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1 0 1 ¯ ¯ fq ,fq ∈T (P ) fq ,fq ∈T (βM (a,C) . . β1 )− 4 1 a∈Lu (P )∪Lr0 (P ) − 54 (βM (a,C) . . β1 ) a∈L0 (P ) (βt(i) . . β1 ) (βt(i) . . β1 ) 0c gi ∈DC a∈L1 3 (βM (a,C) . . β1 )− 4 ) 1 2 1c gi ∈DC Each βi appears with the exponent −1 + xi . 99) where we defined ITi0 ITi1 ILi (C) IL0i (C) IL1i (C) Si0 (C) Si1 (C) := := := := := := := {fj , f¯j ∈ T 0 (P )|j ≥ i} {fj , f¯j ∈ T 1 |j ≥ i} {a ∈ L|M (a, C) ≥ i} {a ∈ L0 (P )|M (a, C) ≥ i} {a ∈ L1 |M (a, C) ≥ i} {gj ∈ DC0c (P )|t(j) ≥ i} {gj ∈ DC1c |t(j) ≥ i}.

114) becomes 2p ||φi ||∞,2 e−(1− |ΓΛ 2p (φ1 , . . 121) Vol. 1, 2000 Continuous Constructive Fermionic Renormalization 53 where K depends only on . )2 . In the case Λ < m, we have a few changes to perform. 122) lq ∈T The factor (Λ/m)n exactly changes Λ2−p−n into Λ2−p m−n = Λ2−p (Λm )−n . The factor (Λm )−n is absorbed in K n¯ since Λm in the hypothesis of Theorem 3 −1/2 remains in the compact X. 104), which was previously bounded by 1, hence not used at all. 123) β1 0 Changing to the variable v = ( /4)m2 β1 Λ−2 we obtain for the final bound a factor m2 Λ−2 /4 (4/ m2 )(p−2)/2 v (p−2)/2 0 dv −v e ≤ (Λm )2−p K p v p!

From now on we work therefore with a fixed process P . We introduce some notations. We define L0 (P ) and L1 as the set of loop half-lines which bear some 0 single or double gradient respectively by some Rgi (P ) or Rg1 operator, Lr0 (P ) as 0 (P ) and Lu (P ) the set of loop half-lines moved to the reference vertex by some Rgi the loop half-lines left unchanged. In the same way we define the sets T 0 (P ), T 1 , T r0 (P ) and T u (P ) for the tree half-lines, and E 0 (P ), E 1 , E r0 (P ) and E u (P ) for the external half-lines.