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.