proposition: Lemma fcard0P (A : pred G) : fclosed face A -> reflect (exists x, x \in A) (0 < fcard face A).
proof: Proof. move=> clFA; rewrite ltnNge leqn0. apply: (iffP pred0Pn) => [[x /andP[]] | [x]] /= Ax; first by exists x. exists (froot face x); rewrite (roots_root cfaceC). by rewrite -(closed_connect clFA (connect_root _ x)). Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq choice) mathcomp(fintype path fingraph) fourcolor(hypermap geometry patch sew snip color chromogram) fourcolor(coloring kempe)