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By (1) and definition of | we have some a in N such that
(3) 3a = n3+2n.
We therefore have
which implies (2) by definition of |.
(n+1)3+2(n+1) = (n3 + 3n2 + 3n + 1) + (2n+2) = (n3 + 2n) + (3n2 + 3n + 3) = (3) 3a + 3(n2 + n + 1) = 3(a + n2 + n + 1)