The last chapter is devoted to computing the contribution of the open subset of non-preserved leaves to the trace formula. The procedure is a bit similar to the case of a compact manifold without preserved leaves, but the new problem is that this open subset is not compact. Then we also consider it as the interior of the compact manifold with boundary obtained by cutting the manifold along the preserved leaves. This allows us to apply techniques from small b-calculus. In this way, we have to use a b-metric, b-pseudo-differential operators, and the b-trace to reach the contribution to the trace formula. The choice of a representative of a 1-cohomology class is made for this computation, which induces an additional term in the formula. It is here that the main analytic tool allows us to choose the right representative so that the resulting Lefschetz distribution is as desired.