have : L ⊬! □^[n]Γ.conj' ⟶ □^[n]p := by
  have := Context.provable_iff.not.mp $ membership_iff.not.mp hp;
  push_neg at this;
  have : L ⊬! (□'^[n]Γ : List (Formula α)).conj' ⟶ □^[n]p := implyLeft_conj_eq_conj'!.not.mp $ FiniteContext.provable_iff.not.mp $ this (□'^[n]Γ) (by
    intro q hq;
    obtain ⟨r, hr₁, hr₂⟩ := by simpa using hq;
    subst hr₂;
    simpa using hΓ₁ r hr₁;
  );
  revert this;
  contrapose;
  simp [neg_neg];
  exact imp_trans! collect_multibox_conj'!;
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p : Formula α
n : ℕ
hp : □^[n]p ∉ Ω.theory
hC : { ctx := □''⁻¹^[n]Ω.theory } ⊢! p
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ □''⁻¹^[n]Ω.theory
hΓ₂ : Γ ⊢[L]! p
this : L ⊢! □^[n]Γ.conj' ⟶ □^[n]p
⊢ False
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
L : DeductionParameter α
inst✝ : L.Normal
Ω Ω₁ Ω₂ : MCT L
p : Formula α
n : ℕ
hp : □^[n]p ∉ Ω.theory
hC : { ctx := □''⁻¹^[n]Ω.theory } ⊢! p
Γ : List (Formula α)
hΓ₁ : ∀ q ∈ Γ, q ∈ □''⁻¹^[n]Ω.theory
hΓ₂ : Γ ⊢[L]! p
this✝ : L ⊢! □^[n]Γ.conj' ⟶ □^[n]p
this : L ⊬! □^[n]Γ.conj' ⟶ □^[n]p
⊢ False