(*========================================================== ============================================================ TOPOLOGICAL FMAPS, QHMAPS, HMAPS -  PROOFS OF THE GENUS THEOREM AND THE EULER RELATION  PART 3:  EQUIVALENCES, CHARACTERISTICS, GENUS, EULER_POINCARE, CONSTRUCTIVE CONDITION OF PLANARITY  (J.-F. Dufourd - June 2006 - FOR PUBLICATION completed in October 2007 / January 2008)  (* COMPLETE, APART FROM LEMMA expf_clos_symm, WHICH IS ADMITTED TO PROVE THE NECESSARY CONDITION OF PLANARITY *)
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(*========================================================== DART EQUIVALENCES IN QHMAPS ==========================================================*)
Require Export Euler2.
(* Face equivalence: *)
Definition eqf(m:fmap)(x y:dart):=
expf m x y \/ expf m y x.
(* Reflexivity of eqf: *)
(* Symmetry of eqf: *)

Lemma symm_eqf : forall (m:fmap)(x y:dart),
eqf m x y -> eqf m y x.