proposition: Lemma BRprodC x y: x *r y = y *r x.
proof: Proof. rewrite /BRprod. case: (equal_or_not x \0r) => ha; first by rewrite !(Y_true ha) !Y_same. rewrite !(Y_false ha). case: (equal_or_not y \0r) => hb;first by rewrite !(Y_true hb) !Y_same. rewrite !(Y_false hb). rewrite (BRprod_auxC x) (BRprod_auxC _ (BRopp y)) (BRprod_auxC y (BRopp x)). case: (inc_or_not x BRps) => hc; first by rewrite !(Y_true hc). by rewrite !(Y_false hc) (BRprod_auxC (BRopp y) (BRopp x)). Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)