intro a
β : Type
collection : β → (W : Type) × ℕ × KripkeModel W × W
newVal : ℕ → Prop
mF : Formula
mR : β
moW : (collection mR).fst
⊢ ∀ (a : ℕ),
    (fun π =>
        (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
            relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
              relate (collection R).snd.2.1 π oldWorld₁ oldWorld₂) ∧
          (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
              R₁ ≠ R₂ →
                ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
            ∀ (R : β) (oldWorld : (collection R).fst),
              ¬relate (combinedModel collection newVal).1 π (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ()))
      (·a)
β : Type
collection : β → (W : Type) × ℕ × KripkeModel W × W
newVal : ℕ → Prop
mF : Formula
mR : β
moW : (collection mR).fst
a : ℕ
⊢ (∀ (R : β) (oldWorld₁ oldWorld₂ : (collection R).fst),
      relate (combinedModel collection newVal).1 (·a) (Sum.inr ⟨R, oldWorld₁⟩) (Sum.inr ⟨R, oldWorld₂⟩) ↔
        relate (collection R).snd.2.1 (·a) oldWorld₁ oldWorld₂) ∧
    (∀ (R₁ R₂ : β) (oldWorld₁ : (collection R₁).fst) (oldWorld₂ : (collection R₂).fst),
        R₁ ≠ R₂ →
          ¬relate (combinedModel collection newVal).1 (·a) (Sum.inr ⟨R₁, oldWorld₁⟩) (Sum.inr ⟨R₂, oldWorld₂⟩)) ∧
      ∀ (R : β) (oldWorld : (collection R).fst),
        ¬relate (combinedModel collection newVal).1 (·a) (Sum.inr ⟨R, oldWorld⟩) (Sum.inl ())