  This is a paper in a series that studies smooth relative Lie algebra
homologies and cohomologies based on the theory of formal manifolds and formal
Lie groups. In two previous papers, we develop the basic theory of formal
manifolds, including generalizations of vector-valued distributions and
generalized functions on smooth manifolds to the setting of formal manifolds.
In this paper, we establish Poincar\'e's lemma for de Rham complexes with
coefficients in formal functions, formal generalized functions, compactly
supported formal densities, or compactly supported formal distributions.
