proposition: Lemma ext_to_prod_prop (a b a' b':Set) (u:a->a')(v:b->b') (x:a)(x':b): J(Ro (u x)) (Ro (v x')) = Ro(ext_to_prodC u v (pairC x x')).
proof: Proof. rewrite /ext_to_prodC prJ_prop. have <-: (x = (pr1C (pairC x x'))). by apply: R_inj; rewrite pr1C_prop prJ_prop; aw. have <- //: (x' = (pr2C (pairC x x'))). by apply: R_inj; rewrite pr2C_prop prJ_prop; aw. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)