-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
a : adicCompletion (R := K[X]) K⟮X⟯ (idealX K)
this :
  ∀ (s : Set (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))),
    Membership.mem (γ := Filter (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))) (𝓝 0) s ↔
      ∃ (γ :
        (MonoidWithZeroHom.ValueGroup₀ (F :=
            Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) Valued.v)ˣ),
        {x : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
            ((Valuation.restrict Valued.v :
                    Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
                      (MonoidWithZeroHom.ValueGroup₀ (F :=
                        adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
                        (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))) :
                  (a : adicCompletion K⟮X⟯ (idealX K)) →
                    MonoidWithZeroHom.ValueGroup₀
                      (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
                x <
              (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
          s
ha : a = 0
S : Set (WithZero (Multiplicative ℤ))
γ : WithZero (Multiplicative ℤ)
γ_ne_zero : γ ≠ 0
γ_le : Set.Iio γ ⊆ S
h :
  ∀ (x : adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
    (hx :
      ((Valued.v : Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) :
            (a : adicCompletion K⟮X⟯ (idealX K)) → WithZero (Multiplicative ℤ))
          x =
        γ),
    ∃ (γ_1 :
      (MonoidWithZeroHom.ValueGroup₀ (F :=
          Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) Valued.v)ˣ),
      Subset.{u_2} (α := Set (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)))
        {x : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
          ((Valuation.restrict Valued.v :
                  Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
                    (MonoidWithZeroHom.ValueGroup₀ (F :=
                      adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
                      (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))) :
                (a : adicCompletion K⟮X⟯ (idealX K)) →
                  MonoidWithZeroHom.ValueGroup₀
                    (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
              x <
            (↑γ_1 : MonoidWithZeroHom.ValueGroup₀ Valued.v)}
        {t : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
          ((Valued.v : Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) :
                (a : adicCompletion K⟮X⟯ (idealX K)) → WithZero (Multiplicative ℤ))
              t <
            γ}
⊢ ∃ (γ_1 :
    (MonoidWithZeroHom.ValueGroup₀ (F :=
        Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) Valued.v)ˣ),
    Subset.{u_2} (α := Set (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)))
      {x : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
        ((Valuation.restrict Valued.v :
                Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
                  (MonoidWithZeroHom.ValueGroup₀ (F :=
                    adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
                    (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))) :
              (a : adicCompletion K⟮X⟯ (idealX K)) →
                MonoidWithZeroHom.ValueGroup₀
                  (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
            x <
          (↑γ_1 : MonoidWithZeroHom.ValueGroup₀ Valued.v)}
      {t : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
        ((Valued.v : Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) :
              (a : adicCompletion K⟮X⟯ (idealX K)) → WithZero (Multiplicative ℤ))
            t <
          γ}

Tactic:
obtain ⟨x, hx⟩ := valuedAdicCompletion_surjective K⟮X⟯ (idealX K) γ

Premises used: RatFunc, IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_surjective, Polynomial.idealX