proposition: Lemma topologysubtype_neighborhood : ∏ (x : ∑ x : T, dom x) (P : (∑ x0 : T, dom x0) → hProp), topologysubtype x P → ∃ Q : (∑ x0 : T, dom x0) → hProp, topologysubtype x Q × (∏ y : ∑ x0 : T, dom x0, Q y → topologysubtype y P).
proof: Proof. intros x A. apply hinhfun. intros B. exists (λ y : ∑ x : T, dom x, pr1 B (pr1 y)). split. - apply hinhpr. exists (pr1 B). split. + exact (pr1 (pr2 B)). + intros. assumption. - intros y By. apply hinhpr. exists (pr1 B). repeat split. + exact By. + exact (pr2 (pr1 (pr2 B))). + exact (pr2 (pr2 B)). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.Topology.Filters UniMath.Algebra.Groups UniMath.Algebra.DivisionRig UniMath.Algebra.ConstructiveStructures