theorem Th48:
  z in cl_Ball(x,r) iff dist(x,z) <= r
proof
  thus z in cl_Ball(x,r) implies dist(x,z) <= r
  proof
    assume z in cl_Ball(x,r);
    then ||.x - z.|| <= r by Th47;
    hence thesis by BHSP_1:def 5;
  end;
  assume dist(x,z) <= r;
  then ||.x - z.|| <= r by BHSP_1:def 5;
  hence thesis;
end;
