Bennett Cleared Taylor two poles scaled eval

import Lean.Elab.Tactic.Omega
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mathlib.Combinatorics.Enumerative.Catalan.Basic
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.PowerSeries.Binomial
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Definitions.Def_D1Bound5_BennettClearedResidues_Definitions
import Definitions.Def_D1Bound5_BennettPoleSeries_Definitions

section
open Finset

namespace D1Bound5

variable {F : Type*} [Field F] [CharZero F]
















theorem _root_.D1Bound5.bennettClearedTaylor_two_poles_scaled_eval
    (s ν a N d e : F) (k r p q : ℕ) :
    s * (bennettClearedTaylor k r ν a
      (bennettPoleSeries d p * bennettPoleSeries e q)).eval N =
    ∑ ht ∈ Finset.antidiagonal r, ∑ uv ∈ Finset.antidiagonal ht.2,
      s * ((Ring.choose ((k : F) + ν) ht.1 *
        ((Ring.choose (-(p : F)) uv.1 * (d⁻¹) ^ (p + uv.1)) *
          (Ring.choose (-(q : F)) uv.2 * (e⁻¹) ^ (q + uv.2)))) *
            (N + a) ^ (k - ht.1)) := by sorry













end D1Bound5
end