Section Grid.
Variable O E E' : Tpoint.
Variable grid_ok : ~ Col O E E'.
(** We are not faithful to Tarski's def because for uniqueness we do not need the assumption that A and B are on line OE as it is implied by the definition of sum. *)
End Grid.
(** Lemma 14.13 *)
(** Parallel projection on the second axis preserves sums. *)

Lemma change_grid_sum_0 :
forall O E E' A B C O' A' B' C',
Par_strict O E O' E' ->
Ar1 O E A B C ->
Ar1 O' E' A' B' C' ->
Pj O O' E E' ->
Pj O O' A A' ->
Pj O O' B B' ->
Pj O O' C C' ->
Sum O E E' A B C ->
A = O ->
Sum O' E' E A' B' C'.