proposition: Lemma to_binop_inv_comm_1 {x y : PA} (f g : PA⟦x, y⟧) : to_binop _ _ (to_inv f) g = to_inv (to_binop _ _ f (to_inv g)).
proof: Proof. apply (to_rcan (to_binop x y f (to_inv g))). rewrite linvax. rewrite (to_commax' f). rewrite to_assoc. rewrite <- (to_assoc g). rewrite rinvax. rewrite to_lunax'. rewrite linvax. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Algebra.Monoids UniMath.Algebra.Groups UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Core.Functors