SET007 Axioms: SET007+279.ax


%------------------------------------------------------------------------------
% File     : SET007+279 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Operations on Submodules 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   82 (  17 unit)
%            Number of atoms       : 1462 (  64 equality)
%            Maximal formula depth :   29 (  12 average)
%            Number of connectives : 1523 ( 143 ~  ;   3  |;1091  &)
%                                         (  15 <=>; 271 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   1 propositional; 0-4 arity)
%            Number of functors    :   24 (   0 constant; 1-6 arity)
%            Number of variables   :  284 (   0 singleton; 271 !;  13 ?)
%            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(fc1_rmod_3,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) )
     => ~ v1_xboole_0(k3_rmod_3(A,B)) ) )).

fof(d2_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( ( v3_vectsp_2(E,A)
                        & m1_rmod_2(E,A,B) )
                     => ( E = k2_rmod_3(A,B,C,D)
                      <=> u1_struct_0(E) = k3_xboole_0(u1_struct_0(C),u1_struct_0(D)) ) ) ) ) ) ) )).

fof(t1_rmod_3,axiom,(
    $true )).

fof(t2_rmod_3,axiom,(
    $true )).

fof(t3_rmod_3,axiom,(
    $true )).

fof(t4_rmod_3,axiom,(
    $true )).

fof(t5_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( r1_rlvect_1(k1_rmod_3(A,B,C,D),E)
                    <=> ? [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                          & ? [G] :
                              ( m1_subset_1(G,u1_struct_0(B))
                              & r1_rlvect_1(C,F)
                              & r1_rlvect_1(D,G)
                              & E = k4_rlvect_1(B,F,G) ) ) ) ) ) ) ) )).

fof(t6_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ( ( r1_rlvect_1(C,E)
                          | r1_rlvect_1(D,E) )
                       => r1_rlvect_1(k1_rmod_3(A,B,C,D),E) ) ) ) ) ) ) )).

fof(t7_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( r1_rlvect_1(k2_rmod_3(A,B,C,D),E)
                    <=> ( r1_rlvect_1(C,E)
                        & r1_rlvect_1(D,E) ) ) ) ) ) ) )).

fof(t8_rmod_3,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] :
              ( ( v3_vectsp_2(C,A)
                & m1_rmod_2(C,A,B) )
             => k1_rmod_3(A,B,C,C) = C ) ) ) )).

fof(t9_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => k1_rmod_3(A,B,C,D) = k1_rmod_3(A,B,D,C) ) ) ) ) )).

fof(t10_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => k1_rmod_3(A,B,C,k1_rmod_3(A,B,D,E)) = k1_rmod_3(A,B,k1_rmod_3(A,B,C,D),E) ) ) ) ) ) )).

fof(t11_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( m1_rmod_2(C,A,k1_rmod_3(A,B,C,D))
                    & m1_rmod_2(D,A,k1_rmod_3(A,B,C,D)) ) ) ) ) ) )).

fof(t12_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( ( v3_vectsp_2(D,A)
                    & m1_rmod_2(D,A,B) )
                 => ( m1_rmod_2(C,A,D)
                  <=> k1_rmod_3(A,B,C,D) = D ) ) ) ) ) )).

fof(t13_rmod_3,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] :
              ( ( v3_vectsp_2(C,A)
                & m1_rmod_2(C,A,B) )
             => ( k1_rmod_3(A,B,k1_rmod_2(A,B),C) = C
                & k1_rmod_3(A,B,C,k1_rmod_2(A,B)) = C ) ) ) ) )).

fof(t14_rmod_3,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)
            & v3_vectsp_2(B,A)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ( k1_rmod_3(A,B,k1_rmod_2(A,B),k2_rmod_2(A,B)) = B
            & k1_rmod_3(A,B,k2_rmod_2(A,B),k1_rmod_2(A,B)) = B ) ) ) )).

fof(t15_rmod_3,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_2(C,A,B)
             => ( k1_rmod_3(A,B,k2_rmod_2(A,B),C) = g1_vectsp_2(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u1_vectsp_2(A,B))
                & k1_rmod_3(A,B,C,k2_rmod_2(A,B)) = g1_vectsp_2(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u1_vectsp_2(A,B)) ) ) ) ) )).

fof(t16_rmod_3,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)
            & v3_vectsp_2(B,A)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => k1_rmod_3(A,B,k2_rmod_2(A,B),k2_rmod_2(A,B)) = B ) ) )).

fof(t17_rmod_3,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] :
              ( ( v3_vectsp_2(C,A)
                & m1_rmod_2(C,A,B) )
             => k2_rmod_3(A,B,C,C) = C ) ) ) )).

fof(t18_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => k2_rmod_3(A,B,C,D) = k2_rmod_3(A,B,D,C) ) ) ) ) )).

fof(t19_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => k2_rmod_3(A,B,C,k2_rmod_3(A,B,D,E)) = k2_rmod_3(A,B,k2_rmod_3(A,B,C,D),E) ) ) ) ) ) )).

fof(t20_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( m1_rmod_2(k2_rmod_3(A,B,C,D),A,C)
                    & m1_rmod_2(k2_rmod_3(A,B,C,D),A,D) ) ) ) ) ) )).

fof(t21_rmod_3,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_2(C,A,B)
             => ( ! [D] :
                    ( ( v3_vectsp_2(D,A)
                      & m1_rmod_2(D,A,B) )
                   => ( m1_rmod_2(D,A,C)
                     => k2_rmod_3(A,B,D,C) = D ) )
                & ! [D] :
                    ( m1_rmod_2(D,A,B)
                   => ( k2_rmod_3(A,B,D,C) = D
                     => m1_rmod_2(D,A,C) ) ) ) ) ) ) )).

fof(t22_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( m1_rmod_2(C,A,D)
                       => m1_rmod_2(k2_rmod_3(A,B,C,E),A,k2_rmod_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t23_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( m1_rmod_2(C,A,D)
                       => m1_rmod_2(k2_rmod_3(A,B,C,E),A,D) ) ) ) ) ) ) )).

fof(t24_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( ( m1_rmod_2(C,A,D)
                          & m1_rmod_2(C,A,E) )
                       => m1_rmod_2(C,A,k2_rmod_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t25_rmod_3,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_2(C,A,B)
             => ( k2_rmod_3(A,B,k1_rmod_2(A,B),C) = k1_rmod_2(A,B)
                & k2_rmod_3(A,B,C,k1_rmod_2(A,B)) = k1_rmod_2(A,B) ) ) ) ) )).

fof(t26_rmod_3,axiom,(
    $true )).

fof(t27_rmod_3,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] :
              ( ( v3_vectsp_2(C,A)
                & m1_rmod_2(C,A,B) )
             => ( k2_rmod_3(A,B,k2_rmod_2(A,B),C) = C
                & k2_rmod_3(A,B,C,k2_rmod_2(A,B)) = C ) ) ) ) )).

fof(t28_rmod_3,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)
            & v3_vectsp_2(B,A)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => k2_rmod_3(A,B,k2_rmod_2(A,B),k2_rmod_2(A,B)) = B ) ) )).

fof(t29_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => m1_rmod_2(k2_rmod_3(A,B,C,D),A,k1_rmod_3(A,B,C,D)) ) ) ) ) )).

fof(t30_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( ( v3_vectsp_2(D,A)
                    & m1_rmod_2(D,A,B) )
                 => k1_rmod_3(A,B,k2_rmod_3(A,B,C,D),D) = D ) ) ) ) )).

fof(t31_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( ( v3_vectsp_2(D,A)
                    & m1_rmod_2(D,A,B) )
                 => k2_rmod_3(A,B,D,k1_rmod_3(A,B,D,C)) = D ) ) ) ) )).

fof(t32_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => m1_rmod_2(k1_rmod_3(A,B,k2_rmod_3(A,B,C,D),k2_rmod_3(A,B,D,E)),A,k2_rmod_3(A,B,D,k1_rmod_3(A,B,C,E))) ) ) ) ) ) )).

fof(t33_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( m1_rmod_2(C,A,D)
                       => k2_rmod_3(A,B,D,k1_rmod_3(A,B,C,E)) = k1_rmod_3(A,B,k2_rmod_3(A,B,C,D),k2_rmod_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t34_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => m1_rmod_2(k1_rmod_3(A,B,C,k2_rmod_3(A,B,D,E)),A,k2_rmod_3(A,B,k1_rmod_3(A,B,D,C),k1_rmod_3(A,B,C,E))) ) ) ) ) ) )).

fof(t35_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( m1_rmod_2(C,A,D)
                       => k1_rmod_3(A,B,D,k2_rmod_3(A,B,C,E)) = k2_rmod_3(A,B,k1_rmod_3(A,B,C,D),k1_rmod_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t36_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( ( v3_vectsp_2(E,A)
                        & m1_rmod_2(E,A,B) )
                     => ( m1_rmod_2(E,A,C)
                       => k1_rmod_3(A,B,E,k2_rmod_3(A,B,D,C)) = k2_rmod_3(A,B,k1_rmod_3(A,B,E,D),C) ) ) ) ) ) ) )).

fof(t37_rmod_3,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] :
              ( ( v3_vectsp_2(C,A)
                & m1_rmod_2(C,A,B) )
             => ! [D] :
                  ( ( v3_vectsp_2(D,A)
                    & m1_rmod_2(D,A,B) )
                 => ( k1_rmod_3(A,B,C,D) = D
                  <=> k2_rmod_3(A,B,C,D) = C ) ) ) ) ) )).

fof(t38_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( ( v3_vectsp_2(D,A)
                    & m1_rmod_2(D,A,B) )
                 => ! [E] :
                      ( ( v3_vectsp_2(E,A)
                        & m1_rmod_2(E,A,B) )
                     => ( m1_rmod_2(C,A,D)
                       => m1_rmod_2(k1_rmod_3(A,B,C,E),A,k1_rmod_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t39_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( m1_rmod_2(C,A,D)
                       => m1_rmod_2(C,A,k1_rmod_3(A,B,D,E)) ) ) ) ) ) ) )).

fof(t40_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( ( m1_rmod_2(C,A,D)
                          & m1_rmod_2(E,A,D) )
                       => m1_rmod_2(k1_rmod_3(A,B,C,E),A,D) ) ) ) ) ) ) )).

fof(t41_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( ~ ( ? [E] :
                            ( m1_rmod_2(E,A,B)
                            & u1_struct_0(E) = k2_xboole_0(u1_struct_0(C),u1_struct_0(D)) )
                        & ~ m1_rmod_2(C,A,D)
                        & ~ m1_rmod_2(D,A,C) )
                    & ~ ( ( m1_rmod_2(C,A,D)
                          | m1_rmod_2(D,A,C) )
                        & ! [E] :
                            ( m1_rmod_2(E,A,B)
                           => u1_struct_0(E) != k2_xboole_0(u1_struct_0(C),u1_struct_0(D)) ) ) ) ) ) ) ) )).

fof(d3_rmod_3,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] :
              ( C = k3_rmod_3(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ? [E] :
                      ( v3_vectsp_2(E,A)
                      & m1_rmod_2(E,A,B)
                      & E = D ) ) ) ) ) )).

fof(t42_rmod_3,axiom,(
    $true )).

fof(t43_rmod_3,axiom,(
    $true )).

fof(t44_rmod_3,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)
            & v3_vectsp_2(B,A)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => r2_hidden(B,k3_rmod_3(A,B)) ) ) )).

fof(d4_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( r1_rmod_3(A,B,C,D)
                  <=> ( g1_vectsp_2(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u1_vectsp_2(A,B)) = k1_rmod_3(A,B,C,D)
                      & k2_rmod_3(A,B,C,D) = k1_rmod_2(A,B) ) ) ) ) ) ) )).

fof(t45_rmod_3,axiom,(
    $true )).

fof(t46_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( r1_rmod_3(A,B,C,D)
                   => r1_rmod_3(A,B,D,C) ) ) ) ) ) )).

fof(t47_rmod_3,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)
            & v3_vectsp_2(B,A)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ( r1_rmod_3(A,B,k1_rmod_2(A,B),k2_rmod_2(A,B))
            & r1_rmod_3(A,B,k2_rmod_2(A,B),k1_rmod_2(A,B)) ) ) ) )).

fof(t48_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m2_rmod_2(E,A,B,C)
                     => ! [F] :
                          ( m2_rmod_2(F,A,B,D)
                         => ( ~ r1_xboole_0(E,F)
                           => m2_rmod_2(k5_subset_1(u1_struct_0(B),E,F),A,B,k2_rmod_3(A,B,C,D)) ) ) ) ) ) ) ) )).

fof(t49_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( r1_rmod_3(A,B,C,D)
                  <=> ! [E] :
                        ( m2_rmod_2(E,A,B,C)
                       => ! [F] :
                            ( m2_rmod_2(F,A,B,D)
                           => ? [G] :
                                ( m1_subset_1(G,u1_struct_0(B))
                                & k5_subset_1(u1_struct_0(B),E,F) = k6_domain_1(u1_struct_0(B),G) ) ) ) ) ) ) ) ) )).

fof(t50_rmod_3,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)
            & v3_vectsp_2(B,A)
            & v5_vectsp_2(B,A)
            & l1_vectsp_2(B,A) )
         => ! [C] :
              ( m1_rmod_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( k1_rmod_3(A,B,C,D) = B
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => ? [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                            & ? [G] :
                                ( m1_subset_1(G,u1_struct_0(B))
                                & r1_rlvect_1(C,F)
                                & r1_rlvect_1(D,G)
                                & E = k4_rlvect_1(B,F,G) ) ) ) ) ) ) ) ) )).

fof(t51_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(B))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_struct_0(B))
                                 => ! [I] :
                                      ( m1_subset_1(I,u1_struct_0(B))
                                     => ( ( r1_rmod_3(A,B,C,D)
                                          & E = k4_rlvect_1(B,F,G)
                                          & E = k4_rlvect_1(B,H,I)
                                          & r1_rlvect_1(C,F)
                                          & r1_rlvect_1(C,H)
                                          & r1_rlvect_1(D,G)
                                          & r1_rlvect_1(D,I) )
                                       => ( F = H
                                          & G = I ) ) ) ) ) ) ) ) ) ) ) )).

fof(t52_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ( B = k1_rmod_3(A,B,C,D)
                   => ( ! [E] :
                          ( m1_subset_1(E,u1_struct_0(B))
                         => ? [F] :
                              ( m1_subset_1(F,u1_struct_0(B))
                              & ? [G] :
                                  ( m1_subset_1(G,u1_struct_0(B))
                                  & ? [H] :
                                      ( m1_subset_1(H,u1_struct_0(B))
                                      & ? [I] :
                                          ( m1_subset_1(I,u1_struct_0(B))
                                          & E = k4_rlvect_1(B,F,G)
                                          & E = k4_rlvect_1(B,H,I)
                                          & r1_rlvect_1(C,F)
                                          & r1_rlvect_1(C,H)
                                          & r1_rlvect_1(D,G)
                                          & r1_rlvect_1(D,I)
                                          & ~ ( F = H
                                              & G = I ) ) ) ) ) )
                      | r1_rmod_3(A,B,C,D) ) ) ) ) ) ) )).

fof(d5_rmod_3,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_2(D,A,B)
                 => ! [E] :
                      ( m1_rmod_2(E,A,B)
                     => ( r1_rmod_3(A,B,D,E)
                       => ! [F] :
                            ( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)))
                           => ( F = k4_rmod_3(A,B,C,D,E)
                            <=> ( C = k4_rlvect_1(B,k2_domain_1(u1_struct_0(B),u1_struct_0(B),F),k3_domain_1(u1_struct_0(B),u1_struct_0(B),F))
                                & r1_rlvect_1(D,k2_domain_1(u1_struct_0(B),u1_struct_0(B),F))
                                & r1_rlvect_1(E,k3_domain_1(u1_struct_0(B),u1_struct_0(B),F)) ) ) ) ) ) ) ) ) ) )).

fof(t53_rmod_3,axiom,(
    $true )).

fof(t54_rmod_3,axiom,(
    $true )).

fof(t55_rmod_3,axiom,(
    $true )).

fof(t56_rmod_3,axiom,(
    $true )).

fof(t57_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ( r1_rmod_3(A,B,C,D)
                       => k2_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,C,D)) = k3_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,D,C)) ) ) ) ) ) ) )).

fof(t58_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ( r1_rmod_3(A,B,C,D)
                       => k3_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,C,D)) = k2_domain_1(u1_struct_0(B),u1_struct_0(B),k4_rmod_3(A,B,E,D,C)) ) ) ) ) ) ) )).

fof(d6_rmod_3,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,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
                & m2_relset_1(C,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) )
             => ( C = k5_rmod_3(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k3_rmod_3(A,B))
                   => ! [E] :
                        ( m1_subset_1(E,k3_rmod_3(A,B))
                       => ! [F] :
                            ( m1_rmod_2(F,A,B)
                           => ! [G] :
                                ( m1_rmod_2(G,A,B)
                               => ( ( D = F
                                    & E = G )
                                 => k2_binop_1(k3_rmod_3(A,B),k3_rmod_3(A,B),k3_rmod_3(A,B),C,D,E) = k1_rmod_3(A,B,F,G) ) ) ) ) ) ) ) ) ) )).

fof(d7_rmod_3,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,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
                & m2_relset_1(C,k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) )
             => ( C = k6_rmod_3(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k3_rmod_3(A,B))
                   => ! [E] :
                        ( m1_subset_1(E,k3_rmod_3(A,B))
                       => ! [F] :
                            ( m1_rmod_2(F,A,B)
                           => ! [G] :
                                ( m1_rmod_2(G,A,B)
                               => ( ( D = F
                                    & E = G )
                                 => k2_binop_1(k3_rmod_3(A,B),k3_rmod_3(A,B),k3_rmod_3(A,B),C,D,E) = k2_rmod_3(A,B,F,G) ) ) ) ) ) ) ) ) ) )).

fof(t59_rmod_3,axiom,(
    $true )).

fof(t60_rmod_3,axiom,(
    $true )).

fof(t61_rmod_3,axiom,(
    $true )).

fof(t62_rmod_3,axiom,(
    $true )).

fof(t63_rmod_3,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) )
         => ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) )).

fof(t64_rmod_3,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) )
         => ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v13_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) )).

fof(t65_rmod_3,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) )
         => ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v14_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) )).

fof(t66_rmod_3,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) )
         => ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v15_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) )).

fof(t67_rmod_3,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) )
         => ( ~ v3_struct_0(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v10_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & v12_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B)))
            & l3_lattices(g3_lattices(k3_rmod_3(A,B),k5_rmod_3(A,B),k6_rmod_3(A,B))) ) ) ) )).

fof(dt_k1_rmod_3,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_2(C,A,B)
        & m1_rmod_2(D,A,B) )
     => ( v3_vectsp_2(k1_rmod_3(A,B,C,D),A)
        & m1_rmod_2(k1_rmod_3(A,B,C,D),A,B) ) ) )).

fof(dt_k2_rmod_3,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_2(C,A,B)
        & m1_rmod_2(D,A,B) )
     => ( v3_vectsp_2(k2_rmod_3(A,B,C,D),A)
        & m1_rmod_2(k2_rmod_3(A,B,C,D),A,B) ) ) )).

fof(dt_k3_rmod_3,axiom,(
    $true )).

fof(dt_k4_rmod_3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ 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(B))
        & m1_rmod_2(D,A,B)
        & m1_rmod_2(E,A,B) )
     => m1_subset_1(k4_rmod_3(A,B,C,D,E),k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B))) ) )).

fof(dt_k5_rmod_3,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) )
     => ( v1_funct_1(k5_rmod_3(A,B))
        & v1_funct_2(k5_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
        & m2_relset_1(k5_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) ) ) )).

fof(dt_k6_rmod_3,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) )
     => ( v1_funct_1(k6_rmod_3(A,B))
        & v1_funct_2(k6_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B))
        & m2_relset_1(k6_rmod_3(A,B),k2_zfmisc_1(k3_rmod_3(A,B),k3_rmod_3(A,B)),k3_rmod_3(A,B)) ) ) )).

fof(d1_rmod_3,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_2(C,A,B)
             => ! [D] :
                  ( m1_rmod_2(D,A,B)
                 => ! [E] :
                      ( ( v3_vectsp_2(E,A)
                        & m1_rmod_2(E,A,B) )
                     => ( E = k1_rmod_3(A,B,C,D)
                      <=> u1_struct_0(E) = a_4_0_rmod_3(A,B,C,D) ) ) ) ) ) ) )).

fof(fraenkel_a_4_0_rmod_3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ 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_2(D,B,C)
        & m1_rmod_2(E,B,C) )
     => ( r2_hidden(A,a_4_0_rmod_3(B,C,D,E))
      <=> ? [F,G] :
            ( m1_subset_1(F,u1_struct_0(C))
            & m1_subset_1(G,u1_struct_0(C))
            & A = k4_rlvect_1(C,F,G)
            & r1_rlvect_1(D,F)
            & r1_rlvect_1(E,G) ) ) ) )).
%------------------------------------------------------------------------------