theorem
  y in Ball(x,r) implies y - z in Ball(x - z,r)
proof
  assume y in Ball(x,r);
  then
A1: dist(x,y) < r by Th41;
  dist(x - z,y - z) = dist(x,y) by BHSP_1:42;
  hence thesis by A1,Th41;
end;
