
theorem Th1:
  for M being MetrSpace, x1,x2 being Point of M, r1,r2 being Real
ex x being Point of M, r being Real
   st Ball(x1,r1) \/ Ball(x2,r2) c= Ball(x,r)
proof
  let M be MetrSpace;
  let x1,x2 be Point of M;
  let r1,r2 be Real;
  reconsider x = x1 as Point of M;
  reconsider r = |.r1.| + |.r2.| + dist(x1,x2) as Real;
  take x;
  take r;
  for a being object
      holds a in Ball(x1,r1) \/ Ball(x2,r2) implies a in Ball( x,r)
  proof
    let a be object;
    assume
A1: a in (Ball(x1,r1) \/ Ball(x2,r2));
    then reconsider a as Point of M;
    now
      per cases by A1,XBOOLE_0:def 3;
      case
A2:     a in Ball(x1,r1);
        r1 <= |.r1.| & 0 <= |.r2.| by ABSVALUE:4,COMPLEX1:46;
        then
A3:     r1 + 0 <= |.r1.| + |.r2.| by XREAL_1:7;
A4:     dist(x,a) < r1 by A2,METRIC_1:11;
        0 <= dist(x1,x2) by METRIC_1:5;
        then r1 + 0 <= |.r1.| + |.r2.| + dist(x1,x2) by A3,XREAL_1:7;
        then dist(x,a) - r < r1 - r1 by A4,XREAL_1:14;
        then
A5:     dist(x,a) - r + r < 0 + r by XREAL_1:8;
        M is non empty by A2;
        hence thesis by A5,METRIC_1:11;
      end;
      case
A6:     a in Ball(x2,r2);
        then dist(x2,a) < r2 by METRIC_1:11;
        then dist(x2,a) - |.r2.| < r2 - r2 by ABSVALUE:4,XREAL_1:14;
        then dist(x,a) <= dist(x1,x2) + dist(x2,a) & dist(x2,a) - |.r2.| +
        |.r2.| < 0 + |.r2.| by METRIC_1:4,XREAL_1:8;
        then dist(x,a) - |.r2.| < dist(x1,x2) + dist(x2,a) - dist(x2,a ) by
XREAL_1:15;
        then dist(x,a) - |.r2.| - |.r1.| < dist(x1,x2) - 0 by COMPLEX1:46
,XREAL_1:14;
        then
A7:     dist(x,a) - (|.r1.| + |.r2.|) + (|.r1.| + |.r2.|) < |.r1.| +
        |.r2.| + dist(x1,x2) by XREAL_1:8;
        M is non empty by A6;
        hence thesis by A7,METRIC_1:11;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
