(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
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(*                                                                          *)
(*                          Constructive Geometry                           *)
(*                             according to                                 *)
(*                            Jan von Plato                                 *)
(*                                                                          *)
(*                               Coq V5.10                                  *)
(*									    *)
(*			        Gilles Kahn 				    *)
(*				   INRIA                                    *)
(*		              Sophia-Antipolis		 	            *)
(*									    *)
(*                                                                          *)
(* Acknowledgments: This work started in Turin in May 95 with Yves Bertot.  *)
(* Thanks to the Newton Institute for providing an exceptional work         *)
(* environment in Summer 1995 so that I could complete this.                *)
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(*                               part3.v                                    *)
(* I don't follow exactly the original in the proofs of the Triangle axioms *)
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Hint Resolve Triangle_axioms_i Triangle_axioms_ii Triangle_axioms_iii.
Hint Resolve thm4_1c thm4_1d.

Theorem Symmetry_of_Apart' :
forall x y : Twolines,
Apart (pt y) (line1 x) \/ Apart (pt y) (line2 x) ->
Apart (pt x) (line1 y) \/ Apart (pt x) (line2 y).