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

Lemma characterization_of_parallelism_F_bis : forall A B C D,
Par A B C D <->
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
let (Cc, _) := coordinates_of_point_F C in
let (Cx, Cy) := Cc in
let (Dc, _) := coordinates_of_point_F D in
let (Dx, Dy) := Dc in
(Ax - Bx) * (Cy - Dy) - (Ay - By) * (Cx - Dx) =F= 0 /\
(~ (Ax =F= Bx) \/ ~ (Ay =F= By)) /\ (~ (Cx =F= Dx) \/ ~ (Cy =F= Dy)).