
theorem
  for V being RealUnitarySpace, v1,v2 being Point of V, r1,r2 being Real
ex u being Point of V, r being Real st Ball(v1,r1) \/ Ball(v2,r2) c= Ball(u,r)
proof
  let V be RealUnitarySpace;
  let v1,v2 be Point of V;
  let r1,r2 be Real;
  reconsider u = v1 as Point of V;
  reconsider r = |.r1.| + |.r2.| + dist(v1,v2) as Real;
  take u;
  take r;
    let a be object;
    assume
A1: a in (Ball(v1,r1) \/ Ball(v2,r2));
    then reconsider a as Point of V;
    now
      per cases by A1,XBOOLE_0:def 3;
      case
        a in Ball(v1,r1);
        then
A2:     dist(u,a) < r1 by BHSP_2:41;
        r1 <= |.r1.| & 0 <= |.r2.| by ABSVALUE:4,COMPLEX1:46;
        then
A3:     r1 + 0 <= |.r1.| + |.r2.| by XREAL_1:7;
        0 <= dist(v1,v2) by BHSP_1:37;
        then r1 + 0 <= |.r1.| + |.r2.| + dist(v1,v2) by A3,XREAL_1:7;
        then dist(u,a) - r < r1 - r1 by A2,XREAL_1:14;
        then dist(u,a) - r + r < 0 + r by XREAL_1:8;
        hence thesis by BHSP_2:41;
      end;
      case
        a in Ball(v2,r2);
        then dist(v2,a) < r2 by BHSP_2:41;
        then dist(v2,a) - |.r2.| < r2 - r2 by ABSVALUE:4,XREAL_1:14;
        then dist(v2,a) - |.r2.| + |.r2.| < 0 + |.r2.| by XREAL_1:8;
        then dist(u,a) - |.r2.| < dist(v1,v2) + dist(v2,a) - dist(v2,a ) by
BHSP_1:35,XREAL_1:15;
        then dist(u,a) - |.r2.| - |.r1.| < dist(v1,v2) - 0 by COMPLEX1:46
,XREAL_1:14;
        then
        dist(u,a) - (|.r1.| + |.r2.|) + (|.r1.| + |.r2.|) < |.r1.| +
        |.r2.| + dist(v1,v2) by XREAL_1:8;
        hence thesis by BHSP_2:41;
      end;
    end;
    hence thesis;
end;
