  We prove that, in certain situations, intersection numbers on formal schemes
that come in profinite families vary locally constantly in the parameter. To
this end, we define the product $S\times M$ of a profinite set $S$ with a
locally noetherian formal scheme $M$ and study intersections thereon. Our
application is to the Arithmetic Fundamental Lemma of W. Zhang where the result
helps to remove a restriction in its recent proof, cf. arXiv:1909.02697.
