Definition Tcircle : Type := Tpoint*Tpoint*Tpoint %type.Definition OnCirc P (C:Tcircle) := 
  match C with 
  (X,A,B) => tarski_axioms.Cong X P A B
  end.Definition CI (J:Tcircle) A C D := J=(A,C,D) /\ C<>D.Definition InCirc P (J:Tcircle) :=
   match J with
  (C,A,B) => 
   exists X Y, Definitions.BetS X C Y /\ tarski_axioms.Cong C Y A B /\
               tarski_axioms.Cong C X A B /\ Definitions.BetS X P Y
  end.Definition OutCirc P (J:Tcircle) :=
   match J with
  (C,A,B) => 
      exists X, Definitions.BetS C X P /\ tarski_axioms.Cong C X A B
 end.

Lemma circle_circle' : circle_circle -> forall C D F G J K P Q R S, CI J C R S -> InCirc P J -> OutCirc Q J -> CI K D F G -> OnCirc P K -> OnCirc Q K -> exists X, OnCirc X J /\ OnCirc X K.