Suppose $V^+(S,t;K)$ is the value of a American option with strike $K$ before the exercise, and $V^-(S,t;K)$ is the value after exercise. Then how to understand the inequality $$V^+(S,t;K)\geq V^-(S,t;K) + (S - K)^+$$
After exercise, the option ceases to exist. It has no value.
If an option was exercised and
then there was still time value in the option and it should not have been exercised.