(** We omit lemma 14.26 which states that we have a division ring. *)

(* Specifications: *)
(** Lemma 14.27. *)
(** Sum and product are preserved by parallel projection on a different x-axis.*)
(** This lemma is used to prove that there is an isomorphism between number lines.*)
Lemma prod_1_l_eq : forall O E E' A B, Prod O E E' A B B -> A = E \/ B = O.
Lemma prod_1_r_eq : forall O E E' A B, Prod O E E' A B A -> B = E \/ A = O.
Lemma change_grid_prod_l_O : forall O E E' B C O' A' B' C',
Par_strict O E O' E' -> Ar1 O E O B C -> Ar1 O' E' A' B' C' ->
Pj O O' E E' -> Pj O O' O A' -> Pj O O' B B' -> Pj O O' C C' ->
Prod O E E' O B C ->
Prod O' E' E A' B' C'.
Lemma change_grid_prod1 : forall O E E' B C O' A' B' C',
Par_strict O E O' E' -> Ar1 O E E B C -> Ar1 O' E' A' B' C' ->
Pj O O' E E' -> Pj O O' E A' -> Pj O O' B B' -> Pj O O' C C' ->
Prod O E E' E B C ->
Prod O' E' E A' B' C'.
Lemma change_grid_prod : 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' ->
Prod O E E' A B C ->
Prod O' E' E A' B' C'.