The cyclically reduced product of two words u, v, denoted u * v, is the cyclically reduced form of the concatenation of u by v. This product is not associative. Recently S. V. Ivanov has proved that the Andrews-Curtis conjecture can be restated in terms of the cyclically reduced product and cyclic permutations instead of the reduced product and conjugations.In a previous paper we have proved that * verifies generalizations of properties of the product in the free group. In another previous paper we have proved that * verifies a generalized version of the associativity property in a special case. In the present paper we prove that a more general version of the associativity property holds for * in the general case.