SET007 Axioms: SET007+789.ax


%------------------------------------------------------------------------------
% File     : SET007+789 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lattice of Fuzzy Sets
% 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    : lfuzzy_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   82 (   2 unit)
%            Number of atoms       :  575 (  53 equality)
%            Maximal formula depth :   22 (   9 average)
%            Number of connectives :  566 (  73 ~  ;   0  |; 312  &)
%                                         (  26 <=>; 155 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   49 (   0 propositional; 1-3 arity)
%            Number of functors    :   72 (  26 constant; 0-7 arity)
%            Number of variables   :  211 (   0 singleton; 183 !;  28 ?)
%            Maximal term depth    :    4 (   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(cc1_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v2_lfuzzy_0(A)
       => ( ~ v3_struct_0(A)
          & v1_lfuzzy_0(A) ) ) ) )).

fof(cc2_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v3_struct_0(A)
       => v1_lfuzzy_0(A) ) ) )).

fof(rc1_lfuzzy_0,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_orders_2(A)
      & v1_lfuzzy_0(A)
      & v2_lfuzzy_0(A) ) )).

fof(cc3_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) )).

fof(fc1_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ( ~ v3_struct_0(k1_lfuzzy_0(A))
        & v1_orders_2(k1_lfuzzy_0(A))
        & v1_lfuzzy_0(k1_lfuzzy_0(A)) ) ) )).

fof(cc4_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_lfuzzy_0(A)
       => ( v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A) ) ) ) )).

fof(cc5_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( ~ v3_struct_0(A)
          & v1_lfuzzy_0(A) )
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v16_waybel_0(A)
          & v1_lattice3(A)
          & v2_lattice3(A)
          & v1_lfuzzy_0(A) ) ) ) )).

fof(cc6_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( ~ v3_struct_0(A)
          & v1_lfuzzy_0(A) )
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v16_waybel_0(A)
          & v1_lattice3(A)
          & v2_lattice3(A)
          & v1_lfuzzy_0(A) ) ) ) )).

fof(cc7_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( ~ v3_struct_0(A)
          & v2_lfuzzy_0(A) )
       => ( ~ v3_struct_0(A)
          & v3_yellow_0(A) ) ) ) )).

fof(cc8_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( ~ v3_struct_0(A)
          & v2_lfuzzy_0(A) )
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v3_yellow_0(A)
          & v16_waybel_0(A)
          & v1_lattice3(A)
          & v2_lattice3(A)
          & v3_lattice3(A)
          & v2_waybel_3(A)
          & v1_lfuzzy_0(A)
          & v2_lfuzzy_0(A) ) ) ) )).

fof(cc9_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v16_waybel_0(A) )
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v16_waybel_0(A)
          & v1_lattice3(A)
          & v2_lattice3(A)
          & v2_waybel_1(A) ) ) ) )).

fof(cc10_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( ( ~ v3_struct_0(A)
          & v2_lfuzzy_0(A) )
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v2_yellow_0(A)
          & v3_yellow_0(A)
          & v16_waybel_0(A)
          & v24_waybel_0(A)
          & ~ v1_yellow_3(A)
          & v1_waybel_2(A)
          & v2_waybel_2(A)
          & v1_lattice3(A)
          & v2_lattice3(A)
          & v3_lattice3(A)
          & v2_waybel_1(A)
          & v9_waybel_1(A)
          & v2_waybel_3(A)
          & v1_lfuzzy_0(A)
          & v2_lfuzzy_0(A) ) ) ) )).

fof(fc2_lfuzzy_0,axiom,
    ( ~ v1_xboole_0(k1_rcomp_1(np__0,np__1))
    & v1_membered(k1_rcomp_1(np__0,np__1))
    & v2_membered(k1_rcomp_1(np__0,np__1)) )).

fof(fc3_lfuzzy_0,axiom,
    ( ~ v3_struct_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v1_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v2_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v3_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v4_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v2_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v3_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v16_waybel_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v24_waybel_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & ~ v1_yellow_3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v1_waybel_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v2_waybel_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v1_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v2_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v3_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v2_waybel_1(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v9_waybel_1(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v2_waybel_3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v1_lfuzzy_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))
    & v2_lfuzzy_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1))) )).

fof(fc4_lfuzzy_0,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v1_lattice3(B)
        & v2_lattice3(B)
        & v3_lattice3(B)
        & v9_waybel_1(B)
        & l1_orders_2(B) )
     => ( ~ v3_struct_0(k6_yellow_1(A,B))
        & v1_orders_2(k6_yellow_1(A,B))
        & v2_orders_2(k6_yellow_1(A,B))
        & v3_orders_2(k6_yellow_1(A,B))
        & v4_orders_2(k6_yellow_1(A,B))
        & v2_yellow_0(k6_yellow_1(A,B))
        & v24_waybel_0(k6_yellow_1(A,B))
        & ~ v1_yellow_3(k6_yellow_1(A,B))
        & v1_waybel_2(k6_yellow_1(A,B))
        & v2_waybel_2(k6_yellow_1(A,B))
        & v1_lattice3(k6_yellow_1(A,B))
        & v2_lattice3(k6_yellow_1(A,B))
        & v3_lattice3(k6_yellow_1(A,B))
        & v2_waybel_1(k6_yellow_1(A,B))
        & v9_waybel_1(k6_yellow_1(A,B)) ) ) )).

fof(fc5_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k4_lfuzzy_0(A))
        & v2_orders_2(k4_lfuzzy_0(A))
        & v3_orders_2(k4_lfuzzy_0(A))
        & v4_orders_2(k4_lfuzzy_0(A))
        & v2_yellow_0(k4_lfuzzy_0(A))
        & v24_waybel_0(k4_lfuzzy_0(A))
        & ~ v1_yellow_3(k4_lfuzzy_0(A))
        & v1_waybel_2(k4_lfuzzy_0(A))
        & v2_waybel_2(k4_lfuzzy_0(A))
        & v1_lattice3(k4_lfuzzy_0(A))
        & v2_lattice3(k4_lfuzzy_0(A))
        & v3_lattice3(k4_lfuzzy_0(A))
        & v2_waybel_1(k4_lfuzzy_0(A))
        & v9_waybel_1(k4_lfuzzy_0(A))
        & v1_monoid_0(k4_lfuzzy_0(A)) ) ) )).

fof(d1_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_lfuzzy_0(A)
      <=> ( r1_tarski(u1_struct_0(A),k1_numbers)
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_xreal_0(C)
                 => ( ( r2_hidden(B,u1_struct_0(A))
                      & r2_hidden(C,u1_struct_0(A)) )
                   => ( r2_hidden(k4_tarski(B,C),u1_orders_2(A))
                    <=> r1_xreal_0(B,C) ) ) ) ) ) ) ) )).

fof(d2_lfuzzy_0,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v2_lfuzzy_0(A)
      <=> ( v1_lfuzzy_0(A)
          & ? [B] :
              ( v1_xreal_0(B)
              & ? [C] :
                  ( v1_xreal_0(C)
                  & r1_xreal_0(B,C)
                  & u1_struct_0(A) = k1_rcomp_1(B,C) ) ) ) ) ) )).

fof(t1_lfuzzy_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ? [B] :
          ( v1_orders_2(B)
          & l1_orders_2(B)
          & u1_struct_0(B) = A
          & v1_lfuzzy_0(B) ) ) )).

fof(t2_lfuzzy_0,axiom,(
    ! [A] :
      ( ( v1_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v1_lfuzzy_0(B)
            & l1_orders_2(B) )
         => ( u1_struct_0(A) = u1_struct_0(B)
           => g1_orders_2(u1_struct_0(A),u1_orders_2(A)) = g1_orders_2(u1_struct_0(B),u1_orders_2(B)) ) ) ) )).

fof(d3_lfuzzy_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( ( v1_orders_2(B)
            & v1_lfuzzy_0(B)
            & l1_orders_2(B) )
         => ( B = k1_lfuzzy_0(A)
          <=> u1_struct_0(B) = A ) ) ) )).

fof(t3_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_xreal_0(B,C)
              <=> r1_orders_2(A,B,C) ) ) ) ) )).

fof(t4_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k13_lattice3(A,B,C) = k2_lfuzzy_0(A,B,C) ) ) ) )).

fof(t5_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k12_lattice3(A,B,C) = k3_lfuzzy_0(A,B,C) ) ) ) )).

fof(t6_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ( ? [B] :
            ( v1_xreal_0(B)
            & r2_hidden(B,u1_struct_0(A))
            & ! [C] :
                ( v1_xreal_0(C)
               => ( r2_hidden(C,u1_struct_0(A))
                 => r1_xreal_0(B,C) ) ) )
      <=> v1_yellow_0(A) ) ) )).

fof(t7_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ( ? [B] :
            ( v1_xreal_0(B)
            & r2_hidden(B,u1_struct_0(A))
            & ! [C] :
                ( v1_xreal_0(C)
               => ( r2_hidden(C,u1_struct_0(A))
                 => r1_xreal_0(C,B) ) ) )
      <=> v2_yellow_0(A) ) ) )).

fof(t8_lfuzzy_0,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_lfuzzy_0(A)
        & l1_orders_2(A) )
     => ! [B] : r1_yellow_0(A,B) ) )).

fof(t9_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_yellow_1(B)
            & v4_waybel_3(B)
            & v5_waybel_3(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ( v2_orders_2(k6_waybel_3(A,B,C))
                  & v3_orders_2(k6_waybel_3(A,B,C))
                  & v4_orders_2(k6_waybel_3(A,B,C))
                  & v1_lattice3(k6_waybel_3(A,B,C))
                  & l1_orders_2(k6_waybel_3(A,B,C)) ) )
           => v1_lattice3(k5_yellow_1(A,B)) ) ) ) )).

fof(t10_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_yellow_1(B)
            & v4_waybel_3(B)
            & v5_waybel_3(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ( v2_orders_2(k6_waybel_3(A,B,C))
                  & v3_orders_2(k6_waybel_3(A,B,C))
                  & v4_orders_2(k6_waybel_3(A,B,C))
                  & v2_lattice3(k6_waybel_3(A,B,C))
                  & l1_orders_2(k6_waybel_3(A,B,C)) ) )
           => v2_lattice3(k5_yellow_1(A,B)) ) ) ) )).

fof(t11_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_yellow_1(B)
            & v4_waybel_3(B)
            & v5_waybel_3(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ( v2_orders_2(k6_waybel_3(A,B,C))
                  & v3_orders_2(k6_waybel_3(A,B,C))
                  & v4_orders_2(k6_waybel_3(A,B,C))
                  & v2_lattice3(k6_waybel_3(A,B,C))
                  & l1_orders_2(k6_waybel_3(A,B,C)) ) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k5_yellow_1(A,B)))
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => k7_waybel_3(A,B,k11_lattice3(k5_yellow_1(A,B),C,D),E) = k11_lattice3(k6_waybel_3(A,B,E),k7_waybel_3(A,B,C,E),k7_waybel_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t12_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_yellow_1(B)
            & v4_waybel_3(B)
            & v5_waybel_3(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ( v2_orders_2(k6_waybel_3(A,B,C))
                  & v3_orders_2(k6_waybel_3(A,B,C))
                  & v4_orders_2(k6_waybel_3(A,B,C))
                  & v1_lattice3(k6_waybel_3(A,B,C))
                  & l1_orders_2(k6_waybel_3(A,B,C)) ) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(k5_yellow_1(A,B)))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(k5_yellow_1(A,B)))
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => k7_waybel_3(A,B,k10_lattice3(k5_yellow_1(A,B),C,D),E) = k10_lattice3(k6_waybel_3(A,B,E),k7_waybel_3(A,B,C,E),k7_waybel_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t13_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_yellow_1(B)
            & v4_waybel_3(B)
            & v5_waybel_3(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ( v2_orders_2(k6_waybel_3(A,B,C))
                  & v3_orders_2(k6_waybel_3(A,B,C))
                  & v4_orders_2(k6_waybel_3(A,B,C))
                  & v1_lattice3(k6_waybel_3(A,B,C))
                  & v2_lattice3(k6_waybel_3(A,B,C))
                  & v3_lattice3(k6_waybel_3(A,B,C))
                  & v9_waybel_1(k6_waybel_3(A,B,C))
                  & l1_orders_2(k6_waybel_3(A,B,C)) ) )
           => ( v3_lattice3(k5_yellow_1(A,B))
              & v9_waybel_1(k5_yellow_1(A,B)) ) ) ) ) )).

fof(d4_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_lfuzzy_0(A) = k6_yellow_1(A,k1_lfuzzy_0(k1_rcomp_1(np__0,np__1))) ) )).

fof(t14_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => u1_struct_0(k4_lfuzzy_0(A)) = k1_funct_2(A,k1_rcomp_1(np__0,np__1)) ) )).

fof(d5_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
         => k5_lfuzzy_0(A,B) = B ) ) )).

fof(d6_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => k6_lfuzzy_0(A,B) = B ) ) )).

fof(t16_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => ( ! [D] :
                    ( m1_subset_1(D,A)
                   => r1_xreal_0(k7_lfuzzy_0(A,B,D),k7_lfuzzy_0(A,C,D)) )
              <=> r1_orders_2(k4_lfuzzy_0(A),k6_lfuzzy_0(A,B),k6_lfuzzy_0(A,C)) ) ) ) ) )).

fof(t17_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k4_lfuzzy_0(A)))
             => ( r1_orders_2(k4_lfuzzy_0(A),B,C)
              <=> ! [D] :
                    ( m1_subset_1(D,A)
                   => r1_xreal_0(k7_lfuzzy_0(A,k5_lfuzzy_0(A,B),D),k7_lfuzzy_0(A,k5_lfuzzy_0(A,C),D)) ) ) ) ) ) )).

fof(t18_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => k2_fuzzy_1(A,B,C) = k13_lattice3(k4_lfuzzy_0(A),k6_lfuzzy_0(A,B),k6_lfuzzy_0(A,C)) ) ) ) )).

fof(t19_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k4_lfuzzy_0(A)))
             => k13_lattice3(k4_lfuzzy_0(A),B,C) = k2_fuzzy_1(A,k5_lfuzzy_0(A,B),k5_lfuzzy_0(A,C)) ) ) ) )).

fof(t20_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => k1_fuzzy_1(A,B,C) = k12_lattice3(k4_lfuzzy_0(A),k6_lfuzzy_0(A,B),k6_lfuzzy_0(A,C)) ) ) ) )).

fof(t21_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k4_lfuzzy_0(A)))
             => k12_lattice3(k4_lfuzzy_0(A),B,C) = k1_fuzzy_1(A,k5_lfuzzy_0(A,B),k5_lfuzzy_0(A,C)) ) ) ) )).

fof(t23_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( m1_fuzzy_1(E,k2_zfmisc_1(A,B))
                     => ! [F] :
                          ( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
                         => ! [G] :
                              ( m1_fuzzy_1(G,k2_zfmisc_1(C,D))
                             => k3_fuzzy_4(A,C,D,k3_fuzzy_4(A,B,C,E,F),G) = k3_fuzzy_4(A,B,D,E,k3_fuzzy_4(B,C,D,F,G)) ) ) ) ) ) ) ) )).

fof(dt_k1_lfuzzy_0,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_orders_2(k1_lfuzzy_0(A))
        & v1_lfuzzy_0(k1_lfuzzy_0(A))
        & l1_orders_2(k1_lfuzzy_0(A)) ) ) )).

fof(dt_k2_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k2_lfuzzy_0(A,B,C),u1_struct_0(A)) ) )).

fof(commutativity_k2_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k2_lfuzzy_0(A,B,C) = k2_lfuzzy_0(A,C,B) ) )).

fof(idempotence_k2_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k2_lfuzzy_0(A,B,B) = B ) )).

fof(redefinition_k2_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k2_lfuzzy_0(A,B,C) = k2_square_1(B,C) ) )).

fof(dt_k3_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k3_lfuzzy_0(A,B,C),u1_struct_0(A)) ) )).

fof(commutativity_k3_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k3_lfuzzy_0(A,B,C) = k3_lfuzzy_0(A,C,B) ) )).

fof(idempotence_k3_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k3_lfuzzy_0(A,B,B) = B ) )).

fof(redefinition_k3_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v1_lfuzzy_0(A)
        & l1_orders_2(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k3_lfuzzy_0(A,B,C) = k1_square_1(B,C) ) )).

fof(dt_k4_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v2_orders_2(k4_lfuzzy_0(A))
        & v3_orders_2(k4_lfuzzy_0(A))
        & v4_orders_2(k4_lfuzzy_0(A))
        & v1_lattice3(k4_lfuzzy_0(A))
        & v2_lattice3(k4_lfuzzy_0(A))
        & v3_lattice3(k4_lfuzzy_0(A))
        & v9_waybel_1(k4_lfuzzy_0(A))
        & l1_orders_2(k4_lfuzzy_0(A)) ) ) )).

fof(dt_k5_lfuzzy_0,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A))) )
     => m1_fuzzy_1(k5_lfuzzy_0(A,B),A) ) )).

fof(dt_k6_lfuzzy_0,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A) )
     => m1_subset_1(k6_lfuzzy_0(A,B),u1_struct_0(k4_lfuzzy_0(A))) ) )).

fof(dt_k7_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k7_lfuzzy_0(A,B,C),u1_struct_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))) ) )).

fof(redefinition_k7_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_subset_1(C,A) )
     => k7_lfuzzy_0(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k8_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
        & m1_subset_1(C,A) )
     => m1_subset_1(k8_lfuzzy_0(A,B,C),u1_struct_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)))) ) )).

fof(redefinition_k8_lfuzzy_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,u1_struct_0(k4_lfuzzy_0(A)))
        & m1_subset_1(C,A) )
     => k8_lfuzzy_0(A,B,C) = k1_funct_1(B,C) ) )).

fof(t15_lfuzzy_0,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v3_lattice3(A)
        & v9_waybel_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k12_lattice3(A,k1_yellow_0(A,B),C) = k1_yellow_0(A,a_3_0_lfuzzy_0(A,B,C)) ) ) ) )).

fof(t22_lfuzzy_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
                 => ! [E] :
                      ( m1_fuzzy_1(E,k2_zfmisc_1(B,C))
                     => ! [F] :
                          ( m1_subset_1(F,A)
                         => ! [G] :
                              ( m1_subset_1(G,C)
                             => k7_lfuzzy_0(k2_zfmisc_1(A,C),k3_fuzzy_4(A,B,C,D,E),k1_domain_1(A,C,F,G)) = k1_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),a_7_0_lfuzzy_0(A,B,C,D,E,F,G)) ) ) ) ) ) ) ) )).

fof(s1_lfuzzy_0,axiom,(
    k1_yellow_0(f1_s1_lfuzzy_0,a_0_0_lfuzzy_0) = k1_yellow_0(f1_s1_lfuzzy_0,a_0_1_lfuzzy_0) )).

fof(s2_lfuzzy_0,axiom,(
    k1_yellow_0(f1_s2_lfuzzy_0,a_0_2_lfuzzy_0) = k1_yellow_0(f1_s2_lfuzzy_0,a_0_3_lfuzzy_0) )).

fof(s3_lfuzzy_0,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s3_lfuzzy_0)
       => ! [B] :
            ( m1_subset_1(B,f2_s3_lfuzzy_0)
           => ( p1_s3_lfuzzy_0(A,B)
             => f3_s3_lfuzzy_0(A,B) = f4_s3_lfuzzy_0(A,B) ) ) )
   => a_0_4_lfuzzy_0 = a_0_5_lfuzzy_0 )).

fof(s4_lfuzzy_0,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s4_lfuzzy_0)
         => ! [B] :
              ( m1_subset_1(B,f2_s4_lfuzzy_0)
             => ( p1_s4_lfuzzy_0(A,B)
              <=> p2_s4_lfuzzy_0(A,B) ) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s4_lfuzzy_0)
         => ! [B] :
              ( m1_subset_1(B,f2_s4_lfuzzy_0)
             => ( p1_s4_lfuzzy_0(A,B)
               => f3_s4_lfuzzy_0(A,B) = f4_s4_lfuzzy_0(A,B) ) ) ) )
   => a_0_6_lfuzzy_0 = a_0_7_lfuzzy_0 )).

fof(s5_lfuzzy_0,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f2_s5_lfuzzy_0)
       => ! [B] :
            ( m1_subset_1(B,f3_s5_lfuzzy_0)
           => ( ( p1_s5_lfuzzy_0(A)
                & p2_s5_lfuzzy_0(B) )
             => f4_s5_lfuzzy_0(A,B) = f5_s5_lfuzzy_0(A,B) ) ) )
   => k1_yellow_0(f1_s5_lfuzzy_0,a_0_8_lfuzzy_0) = k1_yellow_0(f1_s5_lfuzzy_0,a_0_9_lfuzzy_0) )).

fof(fraenkel_a_3_0_lfuzzy_0,axiom,(
    ! [A,B,C,D] :
      ( ( v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v1_lattice3(B)
        & v2_lattice3(B)
        & v3_lattice3(B)
        & v9_waybel_1(B)
        & l1_orders_2(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_lfuzzy_0(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k12_lattice3(B,E,D)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_7_0_lfuzzy_0,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & m1_fuzzy_1(E,k2_zfmisc_1(B,C))
        & m1_fuzzy_1(F,k2_zfmisc_1(C,D))
        & m1_subset_1(G,B)
        & m1_subset_1(H,D) )
     => ( r2_hidden(A,a_7_0_lfuzzy_0(B,C,D,E,F,G,H))
      <=> ? [I] :
            ( m1_subset_1(I,C)
            & A = k12_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k7_lfuzzy_0(k2_zfmisc_1(B,C),E,k1_domain_1(B,C,G,I)),k7_lfuzzy_0(k2_zfmisc_1(C,D),F,k1_domain_1(C,D,I,H))) ) ) ) )).

fof(fraenkel_a_0_0_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_lfuzzy_0)
    <=> ? [B] :
          ( m1_subset_1(B,f2_s1_lfuzzy_0)
          & A = k1_yellow_0(f1_s1_lfuzzy_0,a_1_0_lfuzzy_0(B))
          & p1_s1_lfuzzy_0(B) ) ) )).

fof(fraenkel_a_1_0_lfuzzy_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,f2_s1_lfuzzy_0)
     => ( r2_hidden(A,a_1_0_lfuzzy_0(B))
      <=> ? [C] :
            ( m1_subset_1(C,f3_s1_lfuzzy_0)
            & A = f4_s1_lfuzzy_0(B,C)
            & p2_s1_lfuzzy_0(C) ) ) ) )).

fof(fraenkel_a_0_1_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_lfuzzy_0)
    <=> ? [B,C] :
          ( m1_subset_1(B,f2_s1_lfuzzy_0)
          & m1_subset_1(C,f3_s1_lfuzzy_0)
          & A = f4_s1_lfuzzy_0(B,C)
          & p1_s1_lfuzzy_0(B)
          & p2_s1_lfuzzy_0(C) ) ) )).

fof(fraenkel_a_0_2_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_lfuzzy_0)
    <=> ? [B] :
          ( m1_subset_1(B,f3_s2_lfuzzy_0)
          & A = k1_yellow_0(f1_s2_lfuzzy_0,a_1_1_lfuzzy_0(B))
          & p2_s2_lfuzzy_0(B) ) ) )).

fof(fraenkel_a_1_1_lfuzzy_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,f3_s2_lfuzzy_0)
     => ( r2_hidden(A,a_1_1_lfuzzy_0(B))
      <=> ? [C] :
            ( m1_subset_1(C,f2_s2_lfuzzy_0)
            & A = f4_s2_lfuzzy_0(C,B)
            & p1_s2_lfuzzy_0(C) ) ) ) )).

fof(fraenkel_a_0_3_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_3_lfuzzy_0)
    <=> ? [B,C] :
          ( m1_subset_1(B,f2_s2_lfuzzy_0)
          & m1_subset_1(C,f3_s2_lfuzzy_0)
          & A = f4_s2_lfuzzy_0(B,C)
          & p1_s2_lfuzzy_0(B)
          & p2_s2_lfuzzy_0(C) ) ) )).

fof(fraenkel_a_0_4_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_4_lfuzzy_0)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s3_lfuzzy_0)
          & m1_subset_1(C,f2_s3_lfuzzy_0)
          & A = f3_s3_lfuzzy_0(B,C)
          & p1_s3_lfuzzy_0(B,C) ) ) )).

fof(fraenkel_a_0_5_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_5_lfuzzy_0)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s3_lfuzzy_0)
          & m1_subset_1(C,f2_s3_lfuzzy_0)
          & A = f4_s3_lfuzzy_0(B,C)
          & p1_s3_lfuzzy_0(B,C) ) ) )).

fof(fraenkel_a_0_6_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_6_lfuzzy_0)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s4_lfuzzy_0)
          & m1_subset_1(C,f2_s4_lfuzzy_0)
          & A = f3_s4_lfuzzy_0(B,C)
          & p1_s4_lfuzzy_0(B,C) ) ) )).

fof(fraenkel_a_0_7_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_7_lfuzzy_0)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s4_lfuzzy_0)
          & m1_subset_1(C,f2_s4_lfuzzy_0)
          & A = f4_s4_lfuzzy_0(B,C)
          & p2_s4_lfuzzy_0(B,C) ) ) )).

fof(fraenkel_a_0_8_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_8_lfuzzy_0)
    <=> ? [B] :
          ( m1_subset_1(B,f2_s5_lfuzzy_0)
          & A = k1_yellow_0(f1_s5_lfuzzy_0,a_1_2_lfuzzy_0(B))
          & p1_s5_lfuzzy_0(B) ) ) )).

fof(fraenkel_a_1_2_lfuzzy_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,f2_s5_lfuzzy_0)
     => ( r2_hidden(A,a_1_2_lfuzzy_0(B))
      <=> ? [C] :
            ( m1_subset_1(C,f3_s5_lfuzzy_0)
            & A = f4_s5_lfuzzy_0(B,C)
            & p2_s5_lfuzzy_0(C) ) ) ) )).

fof(fraenkel_a_0_9_lfuzzy_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_9_lfuzzy_0)
    <=> ? [B] :
          ( m1_subset_1(B,f3_s5_lfuzzy_0)
          & A = k1_yellow_0(f1_s5_lfuzzy_0,a_1_3_lfuzzy_0(B))
          & p2_s5_lfuzzy_0(B) ) ) )).

fof(fraenkel_a_1_3_lfuzzy_0,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,f3_s5_lfuzzy_0)
     => ( r2_hidden(A,a_1_3_lfuzzy_0(B))
      <=> ? [C] :
            ( m1_subset_1(C,f2_s5_lfuzzy_0)
            & A = f5_s5_lfuzzy_0(C,B)
            & p1_s5_lfuzzy_0(C) ) ) ) )).
%------------------------------------------------------------------------------