proposition: Lemma ereal_inf_lt S x : ereal_inf S < x -> exists2 y, S y & y < x.
proof: Proof. rewrite lteNl => /ereal_sup_gt[_ [y Sy <-]]. by rewrite lteNl oppeK => xlty; exists y. Qed.
imports: HB(structures) mathcomp(all_ssreflect all_algebra finmap) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(fsbigop cardinality set_interval) reals signed topology constructive_ereal