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In this paper we prove the following results. Let π be a set of prime numbers and G a finite π-soluble group. Consider U, V ≤ G and $H\in {\mathrm{Hall}}_{\pi }\left(G\right)$ such that $H\cap V\in {\mathrm{Hall}}_{\pi }\left(V\right)$ and $1\ne H\cap U\in {\mathrm{Hall}}_{\pi }\left(U\right)$. Suppose also $H\cap U$ is a Hall π-sub-group of some S-permutable subgroup of G. Then $H\cap U\cap V\in {\mathrm{Hall}}_{\pi }(U\cap V)$ and $\langle H\cap U,H\cap V\rangle \in {\mathrm{Hall}}_{\pi }\left(\langle U\cap V\rangle \right)$. Therefore,the set of all S-permutably embedded subgroups of a soluble group G into which a given Hall system Σ reduces is a sublattice of the lattice of all Σ-permutable subgroups of G. Moreover any two subgroups of this sublattice of coprimeorders permute.