proposition: Lemma CR_eq_as_Cauchy_IR_eq : forall (x y:CR), (CRasCauchy_IR x [=] CRasCauchy_IR y) <-> (x==y)%CR.
proof: Proof. intros x y. set (x':=CRasCauchy_IR x). set (y':=CRasCauchy_IR y). rewrite <- (CRasCR x). rewrite <- (CRasCR y). symmetry. apply Cauchy_IR_eq_as_CR_eq. Qed.
imports: CoRN.algebra.RSetoid CoRN.metric2.Metric CoRN.metric2.UniformContinuity CoRN.model.totalorder.QposMinMax CoRN.model.reals.Cauchy_IR CoRN.tactics.CornTac Coq(Lia) CoRN.model.metric2.Qmetric CoRN.model.totalorder.QMinMax Coq.QArith.Qabs CoRN.reals.fast.CRFieldOps