Section NatHomomorphism.
Variable T : Type.

Lemma homo_ltn_in (D : pred nat) (f : nat -> T) (r : T -> T -> Prop) :
(forall y x z, r x y -> r y z -> r x z) ->
{in D &, forall i j k, i < k < j -> k \in D} ->
{in D, forall i, i.+1 \in D -> r (f i) (f i.+1)} ->
{in D &, {homo f : i j / i < j >-> r i j}}.