reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th2:
  for X,Y be RealNormSpace,
        x be Point of X,
        y be Point of Y,
        z be Point of [:X,Y:],
    r1,r2 be Real st 0 < r1 & 0 < r2 & z = [x,y]
  holds
    ex s be Real
    st s = min(r1, r2) & s > 0
     & Ball(z,s) c= [:Ball(x,r1), Ball(y,r2):]
  proof
    let X,Y be RealNormSpace,
          x be Point of X,
          y be Point of Y,
          z be Point of [:X,Y:],
      r1,r2 be Real;

    assume
    A1: 0 < r1 & 0 < r2 & z = [x,y];

    take s = min(r1,r2);

    Ball(z,s) c= [:(Ball (x,r1)),(Ball (y,r2)):]
    proof
      let z1 be object;
      assume
      A3: z1 in Ball(z,s); then
      reconsider z1 as Point of [:X,Y:];
      A4: ex z2 be Point of [:X,Y:] st z2 = z1 & ||.z - z2.|| < s by A3;

      consider x1 be Point of X, y1 be Point of Y such that
      A5: z1 = [x1, y1] by PRVECT_3:18;
      -z1 = [-x1,-y1] by A5,PRVECT_3:18; then
      z - z1 = [x - x1, y - y1] by A1,PRVECT_3:18; then
        ||.x - x1.|| <= ||.z - z1.||
      & ||.y - y1.|| <= ||.z - z1.|| by NDIFF_8:21; then
      A6: ||.x - x1.|| < s & ||.y - y1.|| < s by A4,XXREAL_0:2;
      s <= r1 & s <= r2 by XXREAL_0:17; then
      ||.x - x1.|| < r1 & ||.y - y1.|| < r2 by A6,XXREAL_0:2; then
      x1 in Ball(x,r1) & y1 in Ball(y,r2);
      hence thesis by A5,ZFMISC_1:87;
    end;
    hence thesis by A1,XXREAL_0:15;
  end;
