A tree-composition lemma

A nonzero tuple wholly in ${\tilde{V}}_{+}^n$ or ${\tilde{V}}_{-}^n$ has nonzero sum, because the closed causal cones are pointed. Hence every kernel bounded by $Z_n$ is microcausal in XII-A's sense. The lemma is special to tree compositions on Minkowski space with translation-invariant lines; it is not a statement about loops.