proposition: Lemma separation_capN (G : sgraph) (V1 V2 : {set G}) u v : separation V1 V2 -> u \in V1 -> v \in V2 -> u -- v -> (u \in V1 :&: V2) || (v \in V1 :&: V2).
proof: Proof. move => sepV uV1 vV2 uv; rewrite !inE uV1 vV2 andbT /=. by apply: contraTT uv; rewrite negb_or => /andP[uNV2 vNV1]; rewrite sepV. Qed.
imports: mathcomp(all_ssreflect) GraphTheory(edone preliminaries digraph sgraph) GraphTheory(connectivity minor treewidth arc set_tac)