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).