Definition LeF (x y : F) := LeP O E E' (proj1_sig x) (proj1_sig y).
Instance leF_Antisymmetric : Antisymmetric F EqF LeF.
(* For 8.5 Instance leF_Antisymmetric : Antisymmetric eqF leF. *)

Lemma coordinates_of_point_F : forall P,
{C: F*F | Cd O E SS U1 U2 P (proj1_sig (fst C)) (proj1_sig (snd C))}.