theorem
  A <> {} implies Ball(A,r1) is open
proof
  assume
A1: A <> {};
  let x;
  assume x in Ball(A,r1); then
A2: dist(x,A) < r1 by Th118;
  take r = r1 - dist(x,A);
  thus 0 < r by A2,XREAL_1:50;
  let z;
  assume |.z.| < r; then
A3: |.z.| + dist(x,A) < r + dist(x,A) by XREAL_1:6;
  dist(x + z,A) <= |.z.| + dist(x,A) by A1,Th114;
  then dist(x + z,A) < r + dist(x,A) by A3,XXREAL_0:2;
  hence thesis;
end;
