Pairs of subsets of spheres and Cartesian products thereof with the same distribution of distance

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We prove the following three statements: 1) Let (A, A) be a partition of the spherical surface Sⁿ into two measurable sets. Let stₐ and st A be their measure density functions of distance. Then |stₐ - st A| depends only on the difference of their n-areas. 2) If the spherical surface Sⁿ is divided in two measurable subsets A and A of equal n-surface, then these two subsets have the same distribution of distance. 3) Let there be a pair (S, S') of subsets of a sphere Sⁿ such that stₛ = stₛ'. Then their complementary subsets satisfy st S = st S' and stS, S = stS', S', where stₐ, B is the measure density function of distance between a point in A and a point in B. Furthermore, it is shown that the statements remain true when Sⁿ is substituted by the Cartesian product Sⁿ¹ ×... × Sⁿʳ endowed with the metric which is naturally inherited from its factors.