reserve X, Y for RealNormSpace;

theorem Th11:
  for x be Point of X, r be Real holds Ball(0.X,r)=(-1)* Ball(0.X,r)
proof
  let x be Point of X, r be Real;
  thus Ball(0.X,r) c= (-1)*Ball(0.X,r)
  proof
    let z be object;
    assume
A1: z in Ball(0.X,r);
    then reconsider z1=z as Point of X;
    ex zz1 be Point of X st z1=zz1 & ||.0.X-zz1.|| < r by A1;
    then ||.-z1.|| < r by RLVECT_1:14;
    then ||.-(-z1).|| < r by NORMSP_1:2;
    then ||.0.X-(-z1).|| < r by RLVECT_1:14;
    then
A2: -z1 in Ball(0.X,r);
    (-1)*(-z1) =1*z1 by RLVECT_1:26
      .=z1 by RLVECT_1:def 8;
    hence thesis by A2;
  end;
  let z be object;
  assume
A3: z in (-1)*Ball(0.X,r);
  then reconsider z1=z as Point of X;
  consider y1 be Point of X such that
A4: z1=(-1)* y1 and
A5: y1 in Ball(0.X,r) by A3;
  ex zz1 be Point of X st y1 =zz1 & ||.0.X-zz1.||<r by A5;
  then ||.-y1.|| < r by RLVECT_1:14;
  then ||.-(-y1).|| < r by NORMSP_1:2;
  then
A6: ||.0.X-(-y1).|| < r by RLVECT_1:14;
  -y1 =z1 by A4,RLVECT_1:16;
  hence thesis by A6;
end;
