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. *)

Theorem equivalent_variants_of_circle_circle :
all_equiv
(circle_circle::
circle_circle_two::
circle_circle_bis::
circle_circle_axiom::
euclid_s_prop_1_22::
nil).