proposition: Lemma idealr_closed_nontrivial R S : @idealr_closed R S -> proper_ideal S.
proof: Proof. by case=> S0 S1 hS; split => // a x xS; rewrite -[_ * _]addr0 hS. Qed.
imports: HB(structures) mathcomp(ssreflect eqtype choice ssreflect ssrbool ssrnat) mathcomp(ssrfun seq ssralg generic_quotient)