%--------------------------------------------------------------------------
% File     : SET772+4 : TPTP v8.2.0. Released v2.2.0.
% Domain   : Set Theory (Equivalence relations)
% Problem  : Belonging of the same member of a partition is an equivalence
% Version  : [Pas99] axioms.
% English  :

% Refs     : [Pas99] Pastre (1999), Email to G. Sutcliffe
% Source   : [Pas99]
% Names    :

% Status   : Theorem
% Rating   : 0.75 v8.2.0, 0.78 v8.1.0, 0.75 v7.4.0, 0.63 v7.3.0, 0.66 v7.1.0, 0.61 v7.0.0, 0.73 v6.4.0, 0.77 v6.3.0, 0.71 v6.2.0, 0.76 v6.1.0, 0.83 v6.0.0, 0.87 v5.5.0, 0.85 v5.4.0, 0.93 v5.2.0, 0.95 v5.0.0, 0.96 v3.7.0, 0.90 v3.5.0, 0.89 v3.3.0, 0.86 v3.2.0, 0.91 v3.1.0, 0.89 v2.7.0, 0.83 v2.6.0, 0.86 v2.5.0, 0.88 v2.4.0, 0.75 v2.3.0, 0.67 v2.2.1
% Syntax   : Number of formulae    :   17 (   1 unt;   0 def)
%            Number of atoms       :   76 (   4 equ)
%            Maximal formula atoms :   13 (   4 avg)
%            Number of connectives :   62 (   3   ~;   2   |;  24   &)
%                                         (  16 <=>;  17  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (   7 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    9 (   8 usr;   0 prp; 2-3 aty)
%            Number of functors    :   10 (  10 usr;   1 con; 0-3 aty)
%            Number of variables   :   63 (  58   !;   5   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
%--------------------------------------------------------------------------
%----Include set theory definitions
include('Axioms/SET006+0.ax').
%----Include equivalence relation axioms
include('Axioms/SET006+2.ax').
%--------------------------------------------------------------------------
fof(thIII08,conjecture,
    ! [A,E,R] :
      ( partition(A,E)
     => ( ! [X,Y] :
            ( ( member(X,E)
              & member(Y,E) )
           => ( apply(R,X,Y)
            <=> ? [Z] :
                  ( member(Z,A)
                  & member(X,Z)
                  & member(Y,Z) ) ) )
       => equivalence(R,E) ) ) ).

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