proposition: Lemma setI_covering2 E: {compat intersection_covering2 : x & / covering_s x E}.
proof: Proof. move=> u v c1 c2 z zu. move /setU_P: (c1 _ zu) =>[y zy yu]. move /setU_P: (c2 _ zu) =>[y' xzy' yv]. apply/setU_P;exists (y \cap y'); first by fprops. by apply/ setI_covering2_P; exists y, y'. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)