Theorem 5.24: The labeling algorithm produces a maximum flow Proof: P=xx is labeled when algorithm end, thus v-P=( xx is not labeled when algorithm end. By the labeling algorithm, sEP and tEV-P. Thus E(P,V-P)is a cut. (1)(i)∈E(PvP)(ie.ie∈P.j∈VP) (2)(,i)∈E(e.i∈P.j∈VP) f:=0. By lemma 5.3, the labeling algorithm produces a maximum flow▪ Theorem 5.24: The labeling algorithm produces a maximum flow. ▪ Proof: P={x|x is labeled when algorithm end},thus V-P={ x|x is not labeled when algorithm end}. ▪ By the labeling algorithm, sP and tV-P. Thus E(P,V-P)is a cut. ▪ (1) (i,j) E(P,V-P)(i.e. iP. jV-P) ▪ fij=cij， ▪ (2) (j,i) E (i.e. iP. jV-P) ▪ fji=0. ▪ By lemma 5.3, the labeling algorithm produces a maximum flow