(* Definition of log R+* -> R *)
Definition Rln (y:posreal) : R :=
  let (a,_) := ln_exists (pos y) (cond_pos y) in a.(* Extension on R *)
Definition ln (x:R) : R :=
  match Rlt_dec 0 x with
    | left a => Rln (mkposreal x a)
    | right a => 0
  end.Definition Rlog x y := (ln y)/(ln x).

Theorem ln_increasing : forall x y:R, 0 < x -> x < y -> ln x < ln y.