%--------------------------------------------------------------------------
% File     : GEO100+1 : TPTP v9.0.0. Released v2.4.0.
% Domain   : Geometry (Oriented curves)
% Problem  : Subcurves with common endpoint can be complemented
% Version  : [EHK99] axioms.
% English  : Every proper sub-curve of an open curve that has a common
%            endpoint with the open curve can be complemented by another
%            curve so that their sum constitute the open curve.

% Refs     : [KE99]  Kulik & Eschenbach (1999), A Geometry of Oriented Curv
%          : [EHK99] Eschenbach et al. (1999), Representing Simple Trajecto
% Source   : [KE99]
% Names    : Theorem 2.18 [KE99]
%            T2 [EHK99]

% Status   : Unknown
% Rating   : 1.00 v2.4.0
% Syntax   : Number of formulae    :   17 (   1 unt;   0 def)
%            Number of atoms       :   75 (  13 equ)
%            Maximal formula atoms :   12 (   4 avg)
%            Number of connectives :   64 (   6   ~;   9   |;  27   &)
%                                         (   9 <=>;  13  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   7 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    8 (   7 usr;   0 prp; 1-3 aty)
%            Number of functors    :    1 (   1 usr;   0 con; 2-2 aty)
%            Number of variables   :   58 (  47   !;  11   ?)
% SPC      : FOF_UNK_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%----Include simple curve axioms
include('Axioms/GEO004+0.ax').
%--------------------------------------------------------------------------
fof(theorem_2_18,conjecture,
    ! [C,C1,P] :
      ( ( open(C)
        & part_of(C1,C)
        & C1 != C
        & end_point(P,C1)
        & end_point(P,C) )
     => ? [Q,C2] :
          ( P != Q
          & meet(Q,C1,C2)
          & C = sum(C1,C2) ) ) ).

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