SET007 Axioms: SET007+159.ax


%------------------------------------------------------------------------------
% File     : SET007+159 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Relations of Tolerance
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : toler_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   92 (  43 unit)
%            Number of atoms       :  388 (  16 equality)
%            Maximal formula depth :   15 (   5 average)
%            Number of connectives :  304 (   8 ~  ;   0  |; 199  &)
%                                         (  15 <=>;  82 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   24 (   1 propositional; 0-3 arity)
%            Number of functors    :   19 (   2 constant; 0-6 arity)
%            Number of variables   :  161 (   0 singleton; 154 !;   7 ?)
%            Maximal term depth    :    3 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_toler_1,axiom,
    ( v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_relat_2(k1_xboole_0)
    & v2_relat_2(k1_xboole_0)
    & v3_relat_2(k1_xboole_0)
    & v4_relat_2(k1_xboole_0)
    & v5_relat_2(k1_xboole_0)
    & v6_relat_2(k1_xboole_0)
    & v7_relat_2(k1_xboole_0)
    & v8_relat_2(k1_xboole_0)
    & v1_xboole_0(k1_xboole_0) )).

fof(rc1_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => ? [C] :
          ( m1_toler_1(C,A,B)
          & v1_toler_1(C,A,B) ) ) )).

fof(t1_toler_1,axiom,(
    k3_relat_1(k1_xboole_0) = k1_xboole_0 )).

fof(t2_toler_1,axiom,(
    v1_relat_2(k1_xboole_0) )).

fof(t3_toler_1,axiom,(
    v3_relat_2(k1_xboole_0) )).

fof(t4_toler_1,axiom,(
    v2_relat_2(k1_xboole_0) )).

fof(t5_toler_1,axiom,(
    v4_relat_2(k1_xboole_0) )).

fof(t6_toler_1,axiom,(
    v5_relat_2(k1_xboole_0) )).

fof(t7_toler_1,axiom,(
    v6_relat_2(k1_xboole_0) )).

fof(t8_toler_1,axiom,(
    v7_relat_2(k1_xboole_0) )).

fof(t9_toler_1,axiom,(
    v8_relat_2(k1_xboole_0) )).

fof(t10_toler_1,axiom,(
    $true )).

fof(t11_toler_1,axiom,(
    $true )).

fof(t12_toler_1,axiom,(
    $true )).

fof(t13_toler_1,axiom,(
    ! [A] : k5_relset_1(A,A,k1_eqrel_1(A)) = A )).

fof(t14_toler_1,axiom,(
    $true )).

fof(t15_toler_1,axiom,(
    ! [A,B,C] :
      ( ( r2_hidden(B,A)
        & r2_hidden(C,A) )
     => r2_hidden(k4_tarski(B,C),k1_eqrel_1(A)) ) )).

fof(t16_toler_1,axiom,(
    ! [A,B,C] :
      ( ( r2_hidden(B,k3_relat_1(k1_eqrel_1(A)))
        & r2_hidden(C,k3_relat_1(k1_eqrel_1(A))) )
     => r2_hidden(k4_tarski(B,C),k1_eqrel_1(A)) ) )).

fof(t17_toler_1,axiom,(
    $true )).

fof(t18_toler_1,axiom,(
    $true )).

fof(t19_toler_1,axiom,(
    ! [A] : v7_relat_2(k1_eqrel_1(A)) )).

fof(t20_toler_1,axiom,(
    $true )).

fof(t21_toler_1,axiom,(
    ! [A] : v6_relat_2(k1_eqrel_1(A)) )).

fof(t22_toler_1,axiom,(
    $true )).

fof(t23_toler_1,axiom,(
    $true )).

fof(t24_toler_1,axiom,(
    $true )).

fof(t25_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k5_relset_1(A,A,B) = A ) )).

fof(t26_toler_1,axiom,(
    $true )).

fof(t27_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ( r2_hidden(B,A)
      <=> r2_hidden(k4_tarski(B,B),C) ) ) )).

fof(t28_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => r1_relat_2(B,A) ) )).

fof(t29_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => r3_relat_2(B,A) ) )).

fof(t30_toler_1,axiom,(
    $true )).

fof(t31_toler_1,axiom,(
    $true )).

fof(t32_toler_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,B)
     => ( v3_relat_2(D)
       => v3_relat_2(k1_toler_1(D,C)) ) ) )).

fof(t33_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v3_relat_2(C)
        & v1_partfun1(C,B,B)
        & m2_relset_1(C,B,B) )
     => ( r1_tarski(A,B)
       => ( v1_relat_2(k1_toler_1(C,A))
          & v3_relat_2(k1_toler_1(C,A))
          & v1_partfun1(k1_toler_1(C,A),A,A)
          & m2_relset_1(k1_toler_1(C,A),A,A) ) ) ) )).

fof(d1_toler_1,axiom,(
    $true )).

fof(d2_toler_1,axiom,(
    $true )).

fof(d3_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( m1_toler_1(C,A,B)
        <=> ! [D,E] :
              ( ( r2_hidden(D,C)
                & r2_hidden(E,C) )
             => r2_hidden(k4_tarski(D,E),B) ) ) ) )).

fof(t34_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => m1_toler_1(k1_xboole_0,A,B) ) )).

fof(d4_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( m1_toler_1(C,A,B)
         => ( v1_toler_1(C,A,B)
          <=> ! [D] :
                ~ ( ~ r2_hidden(D,C)
                  & r2_hidden(D,A)
                  & ! [E] :
                      ~ ( r2_hidden(E,C)
                        & ~ r2_hidden(k4_tarski(D,E),B) ) ) ) ) ) )).

fof(t35_toler_1,axiom,(
    $true )).

fof(t36_toler_1,axiom,(
    $true )).

fof(t37_toler_1,axiom,(
    $true )).

fof(t38_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( ( v1_toler_1(k1_xboole_0,A,B)
          & m1_toler_1(k1_xboole_0,A,B) )
       => B = k1_xboole_0 ) ) )).

fof(t39_toler_1,axiom,
    ( v1_relat_2(k1_xboole_0)
    & v3_relat_2(k1_xboole_0)
    & v1_partfun1(k1_xboole_0,k1_xboole_0,k1_xboole_0)
    & m2_relset_1(k1_xboole_0,k1_xboole_0,k1_xboole_0) )).

fof(t40_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C,D] :
          ( r2_hidden(k4_tarski(C,D),B)
         => m1_toler_1(k2_tarski(C,D),A,B) ) ) )).

fof(t41_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( r2_hidden(C,A)
         => m1_toler_1(k1_tarski(C),A,B) ) ) )).

fof(t42_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C,D] :
          ( ( m1_toler_1(C,A,B)
            & m1_toler_1(D,A,B) )
         => m1_toler_1(k3_xboole_0(C,D),A,B) ) ) )).

fof(t43_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v3_relat_2(C)
        & v1_partfun1(C,B,B)
        & m2_relset_1(C,B,B) )
     => ( m1_toler_1(A,B,C)
       => r1_tarski(A,B) ) ) )).

fof(t44_toler_1,axiom,(
    $true )).

fof(t45_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( m1_toler_1(C,A,B)
         => ? [D] :
              ( v1_toler_1(D,A,B)
              & m1_toler_1(D,A,B)
              & r1_tarski(C,D) ) ) ) )).

fof(t46_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C,D] :
          ~ ( r2_hidden(k4_tarski(C,D),B)
            & ! [E] :
                ( ( v1_toler_1(E,A,B)
                  & m1_toler_1(E,A,B) )
               => ~ ( r2_hidden(C,E)
                    & r2_hidden(D,E) ) ) ) ) )).

fof(t47_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ~ ( r2_hidden(C,A)
            & ! [D] :
                ( ( v1_toler_1(D,A,B)
                  & m1_toler_1(D,A,B) )
               => ~ r2_hidden(C,D) ) ) ) )).

fof(t48_toler_1,axiom,(
    $true )).

fof(t49_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => r1_tarski(B,k1_eqrel_1(A)) ) )).

fof(t50_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => r1_tarski(k6_partfun1(A),B) ) )).

fof(t51_toler_1,axiom,(
    ! [A] :
    ? [B] :
      ( v1_relat_2(B)
      & v3_relat_2(B)
      & v1_partfun1(B,k3_tarski(A),k3_tarski(A))
      & m2_relset_1(B,k3_tarski(A),k3_tarski(A))
      & ! [C] :
          ( r2_hidden(C,A)
         => m1_toler_1(C,k3_tarski(A),B) ) ) )).

fof(t52_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,k3_tarski(A),k3_tarski(A))
        & m2_relset_1(B,k3_tarski(A),k3_tarski(A)) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v3_relat_2(C)
            & v1_partfun1(C,k3_tarski(A),k3_tarski(A))
            & m2_relset_1(C,k3_tarski(A),k3_tarski(A)) )
         => ( ( ! [D,E] :
                  ( r2_hidden(k4_tarski(D,E),B)
                <=> ? [F] :
                      ( r2_hidden(F,A)
                      & r2_hidden(D,F)
                      & r2_hidden(E,F) ) )
              & ! [D,E] :
                  ( r2_hidden(k4_tarski(D,E),C)
                <=> ? [F] :
                      ( r2_hidden(F,A)
                      & r2_hidden(D,F)
                      & r2_hidden(E,F) ) ) )
           => B = C ) ) ) )).

fof(t53_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v3_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( ! [D] :
                ( ( v1_toler_1(D,A,B)
                  & m1_toler_1(D,A,B) )
              <=> ( v1_toler_1(D,A,C)
                  & m1_toler_1(D,A,C) ) )
           => B = C ) ) ) )).

fof(t54_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v3_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ! [D] :
          ( r2_hidden(D,k6_eqrel_1(A,C,B))
        <=> r2_hidden(k4_tarski(B,D),C) ) ) )).

fof(t55_toler_1,axiom,(
    $true )).

fof(t56_toler_1,axiom,(
    $true )).

fof(t57_toler_1,axiom,(
    $true )).

fof(t58_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v3_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ! [D] :
          ( ! [E] :
              ( r2_hidden(E,D)
            <=> ( r2_hidden(B,E)
                & v1_toler_1(E,A,C)
                & m1_toler_1(E,A,C) ) )
         => k6_eqrel_1(A,C,B) = k3_tarski(D) ) ) )).

fof(t59_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(C)
        & v3_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ! [D] :
          ( ! [E] :
              ( r2_hidden(E,D)
            <=> ( r2_hidden(B,E)
                & m1_toler_1(E,A,C) ) )
         => k6_eqrel_1(A,C,B) = k3_tarski(D) ) ) )).

fof(d5_toler_1,axiom,(
    $true )).

fof(d6_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( C = k3_toler_1(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> m1_toler_1(D,A,B) ) ) ) )).

fof(d7_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( C = k4_toler_1(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> ( v1_toler_1(D,A,B)
                & m1_toler_1(D,A,B) ) ) ) ) )).

fof(t60_toler_1,axiom,(
    $true )).

fof(t61_toler_1,axiom,(
    $true )).

fof(t62_toler_1,axiom,(
    $true )).

fof(t63_toler_1,axiom,(
    $true )).

fof(t64_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v3_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( r1_tarski(k4_toler_1(A,B),k4_toler_1(A,C))
           => r1_tarski(B,C) ) ) ) )).

fof(t65_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v3_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( k4_toler_1(A,B) = k4_toler_1(A,C)
           => B = C ) ) ) )).

fof(t66_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k3_tarski(k4_toler_1(A,B)) = A ) )).

fof(t67_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k3_tarski(k3_toler_1(A,B)) = A ) )).

fof(t68_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( ! [C] :
            ( r2_hidden(C,A)
           => m1_toler_1(k6_eqrel_1(A,B,C),A,B) )
       => v8_relat_2(B) ) ) )).

fof(t69_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ( v8_relat_2(B)
       => ! [C] :
            ( r2_hidden(C,A)
           => ( v1_toler_1(k6_eqrel_1(A,B,C),A,B)
              & m1_toler_1(k6_eqrel_1(A,B,C),A,B) ) ) ) ) )).

fof(t70_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C,D] :
          ( ( v1_toler_1(D,A,B)
            & m1_toler_1(D,A,B) )
         => ( r2_hidden(C,D)
           => r1_tarski(D,k6_eqrel_1(A,B,C)) ) ) ) )).

fof(t71_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v3_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( r1_tarski(k3_toler_1(A,B),k3_toler_1(A,C))
          <=> r1_tarski(B,C) ) ) ) )).

fof(t72_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => r1_tarski(k4_toler_1(A,B),k3_toler_1(A,B)) ) )).

fof(t73_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] :
          ( ( v1_relat_2(C)
            & v3_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ( ! [D] :
                ( r2_hidden(D,A)
               => r1_tarski(k6_eqrel_1(A,B,D),k6_eqrel_1(A,C,D)) )
           => r1_tarski(B,C) ) ) ) )).

fof(t74_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => r1_tarski(B,k7_relset_1(A,A,A,A,B,B)) ) )).

fof(s1_toler_1,axiom,
    ( ( ! [A] :
          ( r2_hidden(A,f1_s1_toler_1)
         => p1_s1_toler_1(A,A) )
      & ! [A,B] :
          ( ( r2_hidden(A,f1_s1_toler_1)
            & r2_hidden(B,f1_s1_toler_1)
            & p1_s1_toler_1(A,B) )
         => p1_s1_toler_1(B,A) ) )
   => ? [A] :
        ( v1_relat_2(A)
        & v3_relat_2(A)
        & v1_partfun1(A,f1_s1_toler_1,f1_s1_toler_1)
        & m2_relset_1(A,f1_s1_toler_1,f1_s1_toler_1)
        & ! [B,C] :
            ( ( r2_hidden(B,f1_s1_toler_1)
              & r2_hidden(C,f1_s1_toler_1) )
           => ( r2_hidden(k4_tarski(B,C),A)
            <=> p1_s1_toler_1(B,C) ) ) ) )).

fof(dt_m1_toler_1,axiom,(
    $true )).

fof(existence_m1_toler_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A) )
     => ? [C] : m1_toler_1(C,A,B) ) )).

fof(dt_k1_toler_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(A)
     => m2_relset_1(k1_toler_1(A,B),B,B) ) )).

fof(redefinition_k1_toler_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(A)
     => k1_toler_1(A,B) = k2_wellord1(A,B) ) )).

fof(dt_k2_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => ( v1_relat_2(k2_toler_1(A,B,C))
        & v3_relat_2(k2_toler_1(A,B,C))
        & v1_partfun1(k2_toler_1(A,B,C),C,C)
        & m2_relset_1(k2_toler_1(A,B,C),C,C) ) ) )).

fof(redefinition_k2_toler_1,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m1_relset_1(B,A,A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => k2_toler_1(A,B,C) = k2_wellord1(B,C) ) )).

fof(dt_k3_toler_1,axiom,(
    $true )).

fof(dt_k4_toler_1,axiom,(
    $true )).
%------------------------------------------------------------------------------