theorem
  y in Ball(x,r) & r <= q implies y in Ball(x,q)
proof
  assume that
A1: y in Ball(x,r) and
A2: r <= q;
  ||.x - y.|| < r by A1,Th40;
  then ||.x - y.|| < q by A2,XXREAL_0:2;
  hence thesis;
end;
