Any minimum test set (MLTS) for locally exhaustive testing of multiple output combinational circuits (CUTs) has at least 2w test patterns, where w is the maximum number of inputs on which any output depends. In the previous researches, it is clarified that every CUT with up to four outputs has an MLTS with 2w elements. On the other hand, it can be easily shown that every CUT with more than five outputs does not have such an MLTS. It has not been however known whether every CUT with five outputs has such an MLTS or not. In this paper, it is clarified that every CUT with five outputs has such an MLTS. First, some terminologies are introduced as preliminaries. Second, features of 5(w1) dependence matrices of CUTs with five outputs and (w1) inputs are discussed. Third, an equivalence relation between dependence matrices of two CUTs is introduced. The relation means that if it holds and one of the CUTs has an MLTS with 2w elements, then the other CUT also has such an MLTS. Based on the features described above, a theorem is established that there exists a 5w dependence matrix which is equivalent to each of the above 5(w1) matrices. Finally, it is proved by the use of the theorem that every CUT with five outputs has an MLTS with 2 w elements.