Definition not_col : arity Tpoint 3 := fun A B C : Tpoint => ~ Col A B C.

(* Specifications: *)
Lemma not_col_perm_1 : forall A X, app_1_n not_col A X -> app_n_1 not_col X A.
Lemma not_col_perm_2 : forall A B (X : cartesianPower Tpoint 1), app_2_n not_col A B X -> app_2_n not_col B A X.
Lemma concyclic_aux : forall A B C D, Concyclic A B C D -> exists O P, OnCircle A O P /\ OnCircle B O P /\ OnCircle C O P /\ OnCircle D O P /\ Coplanar A B C O.
Lemma not_col_perm_1 : forall A X, app_1_n not_col A X -> app_n_1 not_col X A.
Lemma not_col_perm_2 : forall A B (X : cartesianPower Tpoint 1), app_2_n not_col A B X -> app_2_n not_col B A X.