Half Binomial Coeff cauchy rearrange

import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Analysis.Analytic.Binomial
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Ring.Basic
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Card
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
import Definitions.Def_D1Bound5_HalfBinomialCoefficients_Definitions

section
namespace D1Bound5






theorem _root_.D1Bound5.halfBinomialCoeff_cauchy_rearrange (x y z : ℂ) (n : ℕ) :
    (∑ ij ∈ Finset.antidiagonal n,
      ((halfBinomialCoeff ij.1 : ℂ) * (x * z) ^ ij.1) *
        ((halfBinomialCoeff ij.2 : ℂ) * (y * z) ^ ij.2)) =
    (∑ ij ∈ Finset.antidiagonal n,
      ((halfBinomialCoeff ij.1 * halfBinomialCoeff ij.2 : ℝ) : ℂ) *
        x ^ ij.1 * y ^ ij.2) * z ^ n := by sorry



end D1Bound5
end