Early-exercise point of American Put

Question

Could you please help me? In book Rüdiger Seydel "Tools for Computational Finance" in Chapter 4.5 "American Options as Free Boundary Problem" it is provided the following explanation for case $S>S_f$ , where $S_f$ is contact point:
If $S>S_f$, then early-exercise causes immediate loss, because $-V+K-S<0$. Well, we have options so in case of early exercise we should borrow $1$ risky asset and seal it with price $K$. So our current payoff will be $K$. Why here is $-V+K-S$ ?
Thank you.

Answer 1

By the definition of $S_f$ (eqn 4.22), we have $V^{am}_P(S, t) > (K - S)^+$ whenever $S > S_f(t)$. Rearranging, $-V^{am}_P + (K - S)^+ < 0$, so we have

$-V^{am}_P + K - S \leq -V^{am}_P + (K - S)^+ < 0$