  Mishchenko's theorem states that piecewise smooth and Lie algebroid
cohomology of a transitive Lie algebroid defined over a combinatorial manifold
are isomorphic. In this paper, we describe two applications of that result. The
first application consists in the relationship between piecewise de Rham
cohomology of a Lie groupoid and piecewise smooth cohomology of its Lie
algebroid. For the second application, we combine the classical result dealing
with invariant cohomology in Lie algebroids with the Mishchenko's theorem.
