proposition: Lemma pos_minus_compat (x y z : R) : 0 < z → x < y → x - z < y.
proof: Proof. intros E1 E2. rewrite commutativity. setoid_replace y with (-z + (y + z)) by ring. apply (strictly_order_preserving (-(z) +)). transitivity y; trivial. setoid_replace y with (y + 0) at 1 by ring. now apply (strictly_order_preserving (y +)). Qed.
imports: Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.orders MathClasses.theory.rings MathClasses.orders.semirings