SET007 Axioms: SET007+798.ax


%------------------------------------------------------------------------------
% File     : SET007+798 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Transitive Closure of Fuzzy Relations
% 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_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   77 (   0 unit)
%            Number of atoms       :  416 (  48 equality)
%            Maximal formula depth :   21 (   8 average)
%            Number of connectives :  432 (  93 ~  ;   1  |; 122  &)
%                                         (  17 <=>; 199 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   0 propositional; 1-4 arity)
%            Number of functors    :   40 (   4 constant; 0-5 arity)
%            Number of variables   :  237 (   0 singleton; 228 !;   9 ?)
%            Maximal term depth    :    6 (   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_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ( v1_relat_1(B)
            & v1_seq_1(B) ) ) ) )).

fof(fc1_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_relat_1(k4_fuzzy_4(A,A))
        & v1_funct_1(k4_fuzzy_4(A,A))
        & v1_seq_1(k4_fuzzy_4(A,A))
        & v1_lfuzzy_1(k4_fuzzy_4(A,A),A)
        & v2_lfuzzy_1(k4_fuzzy_4(A,A),A)
        & v3_lfuzzy_1(k4_fuzzy_4(A,A),A)
        & v4_lfuzzy_1(k4_fuzzy_4(A,A),A) ) ) )).

fof(rc1_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_seq_1(B)
          & v1_lfuzzy_1(B,A)
          & v2_lfuzzy_1(B,A)
          & v3_lfuzzy_1(B,A)
          & v4_lfuzzy_1(B,A) ) ) )).

fof(fc2_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v2_lfuzzy_1(B,A)
        & m1_fuzzy_1(B,k2_zfmisc_1(A,A))
        & v2_lfuzzy_1(C,A)
        & m1_fuzzy_1(C,k2_zfmisc_1(A,A)) )
     => ( v1_relat_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v1_funct_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v1_seq_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v2_lfuzzy_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C),A) ) ) )).

fof(fc3_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v2_lfuzzy_1(B,A)
        & m1_fuzzy_1(B,k2_zfmisc_1(A,A))
        & v2_lfuzzy_1(C,A)
        & m1_fuzzy_1(C,k2_zfmisc_1(A,A)) )
     => ( v1_relat_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v1_funct_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v1_seq_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v2_lfuzzy_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C),A) ) ) )).

fof(fc4_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v3_lfuzzy_1(B,A)
        & m1_fuzzy_1(B,k2_zfmisc_1(A,A))
        & v3_lfuzzy_1(C,A)
        & m1_fuzzy_1(C,k2_zfmisc_1(A,A)) )
     => ( v1_relat_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v1_funct_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v1_seq_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
        & v3_lfuzzy_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C),A) ) ) )).

fof(fc5_lfuzzy_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,k2_zfmisc_1(A,A)) )
     => ( v1_relat_1(k5_lfuzzy_1(A,B))
        & v1_funct_1(k5_lfuzzy_1(A,B))
        & v1_seq_1(k5_lfuzzy_1(A,B))
        & v3_lfuzzy_1(k5_lfuzzy_1(A,B),A) ) ) )).

fof(d1_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
             => ! [D] :
                  ( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
                 => ( r1_lfuzzy_1(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,A)
                       => ! [F] :
                            ( m1_subset_1(F,B)
                           => r1_xreal_0(k7_lfuzzy_0(k2_zfmisc_1(A,B),C,k1_domain_1(A,B,E,F)),k7_lfuzzy_0(k2_zfmisc_1(A,B),D,k1_domain_1(A,B,E,F))) ) ) ) ) ) ) ) )).

fof(t1_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => ( ! [D] :
                    ( m1_subset_1(D,A)
                   => k7_lfuzzy_0(A,B,D) = k7_lfuzzy_0(A,C,D) )
               => B = C ) ) ) ) )).

fof(t2_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
             => ! [D] :
                  ( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
                 => ( ! [E] :
                        ( m1_subset_1(E,A)
                       => ! [F] :
                            ( m1_subset_1(F,B)
                           => k7_lfuzzy_0(k2_zfmisc_1(A,B),C,k1_domain_1(A,B,E,F)) = k7_lfuzzy_0(k2_zfmisc_1(A,B),D,k1_domain_1(A,B,E,F)) ) )
                   => C = D ) ) ) ) ) )).

fof(t3_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => ( B = C
              <=> ( r1_fuzzy_1(B,C)
                  & r1_fuzzy_1(C,B) ) ) ) ) ) )).

fof(t4_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => r1_fuzzy_1(B,B) ) ) )).

fof(t5_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => ! [D] :
                  ( m1_fuzzy_1(D,A)
                 => ( ( r1_fuzzy_1(B,C)
                      & r1_fuzzy_1(C,D) )
                   => r1_fuzzy_1(B,D) ) ) ) ) ) )).

fof(t6_lfuzzy_1,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(A,B))
                     => ! [F] :
                          ( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
                         => ! [G] :
                              ( m1_fuzzy_1(G,k2_zfmisc_1(B,C))
                             => ( ( r1_fuzzy_1(D,E)
                                  & r1_fuzzy_1(F,G) )
                               => r1_fuzzy_1(k3_fuzzy_4(A,B,C,D,F),k3_fuzzy_4(A,B,C,E,G)) ) ) ) ) ) ) ) ) )).

fof(t7_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => r1_fuzzy_1(k1_lfuzzy_1(A,B,C),B) ) ) ) )).

fof(t8_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,A)
         => ! [C] :
              ( m1_fuzzy_1(C,A)
             => r1_fuzzy_1(B,k2_lfuzzy_1(A,B,C)) ) ) ) )).

fof(d2_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v1_lfuzzy_1(B,A)
          <=> r1_fuzzy_1(k4_fuzzy_4(A,A),B) ) ) ) )).

fof(d3_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v1_lfuzzy_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,C)) = np__1 ) ) ) ) )).

fof(d4_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v2_lfuzzy_1(B,A)
          <=> k1_fuzzy_4(A,A,B) = B ) ) ) )).

fof(d5_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v2_lfuzzy_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)) = k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,C)) ) ) ) ) ) )).

fof(d6_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v3_lfuzzy_1(B,A)
          <=> r1_fuzzy_1(k3_fuzzy_4(A,A,A,B,B),B) ) ) ) )).

fof(d7_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v3_lfuzzy_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( m1_subset_1(E,A)
                       => r1_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k12_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)),k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,E))),k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,E))) ) ) ) ) ) ) )).

fof(d8_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v4_lfuzzy_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ~ ( k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)) != np__0
                        & k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,C)) != np__0
                        & C != D ) ) ) ) ) ) )).

fof(d9_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v4_lfuzzy_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ~ ( k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)) != np__0
                        & C != D
                        & k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,C)) != np__0 ) ) ) ) ) ) )).

fof(t9_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
             => ( ( v2_lfuzzy_1(B,A)
                  & v2_lfuzzy_1(C,A) )
               => k1_fuzzy_4(A,A,k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C)) = k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C) ) ) ) ) )).

fof(t10_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
             => ( ( v2_lfuzzy_1(B,A)
                  & v2_lfuzzy_1(C,A) )
               => k1_fuzzy_4(A,A,k2_lfuzzy_1(k2_zfmisc_1(A,A),B,C)) = k2_lfuzzy_1(k2_zfmisc_1(A,A),B,C) ) ) ) ) )).

fof(t11_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
             => ( ( v3_lfuzzy_1(B,A)
                  & v3_lfuzzy_1(C,A) )
               => r1_fuzzy_1(k3_fuzzy_4(A,A,A,k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C),k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C)),k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C)) ) ) ) ) )).

fof(t12_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => ( r1_relat_2(B,A)
           => v1_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) )).

fof(t13_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => ( v4_relat_2(B)
           => v4_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) )).

fof(t14_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => ( v3_relat_2(B)
           => v2_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) )).

fof(t15_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_relset_1(B,A,A)
         => ( v8_relat_2(B)
           => v3_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) )).

fof(t16_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v2_lfuzzy_1(k1_fuzzy_3(A,A),A)
        & v4_lfuzzy_1(k1_fuzzy_3(A,A),A)
        & v3_lfuzzy_1(k1_fuzzy_3(A,A),A) ) ) )).

fof(t17_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v2_lfuzzy_1(k2_fuzzy_3(A,A),A)
        & v3_lfuzzy_1(k2_fuzzy_3(A,A),A)
        & v1_lfuzzy_1(k2_fuzzy_3(A,A),A) ) ) )).

fof(t18_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => v2_lfuzzy_1(k2_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B)),A) ) ) )).

fof(t19_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => v2_lfuzzy_1(k1_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B)),A) ) ) )).

fof(t20_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
             => ( ( v2_lfuzzy_1(C,A)
                  & r1_fuzzy_1(B,C) )
               => r1_fuzzy_1(k2_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B)),C) ) ) ) ) )).

fof(t21_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
             => ( ( v2_lfuzzy_1(C,A)
                  & r1_fuzzy_1(C,B) )
               => r1_fuzzy_1(C,k1_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B))) ) ) ) ) )).

fof(d10_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( m1_fuzzy_1(D,k2_zfmisc_1(A,A))
                 => ( D = k4_lfuzzy_1(A,B,C)
                  <=> ? [E] :
                        ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,k1_funct_2(k2_zfmisc_1(A,A),k1_rcomp_1(np__0,np__1)))
                        & m2_relset_1(E,k5_numbers,k1_funct_2(k2_zfmisc_1(A,A),k1_rcomp_1(np__0,np__1)))
                        & D = k1_funct_1(E,C)
                        & k8_funct_2(k5_numbers,k1_funct_2(k2_zfmisc_1(A,A),k1_rcomp_1(np__0,np__1)),E,np__0) = k4_fuzzy_4(A,A)
                        & ! [F] :
                            ( v4_ordinal2(F)
                           => ? [G] :
                                ( m1_fuzzy_1(G,k2_zfmisc_1(A,A))
                                & k1_funct_1(E,F) = G
                                & k1_funct_1(E,k2_xcmplx_0(F,np__1)) = k3_fuzzy_4(A,A,A,G,B) ) ) ) ) ) ) ) ) )).

fof(t22_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => k3_fuzzy_4(A,A,A,k4_fuzzy_4(A,A),B) = B ) ) )).

fof(t23_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => k3_fuzzy_4(A,A,A,B,k4_fuzzy_4(A,A)) = B ) ) )).

fof(t24_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => k4_lfuzzy_1(A,B,np__0) = k4_fuzzy_4(A,A) ) ) )).

fof(t25_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => k4_lfuzzy_1(A,B,np__1) = B ) ) )).

fof(t26_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( v4_ordinal2(C)
             => k4_lfuzzy_1(A,B,k2_xcmplx_0(C,np__1)) = k3_fuzzy_4(A,A,A,k4_lfuzzy_1(A,B,C),B) ) ) ) )).

fof(t27_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => k4_lfuzzy_1(A,B,k2_xcmplx_0(C,D)) = k3_fuzzy_4(A,A,A,k4_lfuzzy_1(A,B,C),k4_lfuzzy_1(A,B,D)) ) ) ) ) )).

fof(t28_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => k4_lfuzzy_1(A,B,k3_xcmplx_0(C,D)) = k4_lfuzzy_1(A,k4_lfuzzy_1(A,B,D),C) ) ) ) ) )).

fof(t30_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => r1_fuzzy_1(B,k5_lfuzzy_1(A,B)) ) ) )).

fof(t31_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ~ r1_xreal_0(C,np__0)
               => r1_fuzzy_1(k4_lfuzzy_1(A,B,C),k5_lfuzzy_1(A,B)) ) ) ) ) )).

fof(t32_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(A))))
         => ! [C] :
              ( m1_subset_1(C,A)
             => k8_lfuzzy_0(A,k1_yellow_0(k4_lfuzzy_0(A),B),C) = k1_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k5_card_3(C,B)) ) ) ) )).

fof(t37_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( v4_ordinal2(C)
             => ( v3_lfuzzy_1(B,A)
               => ( r1_xreal_0(C,np__0)
                  | r1_fuzzy_1(k4_lfuzzy_1(A,B,C),B) ) ) ) ) ) )).

fof(t38_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ( v3_lfuzzy_1(B,A)
           => B = k5_lfuzzy_1(A,B) ) ) ) )).

fof(t39_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ( r1_fuzzy_1(B,C)
                   => r1_fuzzy_1(k4_lfuzzy_1(A,B,D),k4_lfuzzy_1(A,C,D)) ) ) ) ) ) )).

fof(t40_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
             => ( ( v3_lfuzzy_1(C,A)
                  & r1_fuzzy_1(B,C) )
               => r1_fuzzy_1(k5_lfuzzy_1(A,B),C) ) ) ) ) )).

fof(reflexivity_r1_lfuzzy_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_fuzzy_1(C,k2_zfmisc_1(A,B))
        & m1_fuzzy_1(D,k2_zfmisc_1(A,B)) )
     => r1_lfuzzy_1(A,B,C,C) ) )).

fof(redefinition_r1_lfuzzy_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_fuzzy_1(C,k2_zfmisc_1(A,B))
        & m1_fuzzy_1(D,k2_zfmisc_1(A,B)) )
     => ( r1_lfuzzy_1(A,B,C,D)
      <=> r1_fuzzy_1(C,D) ) ) )).

fof(dt_k1_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => m1_fuzzy_1(k1_lfuzzy_1(A,B,C),A) ) )).

fof(commutativity_k1_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => k1_lfuzzy_1(A,B,C) = k1_lfuzzy_1(A,C,B) ) )).

fof(idempotence_k1_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => k1_lfuzzy_1(A,B,B) = B ) )).

fof(redefinition_k1_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => k1_lfuzzy_1(A,B,C) = k1_fuzzy_1(A,B,C) ) )).

fof(dt_k2_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => m1_fuzzy_1(k2_lfuzzy_1(A,B,C),A) ) )).

fof(commutativity_k2_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => k2_lfuzzy_1(A,B,C) = k2_lfuzzy_1(A,C,B) ) )).

fof(idempotence_k2_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => k2_lfuzzy_1(A,B,B) = B ) )).

fof(redefinition_k2_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,A)
        & m1_fuzzy_1(C,A) )
     => k2_lfuzzy_1(A,B,C) = k2_fuzzy_1(A,B,C) ) )).

fof(dt_k3_lfuzzy_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => m1_fuzzy_1(k3_lfuzzy_1(A,B),B) ) )).

fof(redefinition_k3_lfuzzy_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => k3_lfuzzy_1(A,B) = k4_funct_3(A,B) ) )).

fof(dt_k4_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,k2_zfmisc_1(A,A))
        & v4_ordinal2(C) )
     => m1_fuzzy_1(k4_lfuzzy_1(A,B,C),k2_zfmisc_1(A,A)) ) )).

fof(dt_k5_lfuzzy_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_fuzzy_1(B,k2_zfmisc_1(A,A)) )
     => m1_fuzzy_1(k5_lfuzzy_1(A,B),k2_zfmisc_1(A,A)) ) )).

fof(d11_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => k5_lfuzzy_1(A,B) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_2_0_lfuzzy_1(A,B)) ) ) )).

fof(t29_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => k7_lfuzzy_0(k2_zfmisc_1(A,A),k5_lfuzzy_1(A,B),k1_domain_1(A,A,C,D)) = k1_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k5_card_3(k1_domain_1(A,A,C,D),a_2_0_lfuzzy_1(A,B))) ) ) ) ) )).

fof(t33_lfuzzy_1,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,C,k1_yellow_0(A,B)) = k1_yellow_0(A,a_3_0_lfuzzy_1(A,B,C)) ) ) ) )).

fof(t34_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)))))
             => k3_fuzzy_4(A,A,A,B,k5_lfuzzy_0(k2_zfmisc_1(A,A),k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),C))) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_3_1_lfuzzy_1(A,B,C)) ) ) ) )).

fof(t35_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)))))
             => k3_fuzzy_4(A,A,A,k5_lfuzzy_0(k2_zfmisc_1(A,A),k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),C)),B) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_3_2_lfuzzy_1(A,B,C)) ) ) ) )).

fof(t36_lfuzzy_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
         => k3_fuzzy_4(A,A,A,k5_lfuzzy_1(A,B),k5_lfuzzy_1(A,B)) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_2_1_lfuzzy_1(A,B)) ) ) )).

fof(fraenkel_a_2_0_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_fuzzy_1(C,k2_zfmisc_1(B,B)) )
     => ( r2_hidden(A,a_2_0_lfuzzy_1(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k4_lfuzzy_1(B,C,D)
            & ~ r1_xreal_0(D,np__0) ) ) ) )).

fof(fraenkel_a_3_0_lfuzzy_1,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_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k12_lattice3(B,D,E)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_3_1_lfuzzy_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_fuzzy_1(C,k2_zfmisc_1(B,B))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))) )
     => ( r2_hidden(A,a_3_1_lfuzzy_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))
            & A = k3_fuzzy_4(B,B,B,C,k5_lfuzzy_0(k2_zfmisc_1(B,B),E))
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_2_lfuzzy_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_fuzzy_1(C,k2_zfmisc_1(B,B))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))) )
     => ( r2_hidden(A,a_3_2_lfuzzy_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))
            & A = k3_fuzzy_4(B,B,B,k5_lfuzzy_0(k2_zfmisc_1(B,B),E),C)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_2_1_lfuzzy_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_fuzzy_1(C,k2_zfmisc_1(B,B)) )
     => ( r2_hidden(A,a_2_1_lfuzzy_1(B,C))
      <=> ? [D,E] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & m2_subset_1(E,k1_numbers,k5_numbers)
            & A = k3_fuzzy_4(B,B,B,k4_lfuzzy_1(B,C,D),k4_lfuzzy_1(B,C,E))
            & ~ r1_xreal_0(D,np__0)
            & ~ r1_xreal_0(E,np__0) ) ) ) )).
%------------------------------------------------------------------------------