reserve X, Y for RealNormSpace;

theorem Th3:
  for r,a be Real st 0 < a holds Ball(0.X,a*r) = a
  * Ball(0.X,r)
proof
  let r,a be Real;
  assume
A1: 0 < a;
  thus Ball(0.X,a*r) c= a * Ball(0.X,r)
  proof
    let z be object;
    assume
A2: z in Ball(0.X,a*r);
    then reconsider z1=z as Point of X;
    ex zz1 be Point of X st z1=zz1 & ||.0.X-zz1.|| < a*r by A2;
    then ||.-z1.|| < a*r by RLVECT_1:14;
    then ||.z1.|| < a*r by NORMSP_1:2;
    then a"* ||.z1.|| < a"* (a*r) by A1,XREAL_1:68;
    then a"* ||.z1.|| < (a*a")*r;
    then
A3: a"* ||.z1.|| < 1*r by A1,XCMPLX_0:def 7;
    set y1=a"*z1;
    ||.y1.|| = |.a".|* ||.z1.|| by NORMSP_1:def 1
      .= a"* ||.z1.|| by A1,ABSVALUE:def 1;
    then ||.-y1.|| < r by A3,NORMSP_1:2;
    then ||.0.X-y1.|| < r by RLVECT_1:14;
    then
A4: y1 in Ball(0.X,r);
    a*y1=a*a"*z1 by RLVECT_1:def 7
      .= 1*z1 by A1,XCMPLX_0:def 7
      .= z1 by RLVECT_1:def 8;
    hence thesis by A4;
  end;
  let z be object;
  assume
A5: z in a*Ball(0.X,r);
  then reconsider z1 = z as Point of X;
  consider y1 be Point of X such that
A6: z1=a * y1 and
A7: y1 in Ball(0.X,r) by A5;
  ex yy1 be Point of X st y1=yy1 & ||.0.X-yy1.||<r by A7;
  then ||.-y1.|| < r by RLVECT_1:14;
  then
A8: ||.y1.|| < r by NORMSP_1:2;
  ||.z1.|| = |.a.|* ||.y1.|| by A6,NORMSP_1:def 1
    .= a* ||.y1.|| by A1,ABSVALUE:def 1;
  then ||.z1.|| < a*r by A1,A8,XREAL_1:68;
  then ||.-z1.|| < a*r by NORMSP_1:2;
  then ||.0.X-z1.|| < a*r by RLVECT_1:14;
  hence thesis;
end;
