Lemma continuous2d_comp {X : UU} {U V W : TopologicalSpace} (f : X → U) (g : X → V) (h : U → V → W) (F : Filter X) (lf : U) (lg : V) : is_lim f F lf → is_lim g F lg → continuous2d_at h lf lg → is_lim (λ x, h (f x) (g x)) F (h lf lg). Proof. intros Hf Hg. apply (filterlim_comp (λ x, (f x ,, g x))). intros P. apply hinhuniv. intros Hp. generalize (filter_and F _ _ (Hf _ (pr1 (pr2 (pr2 Hp)))) (Hg _ (pr1 (pr2 (pr2 (pr2 Hp)))))). apply (filter_imply F). intros x Hx. apply (pr2 (pr2 (pr2 (pr2 Hp)))). - exact (pr1 Hx). - exact (pr2 Hx). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).