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Show that european call, put with same maturity and strike are equal iff strike=forward price


The book defines put call parity as:
\( P(t)+S(t)-C(t) = Ke^{-r(T-t)} \)
where P(t) is value of the put, C(t) value of the call, S(t) is the value of the underlying asset, K is strike price.

In the proof, they substitute F (price of the forward) for the value of the asset. Why are they allowed to do this? Aren't the forward price and asset value different things?