Let $G$ be a bipartite graph. In order to find a match in the $G$ diagram so that there are no unpaired elements in the set $A$, a necessary and sufficient condition is $|A-N(T)|\leq|B-T|$ that is for every $T\subset B$ . How can I prove this proposition?
Definitions and propositions that I know:
Corner points of graph $G$ that are not endpoints of any edge in a map $M$ are called unpaired vertices. The set formed by the neighbors of all vertices in $T$, $T\subset B$, is denoted by $N(T)$.
Hall Condition: Let $G=(A,B,E)$ be a two-set graph. For each $S\subset A$ set, $|S|\leq|N(S)|$ If so, $G$ is said to satisfy the Hall Condition.
I tried hard, I couldn't do it, please write if you can prove it.