proposition: Lemma tree_connectI (T1 T2 : {set G}) : connected T1 -> connected T2 -> connected (T1 :&: T2).
proof: Proof. move => C1 C2 x y /setIP [X1 X2] /setIP [Y1 Y2]. case: (path_in_connected C1 X1 Y1) => p Ip /subsetP Sp. case: (path_in_connected C2 X2 Y2) => q Iq /subsetP Sq. have {Iq Ip} ? : p = q by apply: tree_G. subst q. apply connectRI with p => z zp. by rewrite !inE Sp ?Sq. Qed.
imports: mathcomp(all_ssreflect) GraphTheory(edone preliminaries digraph sgraph set_tac)