Full ring one three common extension is integer

import Mathlib.Algebra.QuadraticAlgebra.Basic
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Analysis.Complex.Norm
import Mathlib.Data.Finset.Card
import Mathlib.Data.Int.GCD
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination

section
open scoped NumberField ComplexConjugate
open Module

namespace D1Bound5

variable {K : Type*} [Field K] [NumberField K] [NumberField.IsTotallyComplex K]










theorem _root_.D1Bound5.full_ring_one_three_common_extension_is_integer (hK : finrank ℚ K = 2)
    {z : 𝓞 K} (hr : ∃ r : 𝓞 K, r*r = z+1)
    (hs : ∃ s : 𝓞 K, s*s = 3*z+1) : ∃ n : ℤ, z = (n : 𝓞 K) := by sorry

end D1Bound5
end