. intro H p;
  contrapose;
  intro h;
  apply iff_mem_neg.mp;
  apply H;
  apply mem_dn_iff.mpr;
  apply (neg_iff $ mem_multidia_dual).mp;
  exact iff_mem_neg.mpr h;
case mp
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
⊢ (∀ {p : Formula α}, □^[n]p ∈ Ω₁.theory → p ∈ Ω₂.theory) → ∀ {p : Formula α}, p ∈ Ω₂.theory → ◇^[n]p ∈ Ω₁.theory

case mpr
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
⊢ (∀ {p : Formula α}, p ∈ Ω₂.theory → ◇^[n]p ∈ Ω₁.theory) → ∀ {p : Formula α}, □^[n]p ∈ Ω₁.theory → p ∈ Ω₂.theory
case mpr
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
⊢ (∀ {p : Formula α}, p ∈ Ω₂.theory → ◇^[n]p ∈ Ω₁.theory) → ∀ {p : Formula α}, □^[n]p ∈ Ω₁.theory → p ∈ Ω₂.theory