Exercise 7, § 1

Let E be a fully lattice-ordered space, H an isolated subspace of E (Exer. 4). If $B_1$ and $B_2$ are two supplementary bands in E, show that H is the ordered direct sum of $H_1 = H \cap B_1$ and $H_2 = H \cap B_2$, and that the Riesz space $E/H$ is the ordered direct sum of the Riesz spaces $B_1/H_1$ and $B_2/H_2$.