Let us now treat the case that Y is VG, using terminology and results from Section 5. Let us first suppose that the definable functions which take values in VG and which appear in the build-up of f (namely in the forms of generators (2) and (3) of Section 3.1) are linear over X, that W is the definable set
(4.1.2) {(x, y) ∈ X × VG | α(x) ≤ y β(x)}
where is < or no condition and where α, β : X → VG are definable functions, and that all other build-up **data** of f (namely, generators (4) of Section 3.1 and h and e as in (3.2.1)) factor through the projection W → X. Then the conclusion follows from Lemma 4.1.2. Indeed, for any K in Loc 0 , any x ∈ X K and any ψ in D K , f K,ψ (x, y) is a finite sum of terms T i of the form
(4.1.3) c i,K,ψ (x)y a i q b i y K
for integers a i ≥ 0, rational numbers b i and c i in C exp (X). The integrability of f K,ψ (x, y) over y in W K,x is automatic when is < and we get g from Lemma 4.1.2. When is no condition, we regroup the terms if necessary, so that the pairs (a i , b i ) are mutually different for different i. By observing different asymptotic behavior of these terms for growing y, we may consider the subsum i∈J T i with i ∈ J if b i < 0. This time we apply Lemma 4.1.2 to this sub-sum to find g.