You can prove this in many different ways. One way is to use the snake lemma (Lemma 10.4.1), the Jacobi-Zariski sequence (Lemma 10.132.4), combined with the characterization of projective modules as being direct summands of free modules (Lemma 10.76.2). Another proof can be obtained by combining Lemmas 10.135.10, 10.135.9 and 10.135.13.
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