proposition: Lemma face'P (A F: set E): face A F <-> face' A F.
proof: Proof. split => [[FA Fconv Fface]|[FA Fconv Fface]]. split=> // x u v xF uA vA xuv xv; have [xu|xu] := eqVneq x u. by rewrite xu in xF. by move: (Fface x u v xF uA vA xuv xu xv) => []. split => // x u v xF uA vA xuv xu xv; split; [ apply (Fface x u v) | apply (Fface x v u) ] =>//. by rewrite segmentC. Qed.
imports: mathcomp(all_ssreflect all_algebra vector reals ereal classical_sets boolp Rstruct) infotheo(convex Reals_ext) preliminaries Reals