  Let G be any additive abelian group with cyclic torsion subgroup, and let A,
B and C be finite subsets of G with cardinality n>0. We show that there is a
numbering {a_i}_{i=1}^n of the elements of A, a numbering {b_i}_{i=1}^n of the
elements of B and a numbering {c_i}_{i=1}^n of the elements of C, such that all
the sums a_i+b_i+c_i (i=1,...,n) are distinct. Consequently, each subcube of
the Latin cube formed by the Cayley addition table of Z/NZ contains a Latin
transversal. This additive theorem can be further extended via restricted
sumsets in a field.
