proposition: Lemma strictly_preserving_preserves_pos : (∀ x, 0 < x → 0 < f x) → StrictlyOrderPreserving f.
proof: Proof. intros E. repeat (split; try apply _). intros x y F. destruct (decompose_lt F) as [z [Ez1 Ez2]]. apply compose_lt with (f z). now apply E. now rewrite Ez2, preserves_plus. Qed.
imports: Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.orders MathClasses.theory.rings MathClasses.orders.orders MathClasses.orders.maps