Require Export GeoCoq.Axioms.continuity_axioms.
Require Export GeoCoq.Tarski_dev.Annexes.circles.
Require Export GeoCoq.Tarski_dev.Annexes.sums.
Section Elementary_Continuity_Props.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(**    All these properties are known to be equivalent in an arbitrary Hilbert plane (Tarski_2D) as shown by Strommer, but we don't have formalized the proof yet.  We have the following proofs: - five equivalent variants of the circle-circle continuity axiom; - three equivalent variants of the line-circle continuity axiom; - the implication from the circle-circle continuity axiom to the line-circle continuity axiom.  Since we have proved the circle-circle continuity axiom to follow from Tarski's axiom of continuity, all these properties do. *)

Lemma euclid_22_aux : forall A B C D E F A' B' E' F' C1 C2 E1,
SumS A B C D E' F' -> SumS C D E F A' B' -> Le E F E' F' -> Le A B A' B' ->
Out A B C1 -> Cong A C1 C D -> Bet B A C2 -> Cong A C2 C D ->
Out B A E1 -> Cong B E1 E F ->
Bet C1 E1 C2.