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 triangle_inequality1 : forall A B C D E, SumS A B B C D E -> Le A C D E.