(*   Roland Coghetto, 17 March 2018 GNU Lesser General Public License v3.0 See LICENSE GeoCoq 2.3.0  MODIFY, tarski_to_makarios,v Version GeoCoq 2.3.0 MOTIVATION: in [1] Lemma 5: A' implies (RE) CHANGES: {(Point equality decidability),(TE),(IE),(SC),(FS')} |- (RE) i in [1] Lemma 6: if (RE) and (TE) holds, then (FS') is equivalent to (FS). CHANGES: a) {(RE), (TE),(FS)} |- (FS') i b) {(Point equality decidability), (TE), (IE) (SC), (FS')} |- (FS) i SOURCES: [1] Timothy James McKenzie Makarios. A further simplification of Tarski’s axioms of geometry. Note di Matematica, 33(2):123–132, 2013. [2] tarski_to_makarios,v GeoCoq 2.3.0 [3] Mizar Mathematical Library (MML 5.46.1331): gtarski3.miz, Roland Coghetto and Adam Grabowski. [4] Tarski Geometry Axioms, part III. R. Coghetto & A. Grabowski. Formalized Mathematics Vol. 25 N° 4, 2017.(In preparation).  in: Section Tarski83_to_Makarios_variant ------------------------------------ REMOVE: cong_reflexivity, cong_symmetry, cong_left_commutativity. MODIFY: five_segment'.  in: Section Makarios_variant_to_Tarski83 ------------------------------------ REMOVE: between_symmetry, Mcong_left_commutativity. ADD: LmCoghGrab, cong_pre_pseudo_reflexivity. MODIFY: cong_pseudo_reflexivity (Minner_pasch & Mbetween_identity are not used to prove cong_pseudo_reflexivity) *)
(** In this file we formalize the result given in T. J. M. Makarios: A further simplification of Tarski's axioms of geometry, June 2013. *)
Section Tarski83_to_Makarios_variant.
Context `{TnEQD:Tarski_neutral_dimensionless}.

(* Specifications: *)
Lemma five_segment' : forall A A' B B' C C' D D',
Cong A B A' B' ->
Cong B C B' C' ->
Cong A D A' D' ->
Cong B D B' D' ->
Bet A B C -> Bet A' B' C' -> A <> B ->
Cong D C C' D'.
Lemma lower_dim_ex :
exists A B C, ~ (Bet A B C \/ Bet B C A \/ Bet C A B).