A1;
    then ||.-z0.|| < r by RLVECT_1:33;
    then ||.0.X-z0.||< r by RLVECT_1:14;
    then
A2: z0 in Ball(0.X,r);
    z0+y1=z1+((-y1)+y1) by RLVECT_1:def 3;
    then z0+y1=z1+0.X by RLVECT_1:5;
    then z1=z0+y1 by RLVECT_1:4;
    hence thesis by A2;
  end;
  let t be object;
  assume t in y1 + Ball(0.X,r);
  then consider z0 be Point of X such that
A3: t=y1+z0 and
A4: z0 in Ball(0.X,r);
  set z1=z0+y1;
  ex zz0 be Point of X st z0=zz0 & ||.0.X-zz0.|| < r by A4;
  then ||.-z0.|| < r by RLVECT_1:14;
  then ||.z0.|| < r by NORMSP_1:2;
  then ||.z0+ 0.X.|| < r by RLVECT_1:4;
  then ||.z0+ (y1+-y1).|| < r by RLVECT_1:5;
  then ||.z1-y1.|| < r by RLVECT_1:def 3;
  then ||.y1-z1.|| < r by NORMSP_1:7;
  hence thesis by A3;
end;
