proposition: Lemma Cauchy_IR_lt_as_CR_lt_2 : forall (x y:Cauchy_IR), ((Cauchy_IRasCR x) < (Cauchy_IRasCR y))%CR -> x[<]y.
proof: Proof. intros x y H. stepl (CRasCauchy_IR (Cauchy_IRasCR (x))); [| apply Cauchy_IRasCauchy_IR]. stepr (CRasCauchy_IR (Cauchy_IRasCR (y))); [| apply Cauchy_IRasCauchy_IR]. apply CR_lt_as_Cauchy_IR_lt_1. assumption. 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