theorem
  z in Sphere(x,r) iff dist(x,z) = r
proof
  thus z in Sphere(x,r) implies dist(x,z) = r
  proof
    assume z in Sphere(x,r);
    then ||.x - z.|| = r by Th51;
    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;
