proposition: Lemma ub_Rdown E : eqRset (Real.ub (Real.down E)) (Real.ub E).
proof: Proof. move=> z; split=> leEz x => [Ex | [y Ey lexy]]; first by apply: leEz; exists x. exact/(leR_trans lexy)/leEz. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat div order) mathcomp(ssralg ssrnum ssrint rat intdiv) Coq(Morphisms Setoid) fourcolor(real realsyntax)