
theorem
  for V being RealUnitarySpace, v being VECTOR of V, r being Real st the
carrier of V <> {0.V} & r > 0 ex w being VECTOR of V st w <> v & w in Ball(v,r)
proof
  let V be RealUnitarySpace;
  let v be VECTOR of V;
  let r be Real;
  assume that
A1: the carrier of V <> {0.V} and
A2: r > 0;
  consider r9 being Real such that
A3: 0 < r9 and
A4: r9 < r by A2,XREAL_1:5;
  reconsider r9 as Real;
  now
    per cases;
    suppose
A5:   v = 0.V;
      ex w being VECTOR of V st w <> 0.V & ||.w.|| < r
      proof
        not the carrier of V c= {0.V} by A1;
        then NonZero V <> {} by XBOOLE_1:37;
        then consider u being object such that
A6:     u in NonZero V by XBOOLE_0:def 1;
A7:     not u in {0.V} by A6,XBOOLE_0:def 5;
        reconsider u as VECTOR of V by A6;
A8:     u <> 0.V by A7,TARSKI:def 1;
        then
A9:     ||.u.|| <> 0 by BHSP_1:26;
        set a = ||.u.||;
A10:    ||.u.|| >= 0 by BHSP_1:28;
        take w = (r9/a)*u;
A11:    ||.w.|| = |.r9/a.|*||.u.|| by BHSP_1:27
          .= (r9/a)*||.u.|| by A3,A10,ABSVALUE:def 1
          .= r9/(a/a) by XCMPLX_1:81
          .= r9 by A9,XCMPLX_1:51;
        r9/a > 0 by A3,A9,A10,XREAL_1:139;
        hence thesis by A4,A8,A11,RLVECT_1:11;
      end;
      then consider u being VECTOR of V such that
A12:  u <> 0.V and
A13:  ||.u.|| < r;
      ||.v-u.|| = ||. -u .|| by A5
        .= ||. u .|| by BHSP_1:31;
      then u in {y where y is Point of V : ||.v - y.|| < r} by A13;
      then u in Ball(v,r) by BHSP_2:def 5;
      hence thesis by A5,A12;
    end;
    suppose
A14:  v <> 0.V;
      set a = ||.v.||;
A15:  a <> 0 by A14,BHSP_1:26;
      set u9 = (1-r9/a)*v;
A16:  ||.v-u9.|| = ||. 1*v - (1-r9/a)*v .|| by RLVECT_1:def 8
        .= ||. (1-(1-r9/a))*v .|| by RLVECT_1:35
        .= |.r9/a.|*||.v.|| by BHSP_1:27;
      a >= 0 by BHSP_1:28;
      then
A17:  ||.v-u9.|| = r9/a*a by A3,A16,ABSVALUE:def 1
        .= r9/(a/a) by XCMPLX_1:81
        .= r9 by A15,XCMPLX_1:51;
      then v - u9 <> 0.V by A3,BHSP_1:26;
      then
A18:  v <> u9 by RLVECT_1:15;
      u9 in {y where y is Point of V : ||.v - y.|| < r} by A4,A17;
      then u9 in Ball(v,r) by BHSP_2:def 5;
      hence thesis by A18;
    end;
  end;
  hence thesis;
end;
