  We prove a technical lemma, the $C^{1+\alpha }$-Denjoy-Koebe distortion
lemma, estimating the distortion of a long composition of a $C^{1+\alpha }$
one-dimensional mapping $f:M\mapsto M$ with finitely many, non-recurrent, power
law critical points. The proof of this lemma combines the ideas of the
distortion lemmas of Denjoy and Koebe.
