Section HilbertContext_2.
Context `{Hi:Hilbert_neutral_dimensionless}.
Definition Cong A B C D := (CongH A B C D /\ A <> B /\ C <> D) \/ (A = B /\ C = D).
(****************** between_identity ************************)
(************************************************************)
(***************************** cong_identity ***********************)
(************************ cong_inner_transitivity ****************************)
intros.
unfold disjoint.
intro.
destruct H0 as [P []].
assert(HP:= betH_trans0 A P B C H0 H).
spliter.
apply between_only_one' in H1.
spliter.
contradiction.
Qed.
Definition Para := fun l m =>
(~ exists X, IncidL X l /\ IncidL X m) /\ exists p, IncidLP l p /\ IncidLP m p.
Definition ParaP := fun A B C D =>
forall l m, IncidL A l -> IncidL B l -> IncidL C m -> IncidL D m -> Para l m.
Definition is_line A B l := A <> B /\ IncidL A l /\ IncidL B l.
intros.
unfold outH in H.
induction H.
apply between_col in H.
auto.
induction H.
apply between_col in H.
Col.
spliter.
subst C.
Col.
Qed.
Global Instance IncidLP_morphisml : Proper(EqL ==> eq ==> iff) IncidLP.

Lemma same_side_prime__plane : forall A B C D, same_side' A B C D ->
exists p, IncidP A p /\ IncidP B p /\ IncidP C p /\ IncidP D p.