Definition closed_sieve_type_intersection (I : UU) (PI : I → closed_sieve) : closed_sieve. Proof. use make_closed_sieve. - exact (sieve_type_intersection I PI). - abstract (intros x C p H i ; refine (is_closed_closed_sieve (PI i) x C p _) ; intros y q ; exact (H y q i)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.OrderTheory.Lattice.Bounded. Require Import UniMath.OrderTheory.Lattice.Distributive. Require Import UniMath.OrderTheory.Lattice.Heyting. Require Import UniMath.OrderTheory.Lattice.CompleteHeyting. Require Import UniMath.OrderTheory.Lattice.Examples.FromPartialOrder.