induction Γ using List.induction_with_singleton with
| hnil => simpa using dhyp! verum!;
| hsingle p => simp;
| hcons p Γ h ih =>
  simp_all;
  exact imp_trans! distribute_multidia_and! $ by
    apply deduct'!;
    apply iff_provable_list_conj.mpr;
    intro q hq;
    simp at hq;
    cases hq with
    | inl => subst_vars; exact conj₁'! id!;
    | inr hq =>
      obtain ⟨r, hr₁, hr₂⟩ := hq;
      exact (iff_provable_list_conj.mp $ (of'! ih) ⨀ (conj₂'! $ id!)) q (by aesop);
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ (◇'^[n]Γ).conj'
no goals