Prove that if V = W1 ⊕ . . . . . . . ⊕ Wk then Wi ∩ Wj is trivial whenever i ≠ j. This shows that the first half of the proof of Lemma 4.15 extends to the case of more than two subspaces. (Example 4.19 shows that this implication does not reverse, the other half does not extend.)