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)).