∀ {I : Type u} (J J_1 : I → Prop),
  J = J_1 →
    ∀ {inst : (i : I) → Decidable (J i)} [inst_1 : (i : I) → Decidable (J_1 i)] (a a_1 : I → Bool),
      a = a_1 →
        ∀ (a_2 a_3 : I), a_2 = a_3 → Profinite.NobelingProof.Proj J a a_2 = Profinite.NobelingProof.Proj J_1 a_1 a_3