Definition lersif (x y : R) b := if b then x < y else x <= y.Notation "`] -oo , '+oo' [" := (Interval (BInfty _) (BInfty _))
  (at level 0, format "`] -oo ,  '+oo' [") : ring_scope.Definition subr_lersif0 := (subr_lersifr0, subr_lersif0r).Definition lersif_oppE := (lersif_0oppr, lersif_oppr0, lersif_opp2).Definition lersif_add2 := (lersif_add2l, lersif_add2r).Definition lersif_sub_addr := (lersif_subl_addr, lersif_subr_addr).Definition lersif_sub_addl := (lersif_subl_addl, lersif_subr_addl).Notation BOpen := (BOpen_if true).Notation BClose := (BOpen_if false).

Lemma eq_itv_boundP : Equality.axiom eq_itv_bound.