∀ {X : Type u_1} [inst : Order.Frame X] (carrier₁ carrier₂ : Set X) (sInf_mem'₁ : ∀ s ⊆ carrier₁, sInf s ∈ carrier₁)
  (sInf_mem'₂ : ∀ s ⊆ carrier₂, sInf s ∈ carrier₂) (himp_mem'₁ : ∀ (a b : X), b ∈ carrier₁ → a ⇨ b ∈ carrier₁)
  (himp_mem'₂ : ∀ (a b : X), b ∈ carrier₂ → a ⇨ b ∈ carrier₂),
  ({ carrier := carrier₁, sInf_mem' := sInf_mem'₁, himp_mem' := himp_mem'₁ } ≤
      { carrier := carrier₂, sInf_mem' := sInf_mem'₂, himp_mem' := himp_mem'₂ }) =
    (carrier₁ ⊆ carrier₂)