Give a financial interpretation of the following theorems and propositions :

1) For European and American options with the same exercise price X and the same maturity time T, given that an American option gives at least the same rights as the corresponding European option:

C^EC^A ; P^EP^A

2) Theorem . The prices of the European and American call options in a stock that does not pay dividends are equal, C^A = C^E, whenever the strike price X and expiry time T are the same for both options.

3) Proposition .If X < X, then

C^E (X) C^E (X) ,

P^E (X) P^E (X) .

This means that C^E(X) is a strictly decreasing and P^E(X) a strictly increasing function of X.