proposition: Lemma Dcuts_minus_open : Dcuts_def_open Dcuts_minus_val.
proof: Proof. intros r. apply hinhuniv ; intros x. generalize (X_open _ (pr1 (pr2 x))). apply hinhfun ; intros x'. exists (r + (pr1 x' - pr1 x)) ; split. - apply hinhpr ; exists (pr1 x') ; split. + exact (pr1 (pr2 x')). + intros y Yy. pattern (pr1 x') at 2 ; rewrite <- (minusNonnegativeRationals_plus_r _ _ (lt_leNonnegativeRationals _ _ (pr2 (pr2 x')))). rewrite (iscomm_plusNonnegativeRationals r), isassoc_plusNonnegativeRationals. apply plusNonnegativeRationals_ltcompat_l. now apply (pr2 (pr2 x)). - apply plusNonnegativeRationals_lt_r. apply ispositive_minusNonnegativeRationals. exact (pr2 (pr2 x')). Qed.
imports: UniMath.MoreFoundations.Orders UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Notations UniMath.RealNumbers.Sets UniMath.Algebra.ConstructiveStructures UniMath.OrderTheory.Lattice.Lattice UniMath.RealNumbers.Prelim UniMath.RealNumbers.NonnegativeRationals