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                                                    The Basic Approximation Lemma
                        Proof. By the definition of P A , the only arithmetic progressions
                        in S(n;P A ) are the trivial ones, three equal terms, which number is
                                                                                  2
                        at most n. Thus A(n;P A ) ≤ n, and so, by (8), C 3  n 2  + o(n ) ≤ n.
                                                                      P A 2
                                         0.
                        Therefore C P A
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