Unfortunately the professor’s note does not specify the case when the risk-free rate decreases, only when it increases. But if it didn’t specify that case, then it should be intuitive: for call options the relation between RFR and call value is positive, but for put options the relation is negative.
C - S + [X / (1 + Rf)^T] = P. Keeping other things equal, you can clearly see put option and risk free rate are inversely related.
P + S - [X / (1 + Rf)^T] = C. Keeping other things constant, they are directly related.
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