Similarities are shown between the algebras of differential forms and of Clifford algebra-valued multi-vector functions in an open region of Euclidean space. The Poincar´e Lemma and the Dual Poincar´e Lemma are restated and proved in a refined version. In the case of real-analytic differential forms an alternative proof of the Poincar´e Lemma is given using the Euler operator. A position is taken in the debate on the redundancy of either of the two algebras.