Ltac decompose_coordinates' :=
  repeat
  match goal with
    H: _ |- context[ (coordinates_of_point_F ?X) ] =>
      let fx := fresh X "x" in
      let fy := fresh X "y" in
      let HX := fresh "H" X in
      destruct (coordinates_of_point_F X) as [[fx fy] HX]
  end.

Lemma axiom_A2a : forall A B r, A<>B -> exists P, AM_Col A B P /\ ratio A P A B =F= r.