Hedging Strategy Perpetual American Put

According to Shreve (Stochastic Calculus for Finance II ), the perpetual american put needs to satisfy two conditions :

Intuitively speaking, the first one reflects the fact that the value of the put needs to be greater or equal to the intrinsic value in order to avoid arbitrage by exercising right after purchasing the option. The second comes from the fact that the discounted option value is a supermartingale under the risk neutral measure.

A bit farther in the text, he states :

with the 8.3.20 given by :

How is this third condition relevant for the owner? what is the intuition behind?

The explanation given is not clear for me (given below). Any help please? Thank you !!!!