Hint Resolve thm4_1a thm4_1b.

(* Specifications: *)
Theorem thm4_1c :
forall (x : Twolines) (a : Point),
DiPt a (pt x) -> Apart a (line1 x) \/ Apart a (line2 x).
Theorem thm4_1d :
forall (x : Twolines) (a : Point),
Apart a (line1 x) \/ Apart a (line2 x) -> DiPt a (pt x).
Theorem Symmetry_of_Apart :
forall x y : Segment,
Apart (origin x) (ln y) \/ Apart (extremity x) (ln y) ->
Apart (origin y) (ln x) \/ Apart (extremity y) (ln x).