Section SPP_tarski.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

(* Specifications: *)
Lemma impossible_case_5 : forall P Q R S T U I,
BetS P T Q ->
BetS R T S ->
BetS Q U R ->
~ Col P Q S ->
~ Col P R U ->
Par P R Q S ->
Par P S Q R ->
Bet S Q I ->
Bet U I P ->
False.
Lemma impossible_case_6 : forall P Q R S T U I,
BetS P T Q ->
BetS R T S ->
BetS Q U R ->
~ Col P Q S ->
~ Col P R U ->
Par P R Q S ->
Par P S Q R ->
Bet S Q I ->
Bet I P U ->
False.
Lemma impossible_case_7 : forall P Q R S T U I,
BetS P T Q ->
BetS R T S ->
BetS Q U R ->
~ Col P Q S ->
~ Col P R U ->
Par P R Q S ->
Par P S Q R ->
Col P U I ->
Bet Q I S ->
False.
Lemma impossible_case_8 : forall P Q R S T U I,
BetS P T Q ->
BetS R T S ->
BetS Q U R ->
~ Col P Q S ->
~ Col P R U ->
Par P R Q S ->
Par P S Q R ->
Col P U I ->
Bet I S Q ->
False.
Lemma strong_parallel_postulate_implies_tarski_s_euclid_aux :
strong_parallel_postulate ->
(forall A B C D T,
A <> B ->
A <> C ->
A <> D ->
A <> T ->
B <> C ->
B <> D ->
B <> T ->
C <> D ->
C <> T ->
D <> T ->
~ Col A B C ->
Bet A D T ->
Bet B D C ->
exists B', exists B'', exists MB, exists X, Bet A B X /\ Par_strict B C T X /\
BetS B MB T /\ BetS B' MB B'' /\
Cong B MB T MB /\ Cong B' MB B'' MB /\
Col B B' D /\ Bet B'' T X /\
B <> B' /\ B'' <> T).
Lemma strong_parallel_postulate_implies_tarski_s_euclid :
strong_parallel_postulate ->
tarski_s_parallel_postulate.