%------------------------------------------------------------------------------
% File     : GEO179+2 : TPTP v8.2.0. Released v3.3.0.
% Domain   : Geometry (Constructive)
% Problem  : Lemma on symmetry and apartness
% Version  : [vPl95] axioms : Reduced > Especial.
% English  : If two points X and Y are distinct and a point Z is apart from
%            the line connecting X and Y, then the line connecting X and Y
%            is distinct from the line connecting Z and X and distinct from
%            the line connecting Z and Y.

% Refs     : [vPl95] von Plato (1995), The Axioms of Constructive Geometry
%          : [Li97]  Li (1997), Replacing the Axioms for Connecting Lines a
%          : [Li98]  Li (1998), A Shorter and Intuitive Axiom to Replace th
%          : [ROK06] Raths et al. (2006), The ILTP Problem Library for Intu
% Source   : [ILTP]
% Names    :

% Status   : Theorem
% Rating   : 0.00 v6.1.0, 0.08 v6.0.0, 0.50 v5.5.0, 0.17 v5.3.0, 0.26 v5.2.0, 0.21 v5.0.0, 0.05 v4.1.0, 0.06 v4.0.1, 0.11 v4.0.0, 0.15 v3.7.0, 0.00 v3.3.0
% Syntax   : Number of formulae    :   13 (   3 unt;   0 def)
%            Number of atoms       :   38 (   0 equ)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives :   28 (   3   ~;   9   |;   4   &)
%                                         (   0 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   6 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    4 (   4 usr;   0 prp; 2-2 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 2-2 aty)
%            Number of variables   :   33 (  33   !;   0   ?)
% SPC      : FOF_THM_RFO_NEQ

% Comments :  Definitions unfolded, hence Especial.
%------------------------------------------------------------------------------
include('Axioms/GEO008+0.ax').
%------------------------------------------------------------------------------
fof(con,conjecture,
    ! [X,Y,Z] :
      ( ( distinct_points(X,Y)
        & apart_point_and_line(Z,line_connecting(X,Y)) )
     => ( distinct_lines(line_connecting(X,Y),line_connecting(Z,X))
        & distinct_lines(line_connecting(X,Y),line_connecting(Z,Y)) ) ) ).

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