We present a new method to prove a number of the well-known theorems on properties of some classical Banach spaces. It is based on an elementary lemma (Lemma 1) which makes the most of the known proofs more short. Besides, we prove generalizations of the J.Lindenstrauss, A.M.Olevskii and A.Peƚczyński theorem on precise reproducibility of the Haar system and the J.Bourgain and H.Rosenthal theorem on the non-existence of sign-embeddings of the space $L_1$ into $c_0$. 

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