(* Lemma test_on_line_d_13 : forall Y P Q A C D lambda, on_line_d Y P Q lambda ->  Y<>A -> parallel C D Y A -> False -> C**D/Y**A = C**D. *)
(** Test inter_ll in a ratio *)
(** Test cases where Y occurs both in denominator and numerator *)

(* Specifications: *)
Lemma test_inter_ll_1 :  forall Y P Q U V,
inter_ll Y P Q U V ->  V<>Y -> parallel U Y V Y ->
U**Y/V**Y = S U P Q / S V P Q.
Lemma test_inter_ll_2 :  forall Y P Q U V,
inter_ll Y P Q U V ->  U<>Y -> parallel V Y U Y ->
V**Y/U**Y = S V P Q / S U P Q.
Lemma test_inter_ll_3 :  forall Y P Q U V,
inter_ll Y P Q U V ->  Q<>Y -> parallel P Y Q Y ->
P**Y/Q**Y =S P U V / S Q U V.
Lemma test_inter_ll_4 :  forall Y P Q U V,
inter_ll Y P Q U V ->  P<>Y -> parallel Q Y P Y ->
Q**Y/P**Y =S Q U V / S P U V.
Lemma test_inter_ll : forall T P Q A B C D,
inter_ll T A B C D->
parallel T P T Q ->
T<>Q ->
P<>T ->
T ** P / T ** Q = T ** P / T ** Q + 1 - 1.
Lemma test_inter_ll_1b :  forall Y P Q U V,
inter_ll Y P Q U V ->  Y<>V -> parallel Y U Y V ->
Y**U/Y**V = S U P Q / S V P Q.
Lemma test_inter_ll_2b :  forall Y P Q U V,
inter_ll Y P Q U V ->  Y<>U -> parallel V Y Y U ->
V**Y/Y**U = - (S V P Q / S U P Q).
Lemma test_inter_ll_3b :  forall Y P Q U V,
inter_ll Y P Q U V ->  Q<>Y -> parallel Y P Q Y ->
Y**P/Q**Y = - (S P U V / S Q U V).
Lemma test_inter_ll_4b :  forall Y P Q U V,
inter_ll Y P Q U V ->  P<>Y -> parallel Q Y P Y ->
Q**Y/P**Y =S Q U V / S P U V.
Lemma test_inter_ll_5 :  forall Y P Q U V A C,
inter_ll Y P Q U V ->  C<>Y -> parallel A Y C Y ->
A**Y/C**Y = A**Y/C**Y.
Lemma test_inter_ll_gen_1 :  forall Y P Q U V A C D,
inter_ll Y P Q U V ->  C<>D -> parallel A Y C D ->
False ->
A**Y/C**D = 1.
Lemma test_inter_ll_gen_2 :  forall Y P Q U V A C D,
inter_ll Y P Q U V ->  C<>D -> parallel A Y C D -> parallel Y A C D -> A<>Y -> Y<>A ->
A**Y/C**D + Y**A/C**D=0.
Lemma test_inter_ll_gen_3 :  forall Y P Q U V A C D,
inter_ll Y P Q U V ->  A<>Y -> C<>D -> parallel C D A Y ->
C**D/A**Y = C**D/A**Y.
Lemma test_on_parallel_d_1 : forall A C D Y R P Q lambda,
on_parallel_d Y R P Q lambda ->
C<>D ->
parallel A Y C D ->
A**Y/C**D = A**Y/C**D.
Lemma test_on_foot_1 : forall A C D Y R P Q,
on_foot Y R P Q ->
C<>D ->
parallel A Y C D ->
A**Y/C**D = 1-1+ A**Y/C**D.
Lemma test_on_foot_2 : forall A C D Y R P Q,
on_foot Y R P Q ->
False ->
C <> D ->
parallel Y A C D ->
Y**A/C**D = - - (Y**A/C**D).