Link to problem and solution: Here

In part (d) of this problem, they assume P to be also augmented w.r.t M. How's this assumed? And after assuming this, they prove something else, and since that is proved, they say the condition they assumed earlier is true.

It seems very cyclic to me. In the previous part (part (c)), they had proved that P is augmented if it is vertex disjoint. But in this part, it is not vertex disjoint. Am I missing something?