Section Grid.
Variable O E E' : Tpoint.
Variable grid_ok : ~ Col O E E'.
End Grid.
(** We omit lemma 14.26 which states that we have a division ring. *)
(** Lemma 14.28 is something like exists f, prod O E X A B C -> prod O' E' f(X) f(A') f(B') f(C') ? *)
(** Lemma 14.29 *)
(** From Pappus-Pascal we can derive that the product is symmetric, hence we have a field. *)
(* already done : prod_comm *)

Lemma prod_x_axis_unit_change : forall O E E' A B C D U,
Ar2_4 O E E' A B C D -> Col O E U -> U <> O ->
( exists X, Prod O E E' A B X /\ Prod O E E' C D X) ->
( exists Y, Prod O U E' A B Y /\ Prod O U E' C D Y).