Definition strictly_increasing f x y := @ord A x y -> @ord B (f x) (f y).Structure mono : Type := Mono 
           {fun_of: A -> B; _: forall x y, strictly_increasing fun_of x y}.Structure mono : Type := Mono 
           {fun_of: A -> B; _: forall x y, strictly_increasing fun_of x y}.Structure mono : Type := Mono 
           {fun_of: A -> B; _: forall x y, strictly_increasing fun_of x y}.

Lemma trans_ltn_nat : transitive ltn.