Birkhoff-von Neumann Proof inequality explanation

Question

https://staff.fnwi.uva.nl/n.s.walton/Notes/Hall_Birkhoff.pdf

Could someone possible explain how the inequality arises in $(44)$?

Answer 1

This is a consequence of $P_{i,j}\geq 0$ and the fact that the sum on the right runs over more elements. $$ \sum_{i \in U} \sum_{j \in N(U)}{N_{i,j}} \leq $$ $$ \sum_{i = 1}^n \sum_{j \in N(U)}{N_{i,j}} = $$ $$ \sum_{j \in N(U)} \sum_{i = 1}^n {N_{i,j}}. $$