Show that the label changing algorithm must produce an equality subgraph that has a complete matching, by arguing as follows:

(a) Upon changing labeling, since by claim (5) there is a new edge from \(S\) to \(T^{c}\), show that there will either be an augmenting path in the new equality subgraph, or else the set \(S\) must become strictly larger.

(b) In the worst case, a labeling will be reached where \(S\) is the whole left side. Then show that further relabelings can lose no edges and must gain edges. Therefore conclude that a complete matching must be reached.