Theorem 6. 5 .
5Suppose that A, B have property (i) of Theorem 5.4, and that A, B are linked. Suppose that A is an Euler **data** with deg α A d ≤ −2 for all d ≺ 0, and that there exists power series f ∈ R[[K ∨ ]], g = (g 1 , .., g m ), g j ∈ R[[K ∨ ]], all without constant term,such thate f /α HG[B](t) = Ω − Ω H · (t + g) α + O(α −2 ) (6.5)when expanded in powers of α −1 . ThenHG[A](t + g) = e f /α HG[B](t). Proof: By Theorem 6.4, f, g define two mirror transformations µ f , ν g , with HG[B](t) = e f /α HG[B](t) HG[Ã](t) = HG[A](t + g) (6.6) whereB = µ f (B),Ã = ν g (A). Now bothB,Ã have property (i) of Theorem 5.4. (See remark after Theorem 6.4.) Since deg α A d ≤ −2, HG[Ã](t) has the same asymtotic form as HG[B](t) in eqn. (6.5) mod O(α −2 ). It follows that e H·t/α HG[Ã −B](t) ≡ O(α −2 ), or equivalently deg α (Ã d −B d ) ≤ −2. ThusÃ,B satisfy condition (ii) of Theorem 5.4.
