Definition sqrt3 := PythF 1 (PythF 1 1).

Lemma characterization_of_neq_F : forall A B, A <> B <-> let (Ac, _) := coordinates_of_point_F A in let (Ax, Ay) := Ac in let (Bc, _) := coordinates_of_point_F B in let (Bx, By) := Bc in ~ ((Ax - Bx) * (Ax - Bx) + (Ay - By) * (Ay - By) =F= OF).