%--------------------------------------------------------------------------
% File     : GEO135+1 : TPTP v8.2.0. Released v2.4.0.
% Domain   : Geometry (Oriented curves)
% Problem  : Ordering can be determined by betweenness and incidence
% Version  : [EHK99] axioms.
% English  : The ordering of any pair of points R and S on an oriented line
%            o can be determined on the basis of a given pair of points P and
%            Q using betweenness and incidence only.

% Refs     : [KE99]  Kulik & Eschenbach (1999), A Geometry of Oriented Curv
%          : [EHK99] Eschenbach et al. (1999), Representing Simple Trajecto
% Source   : [KE99]
% Names    : Corollary 4.18 [KE99]
%          : T10 [EHK99]

% Status   : Unknown
% Rating   : 1.00 v2.4.0
% Syntax   : Number of formulae    :   28 (   2 unt;   0 def)
%            Number of atoms       :  121 (  18 equ)
%            Maximal formula atoms :   12 (   4 avg)
%            Number of connectives :  104 (  11   ~;  12   |;  42   &)
%                                         (  21 <=>;  18  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (   7 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :   15 (  14 usr;   0 prp; 1-4 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 1-2 aty)
%            Number of variables   :   99 (  85   !;  14   ?)
% SPC      : FOF_UNK_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%----Include simple curve axioms
include('Axioms/GEO004+0.ax').
%----Include axioms of betweenness for simple curves
include('Axioms/GEO004+1.ax').
%----Include oriented curve axioms
include('Axioms/GEO004+2.ax').
%--------------------------------------------------------------------------
fof(corollary_4_18,conjecture,
    ! [O,P,Q] :
      ( ordered_by(O,P,Q)
     => ! [R,S] :
          ( ordered_by(O,R,S)
        <=> ( ( incident_o(S,O)
              & ( between(O,R,P,Q)
                | P = R )
              & R != S
              & ~ between(O,S,R,Q) )
            | ( between(O,P,R,S)
              & ~ between(O,Q,P,R) ) ) ) ) ).

%--------------------------------------------------------------------------