proposition: Lemma sqrt_up_lt_cancel : forall a b, √°a < √°b -> a < b.
proof: Proof. intros a b H. destruct (le_gt_cases b 0) as [Hb|Hb]. - rewrite (sqrt_up_eqn0 _ Hb) in H; generalize (sqrt_up_nonneg a); order. - destruct (le_gt_cases a 0) as [Ha|Ha]; [order|]. rewrite <- (lt_succ_pred 0 a Ha), <- (lt_succ_pred 0 b Hb), <- succ_lt_mono. apply sqrt_lt_cancel, succ_lt_mono. now rewrite <- 2 sqrt_up_eqn. Qed.
imports: NZAxioms NZMulOrder