unfold on_line_d in |- *.
intuition.
Qed.

(* Specifications: *)
Theorem non_zero_denom_on_line_d_1_length_ratio_seg :
forall (Y P Q : Point) (l : F), on_line_d Y P Q l -> P**Q <> 0.
Theorem non_zero_denom_on_line_1_length_ratio :
forall (Y P Q : Point), on_line Y P Q -> P<>Q.