Harshman (UCLA Working Papers in Phonetics 1972; 22: 111-117) has given a proof of uniqueness (identification) of Parafac solutions, when two of the three component matrices are of full column rank, and the third satisfies a few other conditions. Kruskal has given more relaxed sufficient conditions, which do not require any of the component matrices to be of full column rank. However, even when two component matrices are of full column rank, Harshman’s conditions on the third matrix are still less easily satisfied than Kruskal’s. The present paper bridges the gap between the two sets of conditions by utilizing the possibilities of slice mixing in Harshman’s approach. It offers an alternative uniqueness theorem that is sufficiently general for all practical purposes and easy to interpret, with a proof that is easy to understand.