 The intersection of any non-empty family of proper cones is again a proper cone.Each proper cone "C" in a real vector space induces an order on the vector space by defining "x" ≤ "y" if and only if "y" − "x" ∈ "C", and furthermore, the positive cone of this ordered vector space will be "C".Therefore, there exists a one-to-one correspondence between the proper convex cones of "X" and the vector partial orders on "X".