SET007 Axioms: SET007+324.ax


%------------------------------------------------------------------------------
% File     : SET007+324 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Category of Left Modules
% 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    : modcat_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   67 (   3 unit)
%            Number of atoms       : 1111 (  52 equality)
%            Maximal formula depth :   30 (  15 average)
%            Number of connectives : 1158 ( 114 ~  ;   0  |; 853  &)
%                                         (  23 <=>; 168 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   38 (   1 propositional; 0-5 arity)
%            Number of functors    :   41 (   1 constant; 0-6 arity)
%            Number of variables   :  226 (   0 singleton; 207 !;  19 ?)
%            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(rc1_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ? [C] :
          ( m1_subset_1(C,B)
          & ~ v3_struct_0(C)
          & v3_rlvect_1(C)
          & v4_rlvect_1(C)
          & v5_rlvect_1(C)
          & v6_rlvect_1(C)
          & v11_vectsp_1(C,A)
          & v12_vectsp_1(C,A) ) ) )).

fof(rc2_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m3_modcat_1(B,A) )
     => ? [C] :
          ( m1_subset_1(C,B)
          & v2_mod_2(C,A)
          & v3_mod_2(C,A) ) ) )).

fof(fc1_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ~ v1_xboole_0(k2_modcat_1(A,B)) ) )).

fof(fc2_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ( v1_cat_1(k12_modcat_1(A,B))
        & v2_cat_1(k12_modcat_1(A,B)) ) ) )).

fof(d1_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( m1_modcat_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,B)
               => ( ~ v3_struct_0(C)
                  & v3_rlvect_1(C)
                  & v4_rlvect_1(C)
                  & v5_rlvect_1(C)
                  & v6_rlvect_1(C)
                  & v11_vectsp_1(C,A)
                  & v12_vectsp_1(C,A)
                  & l4_vectsp_1(C,A) ) ) ) ) ) )).

fof(d2_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( m3_modcat_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,B)
               => ( v2_mod_2(C,A)
                  & v3_mod_2(C,A)
                  & l1_mod_2(C,A) ) ) ) ) ) )).

fof(t1_modcat_1,axiom,(
    $true )).

fof(t2_modcat_1,axiom,(
    $true )).

fof(t3_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v2_mod_2(B,A)
            & v3_mod_2(B,A)
            & l1_mod_2(B,A) )
         => m3_modcat_1(k1_tarski(B),A) ) ) )).

fof(d3_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v12_vectsp_1(B,A)
            & l4_vectsp_1(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] :
                  ( m3_modcat_1(D,A)
                 => ( m5_modcat_1(D,A,B,C)
                  <=> ! [E] :
                        ( m4_modcat_1(E,A,D)
                       => ( v2_mod_2(E,A)
                          & m1_mod_2(E,A,B,C) ) ) ) ) ) ) ) )).

fof(t4_modcat_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,B)
                & l4_vectsp_1(C,B) )
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v3_rlvect_1(D)
                    & v4_rlvect_1(D)
                    & v5_rlvect_1(D)
                    & v6_rlvect_1(D)
                    & v12_vectsp_1(D,B)
                    & l4_vectsp_1(D,B) )
                 => ( m5_modcat_1(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,A)
                       => ( v2_mod_2(E,B)
                          & m1_mod_2(E,B,C,D) ) ) ) ) ) ) ) )).

fof(t5_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v12_vectsp_1(B,A)
            & l4_vectsp_1(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] :
                  ( ( v2_mod_2(D,A)
                    & m1_mod_2(D,A,B,C) )
                 => m5_modcat_1(k1_tarski(D),A,B,C) ) ) ) ) )).

fof(d4_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v12_vectsp_1(B,A)
            & l4_vectsp_1(B,A) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v12_vectsp_1(C,A)
                & l4_vectsp_1(C,A) )
             => ! [D] :
                  ( m5_modcat_1(D,A,B,C)
                 => ( D = k1_modcat_1(A,B,C)
                  <=> ! [E] :
                        ( r2_hidden(E,D)
                      <=> ( v2_mod_2(E,A)
                          & m1_mod_2(E,A,B,C) ) ) ) ) ) ) ) )).

fof(d5_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v4_group_1(C)
        & v6_vectsp_1(C)
        & v7_vectsp_1(C)
        & v8_vectsp_1(C)
        & l3_vectsp_1(C) )
     => ( r1_modcat_1(A,B,C)
      <=> ? [D,E] :
            ( A = k4_tarski(D,E)
            & ? [F] :
                ( ~ v3_struct_0(F)
                & v3_rlvect_1(F)
                & v4_rlvect_1(F)
                & v5_rlvect_1(F)
                & v6_rlvect_1(F)
                & v11_vectsp_1(F,C)
                & v12_vectsp_1(F,C)
                & l4_vectsp_1(F,C)
                & B = F
                & D = g1_rlvect_1(u1_struct_0(F),u1_rlvect_1(F),u2_struct_0(F))
                & E = u2_vectsp_1(C,F) ) ) ) ) )).

fof(t6_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B,C,D] :
          ( ( r1_modcat_1(B,C,A)
            & r1_modcat_1(B,D,A) )
         => C = D ) ) )).

fof(t8_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => r2_hidden(k1_mod_2(B),k2_modcat_1(A,B)) ) ) )).

fof(t9_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,k2_modcat_1(A,B))
             => ( ~ v3_struct_0(C)
                & v3_rlvect_1(C)
                & v4_rlvect_1(C)
                & v5_rlvect_1(C)
                & v6_rlvect_1(C)
                & v11_vectsp_1(C,B)
                & v12_vectsp_1(C,B)
                & l4_vectsp_1(C,B) ) ) ) ) )).

fof(d7_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m3_modcat_1(C,A)
             => ( C = k4_modcat_1(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,C)
                  <=> ? [E] :
                        ( v11_vectsp_1(E,A)
                        & m2_modcat_1(E,A,B)
                        & ? [F] :
                            ( v11_vectsp_1(F,A)
                            & m2_modcat_1(F,A,B)
                            & v2_mod_2(D,A)
                            & m1_mod_2(D,A,E,F) ) ) ) ) ) ) ) )).

fof(d8_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m4_modcat_1(C,A,k4_modcat_1(A,B))
             => k5_modcat_1(A,B,C) = k2_mod_2(A,C) ) ) ) )).

fof(d9_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m4_modcat_1(C,A,k4_modcat_1(A,B))
             => k6_modcat_1(A,B,C) = k3_mod_2(A,C) ) ) ) )).

fof(d10_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m2_modcat_1(C,A,B)
             => k7_modcat_1(A,B,C) = k6_mod_2(A,C) ) ) ) )).

fof(d11_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k4_modcat_1(A,B),B)
                & m2_relset_1(C,k4_modcat_1(A,B),B) )
             => ( C = k8_modcat_1(A,B)
              <=> ! [D] :
                    ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                   => k8_funct_2(k4_modcat_1(A,B),B,C,D) = k5_modcat_1(A,B,D) ) ) ) ) ) )).

fof(d12_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k4_modcat_1(A,B),B)
                & m2_relset_1(C,k4_modcat_1(A,B),B) )
             => ( C = k9_modcat_1(A,B)
              <=> ! [D] :
                    ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                   => k8_funct_2(k4_modcat_1(A,B),B,C,D) = k6_modcat_1(A,B,D) ) ) ) ) ) )).

fof(d13_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k4_modcat_1(A,B))
                & m2_relset_1(C,B,k4_modcat_1(A,B)) )
             => ( C = k10_modcat_1(A,B)
              <=> ! [D] :
                    ( m2_modcat_1(D,A,B)
                   => k8_funct_2(B,k4_modcat_1(A,B),C,D) = k7_modcat_1(A,B,D) ) ) ) ) ) )).

fof(t10_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m4_modcat_1(C,A,k4_modcat_1(A,B))
             => ! [D] :
                  ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                 => ~ ( k5_modcat_1(A,B,C) = k6_modcat_1(A,B,D)
                      & ! [E] :
                          ( ( v11_vectsp_1(E,A)
                            & m2_modcat_1(E,A,B) )
                         => ! [F] :
                              ( ( v11_vectsp_1(F,A)
                                & m2_modcat_1(F,A,B) )
                             => ! [G] :
                                  ( ( v11_vectsp_1(G,A)
                                    & m2_modcat_1(G,A,B) )
                                 => ~ ( m1_mod_2(C,A,F,G)
                                      & m1_mod_2(D,A,E,F) ) ) ) ) ) ) ) ) ) )).

fof(t11_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m4_modcat_1(C,A,k4_modcat_1(A,B))
             => ! [D] :
                  ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                 => ( k5_modcat_1(A,B,C) = k6_modcat_1(A,B,D)
                   => r2_hidden(k8_mod_2(A,C,D),k4_modcat_1(A,B)) ) ) ) ) ) )).

fof(t12_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m4_modcat_1(C,A,k4_modcat_1(A,B))
             => ! [D] :
                  ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                 => ( k2_mod_2(A,C) = k3_mod_2(A,D)
                   => r2_hidden(k8_mod_2(A,C,D),k4_modcat_1(A,B)) ) ) ) ) ) )).

fof(d14_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k2_zfmisc_1(k4_modcat_1(A,B),k4_modcat_1(A,B)),k4_modcat_1(A,B)) )
             => ( C = k11_modcat_1(A,B)
              <=> ( ! [D] :
                      ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                     => ! [E] :
                          ( m4_modcat_1(E,A,k4_modcat_1(A,B))
                         => ( r2_hidden(k4_tarski(D,E),k1_relat_1(C))
                          <=> k5_modcat_1(A,B,D) = k6_modcat_1(A,B,E) ) ) )
                  & ! [D] :
                      ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                     => ! [E] :
                          ( m4_modcat_1(E,A,k4_modcat_1(A,B))
                         => ( r2_hidden(k4_tarski(D,E),k1_relat_1(C))
                           => k1_funct_1(C,k4_tarski(D,E)) = k8_mod_2(A,D,E) ) ) ) ) ) ) ) ) )).

fof(t13_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ! [C] :
              ( m4_modcat_1(C,A,k4_modcat_1(A,B))
             => ! [D] :
                  ( m4_modcat_1(D,A,k4_modcat_1(A,B))
                 => ( r2_hidden(k4_tarski(C,D),k1_relat_1(k11_modcat_1(A,B)))
                  <=> k2_mod_2(A,C) = k3_mod_2(A,D) ) ) ) ) ) )).

fof(d15_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => k12_modcat_1(A,B) = g1_cat_1(k3_modcat_1(A,B),k4_modcat_1(B,k3_modcat_1(A,B)),k8_modcat_1(B,k3_modcat_1(A,B)),k9_modcat_1(B,k3_modcat_1(A,B)),k11_modcat_1(B,k3_modcat_1(A,B)),k10_modcat_1(B,k3_modcat_1(A,B))) ) ) )).

fof(t14_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u2_cat_1(k12_modcat_1(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u2_cat_1(k12_modcat_1(A,B)))
                 => ( r2_hidden(k4_tarski(D,C),k1_relat_1(u5_cat_1(k12_modcat_1(A,B))))
                  <=> k2_cat_1(k12_modcat_1(A,B),D) = k3_cat_1(k12_modcat_1(A,B),C) ) ) ) ) ) )).

fof(t15_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u2_cat_1(k12_modcat_1(A,B)))
             => ! [D] :
                  ( m4_modcat_1(D,B,k4_modcat_1(B,k3_modcat_1(A,B)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_cat_1(k12_modcat_1(A,B)))
                     => ! [F] :
                          ( m2_modcat_1(F,B,k3_modcat_1(A,B))
                         => ( v2_mod_2(C,B)
                            & m4_modcat_1(C,B,k4_modcat_1(B,k3_modcat_1(A,B)))
                            & m1_subset_1(D,u2_cat_1(k12_modcat_1(A,B)))
                            & v11_vectsp_1(E,B)
                            & m2_modcat_1(E,B,k3_modcat_1(A,B))
                            & m1_subset_1(F,u1_cat_1(k12_modcat_1(A,B))) ) ) ) ) ) ) ) )).

fof(t16_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_cat_1(k12_modcat_1(A,B)))
             => ! [D] :
                  ( m2_modcat_1(D,B,k3_modcat_1(A,B))
                 => ( C = D
                   => k5_cat_1(k12_modcat_1(A,B),C) = k7_modcat_1(B,k3_modcat_1(A,B),D) ) ) ) ) ) )).

fof(t17_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u2_cat_1(k12_modcat_1(A,B)))
             => ! [D] :
                  ( m4_modcat_1(D,B,k4_modcat_1(B,k3_modcat_1(A,B)))
                 => ( C = D
                   => ( k2_cat_1(k12_modcat_1(A,B),C) = k2_mod_2(B,D)
                      & k3_cat_1(k12_modcat_1(A,B),C) = k3_mod_2(B,D) ) ) ) ) ) ) )).

fof(t18_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u2_cat_1(k12_modcat_1(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u2_cat_1(k12_modcat_1(A,B)))
                 => ! [E] :
                      ( m4_modcat_1(E,B,k4_modcat_1(B,k3_modcat_1(A,B)))
                     => ! [F] :
                          ( m4_modcat_1(F,B,k4_modcat_1(B,k3_modcat_1(A,B)))
                         => ( ( C = E
                              & D = F )
                           => ( ( k2_cat_1(k12_modcat_1(A,B),D) = k3_cat_1(k12_modcat_1(A,B),C)
                               => k2_mod_2(B,F) = k3_mod_2(B,E) )
                              & ( k2_mod_2(B,F) = k3_mod_2(B,E)
                               => k2_cat_1(k12_modcat_1(A,B),D) = k3_cat_1(k12_modcat_1(A,B),C) )
                              & ( k2_cat_1(k12_modcat_1(A,B),D) = k3_cat_1(k12_modcat_1(A,B),C)
                               => r2_hidden(k4_tarski(F,E),k1_relat_1(k11_modcat_1(B,k3_modcat_1(A,B)))) )
                              & ( r2_hidden(k4_tarski(F,E),k1_relat_1(k11_modcat_1(B,k3_modcat_1(A,B))))
                               => k2_cat_1(k12_modcat_1(A,B),D) = k3_cat_1(k12_modcat_1(A,B),C) )
                              & ( k2_cat_1(k12_modcat_1(A,B),D) = k3_cat_1(k12_modcat_1(A,B),C)
                               => k4_cat_1(k12_modcat_1(A,B),C,D) = k8_mod_2(B,F,E) )
                              & ( k2_cat_1(k12_modcat_1(A,B),C) = k2_cat_1(k12_modcat_1(A,B),D)
                               => k2_mod_2(B,E) = k2_mod_2(B,F) )
                              & ( k2_mod_2(B,E) = k2_mod_2(B,F)
                               => k2_cat_1(k12_modcat_1(A,B),C) = k2_cat_1(k12_modcat_1(A,B),D) )
                              & ( k3_cat_1(k12_modcat_1(A,B),C) = k3_cat_1(k12_modcat_1(A,B),D)
                               => k3_mod_2(B,E) = k3_mod_2(B,F) )
                              & ( k3_mod_2(B,E) = k3_mod_2(B,F)
                               => k3_cat_1(k12_modcat_1(A,B),C) = k3_cat_1(k12_modcat_1(A,B),D) ) ) ) ) ) ) ) ) ) )).

fof(dt_m1_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_modcat_1(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(existence_m1_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ? [B] : m1_modcat_1(B,A) ) )).

fof(dt_m2_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ! [C] :
          ( m2_modcat_1(C,A,B)
         => ( ~ v3_struct_0(C)
            & v3_rlvect_1(C)
            & v4_rlvect_1(C)
            & v5_rlvect_1(C)
            & v6_rlvect_1(C)
            & v12_vectsp_1(C,A)
            & l4_vectsp_1(C,A) ) ) ) )).

fof(existence_m2_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ? [C] : m2_modcat_1(C,A,B) ) )).

fof(redefinition_m2_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ! [C] :
          ( m2_modcat_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) )).

fof(dt_m3_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m3_modcat_1(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(existence_m3_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ? [B] : m3_modcat_1(B,A) ) )).

fof(dt_m4_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m3_modcat_1(B,A) )
     => ! [C] :
          ( m4_modcat_1(C,A,B)
         => ( v3_mod_2(C,A)
            & l1_mod_2(C,A) ) ) ) )).

fof(existence_m4_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m3_modcat_1(B,A) )
     => ? [C] : m4_modcat_1(C,A,B) ) )).

fof(redefinition_m4_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m3_modcat_1(B,A) )
     => ! [C] :
          ( m4_modcat_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) )).

fof(dt_m5_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A) )
     => ! [D] :
          ( m5_modcat_1(D,A,B,C)
         => m3_modcat_1(D,A) ) ) )).

fof(existence_m5_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A) )
     => ? [D] : m5_modcat_1(D,A,B,C) ) )).

fof(dt_m6_modcat_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A)
        & m5_modcat_1(D,A,B,C) )
     => ! [E] :
          ( m6_modcat_1(E,A,B,C,D)
         => m1_mod_2(E,A,B,C) ) ) )).

fof(existence_m6_modcat_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A)
        & m5_modcat_1(D,A,B,C) )
     => ? [E] : m6_modcat_1(E,A,B,C,D) ) )).

fof(redefinition_m6_modcat_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A)
        & m5_modcat_1(D,A,B,C) )
     => ! [E] :
          ( m6_modcat_1(E,A,B,C,D)
        <=> m1_subset_1(E,D) ) ) )).

fof(dt_k1_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v12_vectsp_1(B,A)
        & l4_vectsp_1(B,A)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v12_vectsp_1(C,A)
        & l4_vectsp_1(C,A) )
     => m5_modcat_1(k1_modcat_1(A,B,C),A,B,C) ) )).

fof(dt_k2_modcat_1,axiom,(
    $true )).

fof(dt_k3_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B) )
     => m1_modcat_1(k3_modcat_1(A,B),B) ) )).

fof(redefinition_k3_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B) )
     => k3_modcat_1(A,B) = k2_modcat_1(A,B) ) )).

fof(dt_k4_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => m3_modcat_1(k4_modcat_1(A,B),A) ) )).

fof(dt_k5_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A)
        & m1_subset_1(C,k4_modcat_1(A,B)) )
     => m2_modcat_1(k5_modcat_1(A,B,C),A,B) ) )).

fof(dt_k6_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A)
        & m1_subset_1(C,k4_modcat_1(A,B)) )
     => m2_modcat_1(k6_modcat_1(A,B,C),A,B) ) )).

fof(dt_k7_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A)
        & m1_subset_1(C,B) )
     => ( v2_mod_2(k7_modcat_1(A,B,C),A)
        & m4_modcat_1(k7_modcat_1(A,B,C),A,k4_modcat_1(A,B)) ) ) )).

fof(dt_k8_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ( v1_funct_1(k8_modcat_1(A,B))
        & v1_funct_2(k8_modcat_1(A,B),k4_modcat_1(A,B),B)
        & m2_relset_1(k8_modcat_1(A,B),k4_modcat_1(A,B),B) ) ) )).

fof(dt_k9_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ( v1_funct_1(k9_modcat_1(A,B))
        & v1_funct_2(k9_modcat_1(A,B),k4_modcat_1(A,B),B)
        & m2_relset_1(k9_modcat_1(A,B),k4_modcat_1(A,B),B) ) ) )).

fof(dt_k10_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ( v1_funct_1(k10_modcat_1(A,B))
        & v1_funct_2(k10_modcat_1(A,B),B,k4_modcat_1(A,B))
        & m2_relset_1(k10_modcat_1(A,B),B,k4_modcat_1(A,B)) ) ) )).

fof(dt_k11_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_modcat_1(B,A) )
     => ( v1_funct_1(k11_modcat_1(A,B))
        & m2_relset_1(k11_modcat_1(A,B),k2_zfmisc_1(k4_modcat_1(A,B),k4_modcat_1(A,B)),k4_modcat_1(A,B)) ) ) )).

fof(dt_k12_modcat_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ( v1_cat_1(k12_modcat_1(A,B))
        & l1_cat_1(k12_modcat_1(A,B)) ) ) )).

fof(t7_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ? [C] :
              ( r2_hidden(C,a_2_0_modcat_1(A,B))
              & r1_modcat_1(C,k1_mod_2(A),A) ) ) ) )).

fof(d6_modcat_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_group_1(B)
            & v6_vectsp_1(B)
            & v7_vectsp_1(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( C = k2_modcat_1(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ? [E] :
                      ( r2_hidden(E,a_2_1_modcat_1(A,B))
                      & r1_modcat_1(E,D,B) ) ) ) ) ) )).

fof(fraenkel_a_2_0_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v4_group_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B)
        & ~ v1_xboole_0(C)
        & v1_classes2(C) )
     => ( r2_hidden(A,a_2_0_modcat_1(B,C))
      <=> ? [D,E] :
            ( m2_grcat_1(D,k19_grcat_1(C))
            & m2_fraenkel(E,k2_zfmisc_1(u1_struct_0(B),k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0),k1_fraenkel(k2_zfmisc_1(u1_struct_0(B),k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0)))
            & A = k4_tarski(D,E) ) ) ) )).

fof(fraenkel_a_2_1_modcat_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_classes2(B)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v4_group_1(C)
        & v6_vectsp_1(C)
        & v7_vectsp_1(C)
        & v8_vectsp_1(C)
        & l3_vectsp_1(C) )
     => ( r2_hidden(A,a_2_1_modcat_1(B,C))
      <=> ? [D,E] :
            ( m2_grcat_1(D,k19_grcat_1(B))
            & m2_fraenkel(E,k2_zfmisc_1(u1_struct_0(C),k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0),k1_fraenkel(k2_zfmisc_1(u1_struct_0(C),k1_tarski(k1_xboole_0)),k1_tarski(k1_xboole_0)))
            & A = k4_tarski(D,E) ) ) ) )).
%------------------------------------------------------------------------------