theorem Th51:
  z in Sphere(x,r) iff ||.x - z.|| = r
proof
  thus z in Sphere(x,r) implies ||.x - z.|| = r
  proof
    assume z in Sphere(x,r);
    then ex y be Point of X st z = y & ||.x - y.|| = r;
    hence thesis;
  end;
  assume ||.x - z.|| = r;
  hence thesis;
end;
