Abstract: N−1 total supply capability (TSC) is an important index for evaluating distribution network. This paper proposes and proves the TSC judgment theorems. Theorem 1: each feeder can find a feeder/feeder segment that has a transfer relationship with it, so that the loads of the two are complementary; and the load of backup feeder in N-supply-one-backup mode and asymmetry multiple-section-and-tie mode is the smallest, and the load of feeder segments with backup provided by the same feeder is equal in the symmetry multiple-section-and-tie mode. Theorem 2: after any feeder N−1, all feeders connected with the fault feeder are fully loaded. Theorem 3: redundancy is the smallest. Redundancy is the value of redundancy formula obtained by strict boundary equality constraints. These three theorems are judged from different angles of load characteristics before N−1, load characteristics after N−1 and mathematical model. Above three theorems are proved in this paper, and verified by typical wiring modes in China and distribution network case. Furthermore, an example of the application of the theorem is also given in this paper. Judgment theorem is a basic work of TSC theory system. Compared with existing research, judgment theorems abstract the load distribution characteristics of TSC points, which can easily determine all the system states reaching TSC, provide a direction for optimizing load distribution in planning and operation, and give full play to the power supply capability of distribution network.