  We prove a ``general shrinking lemma'' that resembles the
Schwarz--Pick--Ahlfors Lemma and its many generalizations, but differs in
applying to maps of a finite disk into a disk, rather than requiring the domain
of the map to be complete. The conclusion is that distances to the origin are
all shrunk, and by a limiting procedure we can recover the original Ahlfors
Lemma, that {\em all} distances are shrunk. The method of proof is also
different in that it relates the shrinking of the Schwarz--Pick--Ahlfors-type
lemmas to the comparison theorems of Riemannian geometry.
