SET007 Axioms: SET007+280.ax


%------------------------------------------------------------------------------
% File     : SET007+280 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Linear Combinations in Right Module over Associative Ring
% 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    : rmod_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  112 (  21 unit)
%            Number of atoms       : 1942 (  81 equality)
%            Maximal formula depth :   26 (  12 average)
%            Number of connectives : 2030 ( 200 ~  ;   1  |;1475  &)
%                                         (  14 <=>; 340 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   35 (   1 propositional; 0-4 arity)
%            Number of functors    :   44 (   3 constant; 0-4 arity)
%            Number of variables   :  369 (   0 singleton; 362 !;   7 ?)
%            Maximal term depth    :    5 (   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(t1_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(B))
                     => ( ( k3_finseq_1(D) = k3_finseq_1(E)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(B))
                                 => ( ( r2_hidden(F,k1_relat_1(D))
                                      & G = k1_funct_1(E,F) )
                                   => k1_funct_1(D,F) = k6_vectsp_2(A,B,C,G) ) ) ) )
                       => k9_rlvect_1(B,D) = k6_vectsp_2(A,B,C,k9_rlvect_1(B,E)) ) ) ) ) ) ) )).

fof(t2_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k6_vectsp_2(A,B,C,k9_rlvect_1(B,k6_finseq_1(u1_struct_0(B)))) = k1_rlvect_1(B) ) ) ) )).

fof(t3_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => k6_vectsp_2(A,B,C,k9_rlvect_1(B,k2_finseq_4(u1_struct_0(B),D,E))) = k4_rlvect_1(B,k6_vectsp_2(A,B,C,D),k6_vectsp_2(A,B,C,E)) ) ) ) ) ) )).

fof(t4_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => k6_vectsp_2(A,B,C,k9_rlvect_1(B,k3_finseq_4(u1_struct_0(B),D,E,F))) = k4_rlvect_1(B,k4_rlvect_1(B,k6_vectsp_2(A,B,C,D),k6_vectsp_2(A,B,C,E)),k6_vectsp_2(A,B,C,F)) ) ) ) ) ) ) )).

fof(d1_rmod_4,axiom,(
    $true )).

fof(d2_rmod_4,axiom,(
    $true )).

fof(d3_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( D = k1_rmod_4(A,B,C)
                  <=> ? [E] :
                        ( m2_finseq_1(E,u1_struct_0(B))
                        & k2_relat_1(E) = C
                        & v2_funct_1(E)
                        & D = k9_rlvect_1(B,E) ) ) ) ) ) ) )).

fof(t5_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => k1_rmod_4(A,B,k1_pre_topc(B)) = k1_rlvect_1(B) ) ) )).

fof(t6_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => k1_rmod_4(A,B,k7_rlvect_2(B,C)) = C ) ) ) )).

fof(t7_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( C != D
                   => k1_rmod_4(A,B,k8_rlvect_2(B,C,D)) = k4_rlvect_1(B,C,D) ) ) ) ) ) )).

fof(t8_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ~ ( C != D
                          & D != E
                          & C != E
                          & k1_rmod_4(A,B,k9_rlvect_2(B,C,D,E)) != k4_rlvect_1(B,k4_rlvect_1(B,C,D),E) ) ) ) ) ) ) )).

fof(t9_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
                 => ( r1_xboole_0(C,D)
                   => k1_rmod_4(A,B,k2_rlvect_2(B,C,D)) = k4_rlvect_1(B,k1_rmod_4(A,B,C),k1_rmod_4(A,B,D)) ) ) ) ) ) )).

fof(t10_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
                 => k1_rmod_4(A,B,k2_rlvect_2(B,C,D)) = k6_rlvect_1(B,k4_rlvect_1(B,k1_rmod_4(A,B,C),k1_rmod_4(A,B,D)),k1_rmod_4(A,B,k3_rlvect_2(B,C,D))) ) ) ) ) )).

fof(t11_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
                 => k1_rmod_4(A,B,k3_rlvect_2(B,C,D)) = k6_rlvect_1(B,k4_rlvect_1(B,k1_rmod_4(A,B,C),k1_rmod_4(A,B,D)),k1_rmod_4(A,B,k2_rlvect_2(B,C,D))) ) ) ) ) )).

fof(t12_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
                 => k1_rmod_4(A,B,k4_rlvect_2(B,C,D)) = k6_rlvect_1(B,k1_rmod_4(A,B,k2_rlvect_2(B,C,D)),k1_rmod_4(A,B,D)) ) ) ) ) )).

fof(t13_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
                 => k1_rmod_4(A,B,k4_rlvect_2(B,C,D)) = k6_rlvect_1(B,k1_rmod_4(A,B,C),k1_rmod_4(A,B,k3_rlvect_2(B,C,D))) ) ) ) ) )).

fof(t14_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
                 => k1_rmod_4(A,B,k5_rlvect_2(B,C,D)) = k6_rlvect_1(B,k1_rmod_4(A,B,k2_rlvect_2(B,C,D)),k1_rmod_4(A,B,k3_rlvect_2(B,C,D))) ) ) ) ) )).

fof(t15_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
                 => k1_rmod_4(A,B,k5_rlvect_2(B,C,D)) = k4_rlvect_1(B,k1_rmod_4(A,B,k4_rlvect_2(B,C,D)),k1_rmod_4(A,B,k4_rlvect_2(B,D,C))) ) ) ) ) )).

fof(d4_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m2_fraenkel(C,u1_struct_0(B),u1_struct_0(A),k1_fraenkel(u1_struct_0(B),u1_struct_0(A)))
             => ( m1_rmod_4(C,A,B)
              <=> ? [D] :
                    ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                    & ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => ( ~ r2_hidden(E,D)
                         => k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E) = k1_rlvect_1(A) ) ) ) ) ) ) ) )).

fof(t16_rmod_4,axiom,(
    $true )).

fof(t17_rmod_4,axiom,(
    $true )).

fof(t18_rmod_4,axiom,(
    $true )).

fof(t19_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C,D] :
              ( m1_rmod_4(D,A,B)
             => ( r2_hidden(C,k2_rmod_4(A,B,D))
              <=> ? [E] :
                    ( m1_subset_1(E,u1_struct_0(B))
                    & C = E
                    & k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,E) != k1_rlvect_1(A) ) ) ) ) ) )).

fof(t20_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ( k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C) = k1_rlvect_1(A)
                  <=> ~ r2_hidden(C,k2_rmod_4(A,B,D)) ) ) ) ) ) )).

fof(d6_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ( C = k3_rmod_4(A,B)
              <=> k2_rmod_4(A,B,C) = k1_xboole_0 ) ) ) ) )).

fof(t21_rmod_4,axiom,(
    $true )).

fof(t22_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => k8_funct_2(u1_struct_0(B),u1_struct_0(A),k3_rmod_4(A,B),C) = k1_rlvect_1(A) ) ) ) )).

fof(d7_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ( m2_rmod_4(D,A,B,C)
                  <=> r1_tarski(k2_rmod_4(A,B,D),C) ) ) ) ) ) )).

fof(t23_rmod_4,axiom,(
    $true )).

fof(t24_rmod_4,axiom,(
    $true )).

fof(t25_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ! [E] :
                      ( m2_rmod_4(E,A,B,C)
                     => ( r1_tarski(C,D)
                       => m2_rmod_4(E,A,B,D) ) ) ) ) ) ) )).

fof(t26_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => m2_rmod_4(k3_rmod_4(A,B),A,B,C) ) ) ) )).

fof(t27_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m2_rmod_4(C,A,B,k1_subset_1(u1_struct_0(B)))
             => C = k3_rmod_4(A,B) ) ) ) )).

fof(t28_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => m2_rmod_4(C,A,B,k2_rmod_4(A,B,C)) ) ) ) )).

fof(d8_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(B))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(B))
                     => ( E = k4_rmod_4(A,B,C,D)
                      <=> ( k3_finseq_1(E) = k3_finseq_1(C)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ( r2_hidden(F,k1_relat_1(E))
                               => k1_funct_1(E,F) = k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,k4_finseq_4(k5_numbers,u1_struct_0(B),C,F)),k4_finseq_4(k5_numbers,u1_struct_0(B),C,F)) ) ) ) ) ) ) ) ) ) )).

fof(t29_rmod_4,axiom,(
    $true )).

fof(t30_rmod_4,axiom,(
    $true )).

fof(t31_rmod_4,axiom,(
    $true )).

fof(t32_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m2_finseq_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,u1_struct_0(B),u1_struct_0(A))
                            & m2_relset_1(F,u1_struct_0(B),u1_struct_0(A)) )
                         => ( ( r2_hidden(C,k1_relat_1(E))
                              & D = k1_funct_1(E,C) )
                           => k1_funct_1(k4_rmod_4(A,B,E,F),C) = k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),F,D),D) ) ) ) ) ) ) ) )).

fof(t33_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(B),u1_struct_0(A))
                & m2_relset_1(C,u1_struct_0(B),u1_struct_0(A)) )
             => k4_rmod_4(A,B,k6_finseq_1(u1_struct_0(B)),C) = k6_finseq_1(u1_struct_0(B)) ) ) ) )).

fof(t34_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
                 => k4_rmod_4(A,B,k12_finseq_1(u1_struct_0(B),C),D) = k12_finseq_1(u1_struct_0(B),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),C)) ) ) ) ) )).

fof(t35_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(A))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(A)) )
                     => k4_rmod_4(A,B,k2_finseq_4(u1_struct_0(B),C,D),E) = k2_finseq_4(u1_struct_0(B),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C),C),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,D),D)) ) ) ) ) ) )).

fof(t36_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,u1_struct_0(B),u1_struct_0(A))
                            & m2_relset_1(F,u1_struct_0(B),u1_struct_0(A)) )
                         => k4_rmod_4(A,B,k3_finseq_4(u1_struct_0(B),C,D,E),F) = k3_finseq_4(u1_struct_0(B),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),F,C),C),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),F,D),D),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),F,E),E)) ) ) ) ) ) ) )).

fof(t37_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(B),u1_struct_0(A))
                        & m2_relset_1(E,u1_struct_0(B),u1_struct_0(A)) )
                     => k4_rmod_4(A,B,k8_finseq_1(u1_struct_0(B),C,D),E) = k8_finseq_1(u1_struct_0(B),k4_rmod_4(A,B,C,E),k4_rmod_4(A,B,D,E)) ) ) ) ) ) )).

fof(d9_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( D = k5_rmod_4(A,B,C)
                  <=> ? [E] :
                        ( m2_finseq_1(E,u1_struct_0(B))
                        & v2_funct_1(E)
                        & k2_relat_1(E) = k2_rmod_4(A,B,C)
                        & D = k9_rlvect_1(B,k4_rmod_4(A,B,E,C)) ) ) ) ) ) ) )).

fof(t38_rmod_4,axiom,(
    $true )).

fof(t39_rmod_4,axiom,(
    $true )).

fof(t40_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( k1_rlvect_1(A) != k2_group_1(A)
               => ( ( C != k1_xboole_0
                    & v1_rmod_2(C,A,B) )
                <=> ! [D] :
                      ( m2_rmod_4(D,A,B,C)
                     => r2_hidden(k5_rmod_4(A,B,D),C) ) ) ) ) ) ) )).

fof(t41_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => k5_rmod_4(A,B,k3_rmod_4(A,B)) = k1_rlvect_1(B) ) ) )).

fof(t42_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m2_rmod_4(C,A,B,k1_subset_1(u1_struct_0(B)))
             => k5_rmod_4(A,B,C) = k1_rlvect_1(B) ) ) ) )).

fof(t43_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m2_rmod_4(D,A,B,k7_rlvect_2(B,C))
                 => k5_rmod_4(A,B,D) = k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),C) ) ) ) ) )).

fof(t44_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( C != D
                   => ! [E] :
                        ( m2_rmod_4(E,A,B,k8_rlvect_2(B,C,D))
                       => k5_rmod_4(A,B,E) = k4_rlvect_1(B,k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C),C),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,D),D)) ) ) ) ) ) ) )).

fof(t45_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ( k2_rmod_4(A,B,C) = k1_xboole_0
               => k5_rmod_4(A,B,C) = k1_rlvect_1(B) ) ) ) ) )).

fof(t46_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ( k2_rmod_4(A,B,D) = k7_rlvect_2(B,C)
                   => k5_rmod_4(A,B,D) = k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),C) ) ) ) ) ) )).

fof(t47_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => ( k2_rmod_4(A,B,E) = k8_rlvect_2(B,C,D)
                       => ( C = D
                          | k5_rmod_4(A,B,E) = k4_rlvect_1(B,k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C),C),k6_vectsp_2(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,D),D)) ) ) ) ) ) ) ) )).

fof(d10_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ( r1_rmod_4(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E) = k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,E) ) ) ) ) ) ) )).

fof(d11_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => ( E = k6_rmod_4(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                           => k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,F) = k4_rlvect_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,F),k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,F)) ) ) ) ) ) ) ) )).

fof(t48_rmod_4,axiom,(
    $true )).

fof(t49_rmod_4,axiom,(
    $true )).

fof(t50_rmod_4,axiom,(
    $true )).

fof(t51_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => r1_tarski(k2_rmod_4(A,B,k6_rmod_4(A,B,C,D)),k2_rlvect_2(B,k2_rmod_4(A,B,C),k2_rmod_4(A,B,D))) ) ) ) ) )).

fof(t52_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => ( ( m2_rmod_4(D,A,B,C)
                          & m2_rmod_4(E,A,B,C) )
                       => m2_rmod_4(k6_rmod_4(A,B,D,E),A,B,C) ) ) ) ) ) ) )).

fof(t53_rmod_4,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)
        & v7_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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => r1_rmod_4(A,B,k6_rmod_4(A,B,C,D),k6_rmod_4(A,B,D,C)) ) ) ) ) )).

fof(t54_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => r1_rmod_4(A,B,k6_rmod_4(A,B,C,k6_rmod_4(A,B,D,E)),k6_rmod_4(A,B,k6_rmod_4(A,B,C,D),E)) ) ) ) ) ) )).

fof(t55_rmod_4,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)
        & v7_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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ( r1_rmod_4(A,B,k6_rmod_4(A,B,C,k3_rmod_4(A,B)),C)
                & r1_rmod_4(A,B,k6_rmod_4(A,B,k3_rmod_4(A,B),C),C) ) ) ) ) )).

fof(d12_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => ( E = k7_rmod_4(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                           => k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,F) = k1_group_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,F),C) ) ) ) ) ) ) ) )).

fof(t56_rmod_4,axiom,(
    $true )).

fof(t57_rmod_4,axiom,(
    $true )).

fof(t58_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => r1_tarski(k2_rmod_4(A,B,k7_rmod_4(A,B,C,D)),k2_rmod_4(A,B,D)) ) ) ) ) )).

fof(t59_rmod_4,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)
        & v7_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & v2_vectsp_2(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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( D != k1_rlvect_1(A)
                   => k2_rmod_4(A,B,k7_rmod_4(A,B,D,C)) = k2_rmod_4(A,B,C) ) ) ) ) ) )).

fof(t60_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => r1_rmod_4(A,B,k7_rmod_4(A,B,k1_rlvect_1(A),C),k3_rmod_4(A,B)) ) ) ) )).

fof(t61_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => ( m2_rmod_4(E,A,B,D)
                       => m2_rmod_4(k7_rmod_4(A,B,C,E),A,B,D) ) ) ) ) ) ) )).

fof(t62_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => r1_rmod_4(A,B,k7_rmod_4(A,B,k4_rlvect_1(A,C,D),E),k6_rmod_4(A,B,k7_rmod_4(A,B,C,E),k7_rmod_4(A,B,D,E))) ) ) ) ) ) )).

fof(t63_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => r1_rmod_4(A,B,k7_rmod_4(A,B,C,k6_rmod_4(A,B,D,E)),k6_rmod_4(A,B,k7_rmod_4(A,B,C,D),k7_rmod_4(A,B,C,E))) ) ) ) ) ) )).

fof(t64_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => r1_rmod_4(A,B,k7_rmod_4(A,B,D,k7_rmod_4(A,B,C,E)),k7_rmod_4(A,B,k1_group_1(A,C,D),E)) ) ) ) ) ) )).

fof(t65_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => r1_rmod_4(A,B,k7_rmod_4(A,B,k2_group_1(A),C),C) ) ) ) )).

fof(d13_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => k8_rmod_4(A,B,C) = k7_rmod_4(A,B,k5_rlvect_1(A,k2_group_1(A)),C) ) ) ) )).

fof(t66_rmod_4,axiom,(
    $true )).

fof(t67_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => k8_funct_2(u1_struct_0(B),u1_struct_0(A),k8_rmod_4(A,B,D),C) = k5_rlvect_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C)) ) ) ) ) )).

fof(t68_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ( r1_rmod_4(A,B,k6_rmod_4(A,B,C,D),k3_rmod_4(A,B))
                   => r1_rmod_4(A,B,D,k8_rmod_4(A,B,C)) ) ) ) ) ) )).

fof(t69_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => k2_rmod_4(A,B,k8_rmod_4(A,B,C)) = k2_rmod_4(A,B,C) ) ) ) )).

fof(t70_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ( m2_rmod_4(D,A,B,C)
                   => m2_rmod_4(k8_rmod_4(A,B,D),A,B,C) ) ) ) ) ) )).

fof(d14_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => k9_rmod_4(A,B,C,D) = k6_rmod_4(A,B,C,k8_rmod_4(A,B,D)) ) ) ) ) )).

fof(t71_rmod_4,axiom,(
    $true )).

fof(t72_rmod_4,axiom,(
    $true )).

fof(t73_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => k8_funct_2(u1_struct_0(B),u1_struct_0(A),k9_rmod_4(A,B,D,E),C) = k6_rlvect_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C)) ) ) ) ) ) )).

fof(t74_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => r1_tarski(k2_rmod_4(A,B,k9_rmod_4(A,B,C,D)),k2_rlvect_2(B,k2_rmod_4(A,B,C),k2_rmod_4(A,B,D))) ) ) ) ) )).

fof(t75_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => ! [E] :
                      ( m1_rmod_4(E,A,B)
                     => ( ( m2_rmod_4(D,A,B,C)
                          & m2_rmod_4(E,A,B,C) )
                       => m2_rmod_4(k9_rmod_4(A,B,D,E),A,B,C) ) ) ) ) ) ) )).

fof(t76_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => r1_rmod_4(A,B,k9_rmod_4(A,B,C,C),k3_rmod_4(A,B)) ) ) ) )).

fof(t77_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => k5_rmod_4(A,B,k6_rmod_4(A,B,C,D)) = k4_rlvect_1(B,k5_rmod_4(A,B,C),k5_rmod_4(A,B,D)) ) ) ) ) )).

fof(t78_rmod_4,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)
        & v7_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & v2_vectsp_2(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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k5_rmod_4(A,B,k7_rmod_4(A,B,D,C)) = k6_vectsp_2(A,B,D,k5_rmod_4(A,B,C)) ) ) ) ) )).

fof(t79_rmod_4,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)
        & v7_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & v2_vectsp_2(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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => k5_rmod_4(A,B,k8_rmod_4(A,B,C)) = k5_rlvect_1(B,k5_rmod_4(A,B,C)) ) ) ) )).

fof(t80_rmod_4,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)
        & v7_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & v2_vectsp_2(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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => ! [D] :
                  ( m1_rmod_4(D,A,B)
                 => k5_rmod_4(A,B,k9_rmod_4(A,B,C,D)) = k6_rlvect_1(B,k5_rmod_4(A,B,C),k5_rmod_4(A,B,D)) ) ) ) ) )).

fof(dt_m1_rmod_4,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)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A) )
     => ! [C] :
          ( m1_rmod_4(C,A,B)
         => m2_fraenkel(C,u1_struct_0(B),u1_struct_0(A),k1_fraenkel(u1_struct_0(B),u1_struct_0(A))) ) ) )).

fof(existence_m1_rmod_4,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)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A) )
     => ? [C] : m1_rmod_4(C,A,B) ) )).

fof(dt_m2_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ! [D] :
          ( m2_rmod_4(D,A,B,C)
         => m1_rmod_4(D,A,B) ) ) )).

fof(existence_m2_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ? [D] : m2_rmod_4(D,A,B,C) ) )).

fof(symmetry_r1_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B)
        & m1_rmod_4(D,A,B) )
     => ( r1_rmod_4(A,B,C,D)
       => r1_rmod_4(A,B,D,C) ) ) )).

fof(reflexivity_r1_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B)
        & m1_rmod_4(D,A,B) )
     => r1_rmod_4(A,B,C,C) ) )).

fof(redefinition_r1_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B)
        & m1_rmod_4(D,A,B) )
     => ( r1_rmod_4(A,B,C,D)
      <=> C = D ) ) )).

fof(dt_k1_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & v1_finset_1(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => m1_subset_1(k1_rmod_4(A,B,C),u1_struct_0(B)) ) )).

fof(dt_k2_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B) )
     => ( v1_finset_1(k2_rmod_4(A,B,C))
        & m1_subset_1(k2_rmod_4(A,B,C),k1_zfmisc_1(u1_struct_0(B))) ) ) )).

fof(dt_k3_rmod_4,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)
        & ~ v3_struct_0(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A) )
     => m1_rmod_4(k3_rmod_4(A,B),A,B) ) )).

fof(dt_k4_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_finseq_1(C,u1_struct_0(B))
        & v1_funct_1(D)
        & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
        & m1_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
     => m2_finseq_1(k4_rmod_4(A,B,C,D),u1_struct_0(B)) ) )).

fof(dt_k5_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B) )
     => m1_subset_1(k5_rmod_4(A,B,C),u1_struct_0(B)) ) )).

fof(dt_k6_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B)
        & m1_rmod_4(D,A,B) )
     => m1_rmod_4(k6_rmod_4(A,B,C,D),A,B) ) )).

fof(dt_k7_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_subset_1(C,u1_struct_0(A))
        & m1_rmod_4(D,A,B) )
     => m1_rmod_4(k7_rmod_4(A,B,C,D),A,B) ) )).

fof(dt_k8_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B) )
     => m1_rmod_4(k8_rmod_4(A,B,C),A,B) ) )).

fof(involutiveness_k8_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B) )
     => k8_rmod_4(A,B,k8_rmod_4(A,B,C)) = C ) )).

fof(dt_k9_rmod_4,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)
        & v5_vectsp_2(B,A)
        & l1_vectsp_2(B,A)
        & m1_rmod_4(C,A,B)
        & m1_rmod_4(D,A,B) )
     => m1_rmod_4(k9_rmod_4(A,B,C,D),A,B) ) )).

fof(d5_rmod_4,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)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_4(C,A,B)
             => k2_rmod_4(A,B,C) = a_3_0_rmod_4(A,B,C) ) ) ) )).

fof(fraenkel_a_3_0_rmod_4,axiom,(
    ! [A,B,C,D] :
      ( ( ~ 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)
        & ~ v3_struct_0(C)
        & v3_rlvect_1(C)
        & v4_rlvect_1(C)
        & v5_rlvect_1(C)
        & v6_rlvect_1(C)
        & v5_vectsp_2(C,B)
        & l1_vectsp_2(C,B)
        & m1_rmod_4(D,B,C) )
     => ( r2_hidden(A,a_3_0_rmod_4(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(C))
            & A = E
            & k8_funct_2(u1_struct_0(C),u1_struct_0(B),D,E) != k1_rlvect_1(B) ) ) ) )).
%------------------------------------------------------------------------------