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

Lemma characterization_of_parallelism_F : 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 - Bx) * (Ax - Bx) + (Ay - By) * (Ay - By) =F= OF) /\
~ ((Cx - Dx) * (Cx - Dx) + (Cy - Dy) * (Cy - Dy) =F= OF).