Exercise 4, § 1

Let E be a Riesz space, H a linear subspace of E. One says that H is an isolated subspace of E if the relations $x \in H, |y| \leq |x|$ imply $y \in H$. In this case, let P be the set of elements $x$ of the quotient space $E/H$ such that there exists at least one element $x \geq 0$ in the class $x$. Show that P is the set of elements $\geq 0$ for an order structure on $E/H$, compatible with the vector space structure of $E/H$, for which $E/H$ is a Riesz space (cf. A, VI, §1, Exer. 4).