case (Rle_dec (arcsinh y) (arcsinh x));[ | apply Rnot_le_lt ].
intros abs; case (Rlt_not_le _ _ xy).
rewrite <- (sinh_arcsinh y), <- (sinh_arcsinh x).
destruct abs as [lt | q];[| rewrite q; lra].
apply Rlt_le, sinh_lt; assumption.
Qed.

Lemma arcsinh_le : forall x y, x <= y -> arcsinh x <= arcsinh y.
intros x y [xy | xqy].