SET007 Axioms: SET007+678.ax


%------------------------------------------------------------------------------
% File     : SET007+678 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Ring Ideals
% 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    : ideal_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  226 (   0 unit)
%            Number of atoms       : 2160 (  96 equality)
%            Maximal formula depth :   28 (  10 average)
%            Number of connectives : 2414 ( 480 ~  ;   1  |;1378  &)
%                                         (  37 <=>; 518 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   69 (   0 propositional; 1-4 arity)
%            Number of functors    :   79 (   5 constant; 0-5 arity)
%            Number of variables   :  686 (   0 singleton; 636 !;  50 ?)
%            Maximal term depth    :    8 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_ideal_1,axiom,(
    ? [A] :
      ( l1_rlvect_1(A)
      & ~ v3_struct_0(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v1_algstr_1(A) ) )).

fof(rc2_ideal_1,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & v1_binom(A)
      & v1_algstr_1(A)
      & v2_algstr_1(A)
      & v3_algstr_1(A)
      & ~ v3_realset2(A) ) )).

fof(rc3_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_ideal_1(B,A) ) ) )).

fof(rc4_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v2_ideal_1(B,A) ) ) )).

fof(rc5_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v3_ideal_1(B,A) ) ) )).

fof(rc6_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_ideal_1(B,A)
          & v2_ideal_1(B,A)
          & v3_ideal_1(B,A) ) ) )).

fof(rc7_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_ideal_1(B,A)
          & v3_ideal_1(B,A) ) ) )).

fof(rc8_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_ideal_1(B,A)
          & v2_ideal_1(B,A) ) ) )).

fof(cc1_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( ~ v1_xboole_0(B)
              & v2_ideal_1(B,A) )
           => ( ~ v1_xboole_0(B)
              & v3_ideal_1(B,A) ) ) ) ) )).

fof(cc2_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( ~ v1_xboole_0(B)
              & v3_ideal_1(B,A) )
           => ( ~ v1_xboole_0(B)
              & v2_ideal_1(B,A) ) ) ) ) )).

fof(fc1_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => v1_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) )).

fof(fc2_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => v2_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) )).

fof(fc3_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => v3_ideal_1(k1_tarski(k1_rlvect_1(A)),A) ) )).

fof(rc9_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v5_ideal_1(B,A) ) ) )).

fof(rc10_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & ~ v3_realset2(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_ideal_1(B,A)
          & v2_ideal_1(B,A)
          & v3_ideal_1(B,A)
          & v5_ideal_1(B,A) ) ) )).

fof(fc4_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v3_struct_0(k3_ideal_1(A,B))
        & v4_rlvect_1(k3_ideal_1(A,B)) ) ) )).

fof(fc5_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v3_struct_0(k3_ideal_1(A,B))
        & v4_rlvect_1(k3_ideal_1(A,B))
        & v5_rlvect_1(k3_ideal_1(A,B)) ) ) )).

fof(fc6_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v3_struct_0(k3_ideal_1(A,B))
        & v3_rlvect_1(k3_ideal_1(A,B)) ) ) )).

fof(fc7_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v3_struct_0(k3_ideal_1(A,B))
        & v3_rlvect_1(k3_ideal_1(A,B))
        & v4_rlvect_1(k3_ideal_1(A,B))
        & v5_rlvect_1(k3_ideal_1(A,B))
        & v6_rlvect_1(k3_ideal_1(A,B))
        & v1_binom(k3_ideal_1(A,B))
        & v1_algstr_1(k3_ideal_1(A,B))
        & v2_algstr_1(k3_ideal_1(A,B))
        & v3_algstr_1(k3_ideal_1(A,B))
        & v4_algstr_1(k3_ideal_1(A,B))
        & v5_algstr_1(k3_ideal_1(A,B))
        & v6_algstr_1(k3_ideal_1(A,B)) ) ) )).

fof(rc11_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] :
          ( m1_ideal_1(C,A,B)
          & ~ v1_xboole_0(C)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_finset_1(C)
          & v1_finseq_1(C)
          & v1_polynom1(C) ) ) )).

fof(rc12_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] :
          ( m2_ideal_1(C,A,B)
          & ~ v1_xboole_0(C)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_finset_1(C)
          & v1_finseq_1(C)
          & v1_polynom1(C) ) ) )).

fof(rc13_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] :
          ( m3_ideal_1(C,A,B)
          & ~ v1_xboole_0(C)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_finset_1(C)
          & v1_finseq_1(C)
          & v1_polynom1(C) ) ) )).

fof(fc8_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ~ v1_xboole_0(k10_ideal_1(A,B,C)) ) )).

fof(fc9_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A)
        & v1_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,u1_struct_0(A)) )
     => v1_ideal_1(k10_ideal_1(A,B,C),A) ) )).

fof(fc10_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l3_vectsp_1(A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,u1_struct_0(A)) )
     => v3_ideal_1(k10_ideal_1(A,B,C),A) ) )).

fof(fc11_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ~ v1_xboole_0(k11_ideal_1(A,B,C)) ) )).

fof(fc12_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & l1_rlvect_1(A)
        & v1_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v1_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => v1_ideal_1(k11_ideal_1(A,B,C),A) ) )).

fof(fc13_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v5_vectsp_1(A)
        & l3_vectsp_1(A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v3_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => v3_ideal_1(k11_ideal_1(A,B,C),A) ) )).

fof(fc14_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & l3_vectsp_1(A)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => v2_ideal_1(k11_ideal_1(A,B,C),A) ) )).

fof(fc15_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ~ v1_xboole_0(k3_xboole_0(B,C)) ) )).

fof(fc16_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & v1_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v1_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => v1_ideal_1(k3_xboole_0(B,C),A) ) )).

fof(fc17_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => v2_ideal_1(k3_xboole_0(B,C),A) ) )).

fof(fc18_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v3_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => v3_ideal_1(k3_xboole_0(B,C),A) ) )).

fof(fc19_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ~ v1_xboole_0(k14_ideal_1(A,B,C)) ) )).

fof(fc20_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k14_ideal_1(A,B,C))
        & v1_ideal_1(k14_ideal_1(A,B,C),A) ) ) )).

fof(fc21_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v3_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k14_ideal_1(A,B,C))
        & v3_ideal_1(k14_ideal_1(A,B,C),A) ) ) )).

fof(fc22_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k14_ideal_1(A,B,C))
        & v2_ideal_1(k14_ideal_1(A,B,C),A) ) ) )).

fof(fc23_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ~ v1_xboole_0(k16_ideal_1(A,B,C)) ) )).

fof(fc24_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & v1_ideal_1(C,A)
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k16_ideal_1(A,B,C))
        & v1_ideal_1(k16_ideal_1(A,B,C),A) ) ) )).

fof(fc25_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k16_ideal_1(A,B,C))
        & v2_ideal_1(k16_ideal_1(A,B,C),A)
        & v3_ideal_1(k16_ideal_1(A,B,C),A) ) ) )).

fof(fc26_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & v2_ideal_1(C,A)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k16_ideal_1(A,B,C))
        & v2_ideal_1(k16_ideal_1(A,B,C),A)
        & v3_ideal_1(k16_ideal_1(A,B,C),A) ) ) )).

fof(fc27_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ~ v1_xboole_0(k17_ideal_1(A,B)) ) )).

fof(fc28_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k17_ideal_1(A,B))
        & v1_ideal_1(k17_ideal_1(A,B),A) ) ) )).

fof(fc29_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k17_ideal_1(A,B))
        & v2_ideal_1(k17_ideal_1(A,B),A)
        & v3_ideal_1(k17_ideal_1(A,B),A) ) ) )).

fof(fc30_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v2_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k17_ideal_1(A,B))
        & v2_ideal_1(k17_ideal_1(A,B),A)
        & v3_ideal_1(k17_ideal_1(A,B),A) ) ) )).

fof(rc14_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v1_ideal_1(B,A)
          & v2_ideal_1(B,A)
          & v3_ideal_1(B,A)
          & v6_ideal_1(B,A) ) ) )).

fof(fc31_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k7_ideal_1(A,B))
        & v1_ideal_1(k7_ideal_1(A,B),A)
        & v2_ideal_1(k7_ideal_1(A,B),A)
        & v3_ideal_1(k7_ideal_1(A,B),A)
        & v6_ideal_1(k7_ideal_1(A,B),A) ) ) )).

fof(rc15_ideal_1,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v1_int_3(A)
      & v1_binom(A)
      & v1_algstr_1(A)
      & v2_algstr_1(A)
      & v3_algstr_1(A)
      & v4_algstr_1(A)
      & v5_algstr_1(A)
      & v6_algstr_1(A)
      & ~ v3_realset2(A) ) )).

fof(fc32_ideal_1,axiom,
    ( ~ v3_struct_0(k1_int_3)
    & v2_group_1(k1_int_3)
    & v4_group_1(k1_int_3)
    & v7_group_1(k1_int_3)
    & v3_rlvect_1(k1_int_3)
    & v4_rlvect_1(k1_int_3)
    & v5_rlvect_1(k1_int_3)
    & v6_rlvect_1(k1_int_3)
    & v3_vectsp_1(k1_int_3)
    & v4_vectsp_1(k1_int_3)
    & v5_vectsp_1(k1_int_3)
    & v6_vectsp_1(k1_int_3)
    & v7_vectsp_1(k1_int_3)
    & v8_vectsp_1(k1_int_3)
    & ~ v10_vectsp_1(k1_int_3)
    & v1_int_3(k1_int_3)
    & v1_binom(k1_int_3)
    & v2_vectsp_2(k1_int_3)
    & v3_gcd_1(k1_int_3)
    & v1_algstr_1(k1_int_3)
    & v2_algstr_1(k1_int_3)
    & v3_algstr_1(k1_int_3)
    & v4_algstr_1(k1_int_3)
    & v5_algstr_1(k1_int_3)
    & v6_algstr_1(k1_int_3)
    & ~ v3_realset2(k1_int_3)
    & v7_ideal_1(k1_int_3) )).

fof(rc16_ideal_1,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v1_binom(A)
      & v1_algstr_1(A)
      & v2_algstr_1(A)
      & v3_algstr_1(A)
      & v4_algstr_1(A)
      & v5_algstr_1(A)
      & v6_algstr_1(A)
      & ~ v3_realset2(A)
      & v7_ideal_1(A) ) )).

fof(fc33_ideal_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & ~ v1_xboole_0(D)
        & m1_subset_1(D,k1_zfmisc_1(A)) )
     => ( ~ v1_xboole_0(k7_relat_1(C,D))
        & v1_relat_1(k7_relat_1(C,D)) ) ) )).

fof(t1_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v1_algstr_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k9_rlvect_1(A,k2_finseq_4(u1_struct_0(A),B,C)) = k2_rlvect_1(A,B,C) ) ) ) )).

fof(d1_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_ideal_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( r2_hidden(C,B)
                        & r2_hidden(D,B) )
                     => r2_hidden(k2_rlvect_1(A,C,D),B) ) ) ) ) ) ) )).

fof(d2_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_ideal_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,B)
                     => r2_hidden(k1_group_1(A,C,D),B) ) ) ) ) ) ) )).

fof(d3_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_ideal_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,B)
                     => r2_hidden(k1_group_1(A,D,C),B) ) ) ) ) ) ) )).

fof(t2_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => r2_hidden(k1_rlvect_1(A),B) ) ) )).

fof(t3_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => r2_hidden(k1_rlvect_1(A),B) ) ) )).

fof(t4_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A) ) )).

fof(t5_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => v2_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A) ) )).

fof(t6_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => v3_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A) ) )).

fof(t7_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v1_xboole_0(k1_struct_0(A,k1_rlvect_1(A)))
        & v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
        & v2_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
        & v3_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
        & m1_subset_1(k1_struct_0(A,k1_rlvect_1(A)),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t8_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v1_xboole_0(k1_struct_0(A,k1_rlvect_1(A)))
        & v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
        & v2_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
        & m1_subset_1(k1_struct_0(A,k1_rlvect_1(A)),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t9_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v1_xboole_0(k1_struct_0(A,k1_rlvect_1(A)))
        & v1_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
        & v3_ideal_1(k1_struct_0(A,k1_rlvect_1(A)),A)
        & m1_subset_1(k1_struct_0(A,k1_rlvect_1(A)),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t10_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ~ v1_xboole_0(u1_struct_0(A))
        & v1_ideal_1(u1_struct_0(A),A)
        & v2_ideal_1(u1_struct_0(A),A)
        & v3_ideal_1(u1_struct_0(A),A)
        & m1_subset_1(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t11_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ~ v1_xboole_0(u1_struct_0(A))
        & v1_ideal_1(u1_struct_0(A),A)
        & v2_ideal_1(u1_struct_0(A),A)
        & m1_subset_1(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t12_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ~ v1_xboole_0(u1_struct_0(A))
        & v1_ideal_1(u1_struct_0(A),A)
        & v3_ideal_1(u1_struct_0(A),A)
        & m1_subset_1(u1_struct_0(A),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(d4_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v1_realset1(B)
          <=> B = k1_struct_0(A,k1_rlvect_1(A)) ) ) ) )).

fof(d5_ideal_1,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v5_ideal_1(B,A)
          <=> B != u1_struct_0(A) ) ) ) )).

fof(t13_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,B)
               => r2_hidden(k5_rlvect_1(A,C),B) ) ) ) ) )).

fof(t14_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_vectsp_1(A)
        & v6_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,B)
               => r2_hidden(k5_rlvect_1(A,C),B) ) ) ) ) )).

fof(t15_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_hidden(C,B)
                      & r2_hidden(D,B) )
                   => r2_hidden(k6_rlvect_1(A,C,D),B) ) ) ) ) ) )).

fof(t16_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_vectsp_1(A)
        & v6_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_hidden(C,B)
                      & r2_hidden(D,B) )
                   => r2_hidden(k6_rlvect_1(A,C,D),B) ) ) ) ) ) )).

fof(t17_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_struct_0(C,A,B)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => r2_hidden(k5_binom(A,C,D),B) ) ) ) ) )).

fof(t18_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_struct_0(C,A,B)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( D != np__0
                   => r2_hidden(k2_binom(A,C,D),B) ) ) ) ) ) )).

fof(d6_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k1_ideal_1(A,B) = k1_realset1(u1_rlvect_1(A),B) ) ) )).

fof(d7_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k2_ideal_1(A,B) = k1_realset1(u1_group_1(A),B) ) ) )).

fof(d8_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k3_ideal_1(A,B) = g1_rlvect_1(B,k1_ideal_1(A,B),k1_funct_7(k1_rlvect_1(A),B)) ) ) )).

fof(t19_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v5_ideal_1(B,A)
          <=> ~ r2_hidden(k2_group_1(A),B) ) ) ) )).

fof(t20_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_vectsp_1(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v5_ideal_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( v1_gcd_1(C,A)
                    & r2_hidden(C,B) ) ) ) ) ) )).

fof(t21_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v2_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v5_ideal_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( v1_gcd_1(C,A)
                    & r2_hidden(C,B) ) ) ) ) ) )).

fof(t22_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ( ~ v3_struct_0(A)
          & v4_group_1(A)
          & v7_group_1(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A)
          & v7_vectsp_1(A)
          & v8_vectsp_1(A)
          & v9_vectsp_1(A)
          & ~ v10_vectsp_1(A)
          & l3_vectsp_1(A) )
      <=> ! [B] :
            ( ( ~ v1_xboole_0(B)
              & v1_ideal_1(B,A)
              & v2_ideal_1(B,A)
              & v3_ideal_1(B,A)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ( B = k1_struct_0(A,k1_rlvect_1(A))
              | B = u1_struct_0(A) ) ) ) ) )).

fof(d9_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( m1_ideal_1(C,A,B)
              <=> ! [D] :
                    ~ ( r2_hidden(D,k4_relset_1(k5_numbers,u1_struct_0(A),C))
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ! [G] :
                                  ( m1_struct_0(G,A,B)
                                 => k4_finseq_4(k5_numbers,u1_struct_0(A),C,D) != k1_group_1(A,k1_group_1(A,E,G),F) ) ) ) ) ) ) ) ) )).

fof(d10_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( m2_ideal_1(C,A,B)
              <=> ! [D] :
                    ~ ( r2_hidden(D,k4_relset_1(k5_numbers,u1_struct_0(A),C))
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ! [F] :
                              ( m1_struct_0(F,A,B)
                             => k4_finseq_4(k5_numbers,u1_struct_0(A),C,D) != k1_group_1(A,E,F) ) ) ) ) ) ) ) )).

fof(d11_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( m3_ideal_1(C,A,B)
              <=> ! [D] :
                    ~ ( r2_hidden(D,k4_relset_1(k5_numbers,u1_struct_0(A),C))
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ! [F] :
                              ( m1_struct_0(F,A,B)
                             => k4_finseq_4(k5_numbers,u1_struct_0(A),C,D) != k1_group_1(A,F,E) ) ) ) ) ) ) ) )).

fof(t23_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_ideal_1(D,A,B)
                 => m1_ideal_1(k6_polynom1(A,D,C),A,B) ) ) ) ) )).

fof(t24_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_ideal_1(D,A,B)
                 => m1_ideal_1(k7_polynom1(A,D,C),A,B) ) ) ) ) )).

fof(t25_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_vectsp_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_ideal_1(C,A,B)
             => m1_ideal_1(C,A,B) ) ) ) )).

fof(t26_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_ideal_1(D,A,B)
                 => m2_ideal_1(k6_polynom1(A,D,C),A,B) ) ) ) ) )).

fof(t27_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_ideal_1(D,A,B)
                 => m1_ideal_1(k7_polynom1(A,D,C),A,B) ) ) ) ) )).

fof(t28_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m3_ideal_1(C,A,B)
             => m1_ideal_1(C,A,B) ) ) ) )).

fof(t29_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m3_ideal_1(D,A,B)
                 => m3_ideal_1(k7_polynom1(A,D,C),A,B) ) ) ) ) )).

fof(t30_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m3_ideal_1(D,A,B)
                 => m1_ideal_1(k6_polynom1(A,D,C),A,B) ) ) ) ) )).

fof(t31_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_ideal_1(C,A,B)
             => ( m2_ideal_1(C,A,B)
                & m3_ideal_1(C,A,B) ) ) ) ) )).

fof(t32_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_ideal_1(C,A,B) )
             => ? [D] :
                  ( m1_ideal_1(D,A,B)
                  & ? [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                      & C = k8_finseq_1(u1_struct_0(A),D,k12_finseq_1(u1_struct_0(A),E))
                      & m1_ideal_1(k12_finseq_1(u1_struct_0(A),E),A,B) ) ) ) ) ) )).

fof(t33_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_ideal_1(C,A,B) )
             => ? [D] :
                  ( m2_ideal_1(D,A,B)
                  & ? [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                      & C = k8_finseq_1(u1_struct_0(A),D,k12_finseq_1(u1_struct_0(A),E))
                      & m2_ideal_1(k12_finseq_1(u1_struct_0(A),E),A,B) ) ) ) ) ) )).

fof(t34_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m3_ideal_1(C,A,B) )
             => ? [D] :
                  ( m3_ideal_1(D,A,B)
                  & ? [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                      & C = k8_finseq_1(u1_struct_0(A),D,k12_finseq_1(u1_struct_0(A),E))
                      & m3_ideal_1(k12_finseq_1(u1_struct_0(A),E),A,B) ) ) ) ) ) )).

fof(d12_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_ideal_1(C,A,B)
             => ! [D] :
                  ( m2_finseq_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)))
                 => ( r1_ideal_1(A,B,C,D)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(C)
                      & ! [E] :
                          ( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),C))
                         => ( k1_funct_1(C,E) = k1_group_1(A,k1_group_1(A,k5_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E)),k6_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E))),k7_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E)))
                            & r2_hidden(k6_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),D,E)),B) ) ) ) ) ) ) ) ) )).

fof(t35_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_ideal_1(C,A,B)
             => ? [D] :
                  ( m2_finseq_1(D,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)))
                  & r1_ideal_1(A,B,C,D) ) ) ) ) )).

fof(t36_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_vectsp_1(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( m1_ideal_1(D,A,C)
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,u1_struct_0(A),u1_struct_0(B))
                            & m2_relset_1(F,u1_struct_0(A),u1_struct_0(B)) )
                         => ! [G] :
                              ( m2_finseq_1(G,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)))
                             => ~ ( r1_tarski(k4_pre_topc(A,B,F,C),E)
                                  & r1_ideal_1(A,C,D,G)
                                  & ! [H] :
                                      ( m1_ideal_1(H,B,E)
                                     => ~ ( k3_finseq_1(D) = k3_finseq_1(H)
                                          & ! [I] :
                                              ( r2_hidden(I,k4_relset_1(k5_numbers,u1_struct_0(B),H))
                                             => k1_funct_1(H,I) = k1_group_1(B,k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k5_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),G,I))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k6_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),G,I)))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k7_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),G,I)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d13_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_ideal_1(C,A,B)
             => ! [D] :
                  ( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                 => ( r2_ideal_1(A,B,C,D)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(C)
                      & ! [E] :
                          ( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),C))
                         => ( k1_funct_1(C,E) = k1_group_1(A,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)))
                            & r2_hidden(k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),B) ) ) ) ) ) ) ) ) )).

fof(t37_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_ideal_1(C,A,B)
             => ? [D] :
                  ( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                  & r2_ideal_1(A,B,C,D) ) ) ) ) )).

fof(t38_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_vectsp_1(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( m2_ideal_1(D,A,C)
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,u1_struct_0(A),u1_struct_0(B))
                            & m2_relset_1(F,u1_struct_0(A),u1_struct_0(B)) )
                         => ! [G] :
                              ( m2_finseq_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                             => ~ ( r1_tarski(k4_pre_topc(A,B,F,C),E)
                                  & r2_ideal_1(A,C,D,G)
                                  & ! [H] :
                                      ( m2_ideal_1(H,B,E)
                                     => ~ ( k3_finseq_1(D) = k3_finseq_1(H)
                                          & ! [I] :
                                              ( r2_hidden(I,k4_relset_1(k5_numbers,u1_struct_0(B),H))
                                             => k1_funct_1(H,I) = k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d14_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m3_ideal_1(C,A,B)
             => ! [D] :
                  ( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                 => ( r3_ideal_1(A,B,C,D)
                  <=> ( k3_finseq_1(D) = k3_finseq_1(C)
                      & ! [E] :
                          ( r2_hidden(E,k4_relset_1(k5_numbers,u1_struct_0(A),C))
                         => ( k1_funct_1(C,E) = k1_group_1(A,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)))
                            & r2_hidden(k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),D,E)),B) ) ) ) ) ) ) ) ) )).

fof(t39_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m3_ideal_1(C,A,B)
             => ? [D] :
                  ( m2_finseq_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                  & r3_ideal_1(A,B,C,D) ) ) ) ) )).

fof(t40_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_vectsp_1(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( m3_ideal_1(D,A,C)
                 => ! [E] :
                      ( ( ~ v1_xboole_0(E)
                        & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,u1_struct_0(A),u1_struct_0(B))
                            & m2_relset_1(F,u1_struct_0(A),u1_struct_0(B)) )
                         => ! [G] :
                              ( m2_finseq_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))
                             => ~ ( r1_tarski(k4_pre_topc(A,B,F,C),E)
                                  & r3_ideal_1(A,C,D,G)
                                  & ! [H] :
                                      ( m3_ideal_1(H,B,E)
                                     => ~ ( k3_finseq_1(D) = k3_finseq_1(H)
                                          & ! [I] :
                                              ( r2_hidden(I,k4_relset_1(k5_numbers,u1_struct_0(B),H))
                                             => k1_funct_1(H,I) = k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k2_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),F,k3_domain_1(u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),G,I)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t41_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_ideal_1(C,A,B)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => m1_ideal_1(k2_partfun1(k5_numbers,u1_struct_0(A),C,k2_finseq_1(D)),A,B) ) ) ) ) )).

fof(t42_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_ideal_1(C,A,B)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => m2_ideal_1(k2_partfun1(k5_numbers,u1_struct_0(A),C,k2_finseq_1(D)),A,B) ) ) ) ) )).

fof(t43_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m3_ideal_1(C,A,B)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => m3_ideal_1(k2_partfun1(k5_numbers,u1_struct_0(A),C,k2_finseq_1(D)),A,B) ) ) ) ) )).

fof(d15_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ~ v1_xboole_0(B)
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & v1_ideal_1(C,A)
                  & v2_ideal_1(C,A)
                  & v3_ideal_1(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ( C = k7_ideal_1(A,B)
                <=> ( r1_tarski(B,C)
                    & ! [D] :
                        ( ( ~ v1_xboole_0(D)
                          & v1_ideal_1(D,A)
                          & v2_ideal_1(D,A)
                          & v3_ideal_1(D,A)
                          & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                       => ( r1_tarski(B,D)
                         => r1_tarski(C,D) ) ) ) ) ) ) ) ) )).

fof(d16_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ~ v1_xboole_0(B)
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & v1_ideal_1(C,A)
                  & v2_ideal_1(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ( C = k8_ideal_1(A,B)
                <=> ( r1_tarski(B,C)
                    & ! [D] :
                        ( ( ~ v1_xboole_0(D)
                          & v1_ideal_1(D,A)
                          & v2_ideal_1(D,A)
                          & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                       => ( r1_tarski(B,D)
                         => r1_tarski(C,D) ) ) ) ) ) ) ) ) )).

fof(d17_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ~ v1_xboole_0(B)
           => ! [C] :
                ( ( ~ v1_xboole_0(C)
                  & v1_ideal_1(C,A)
                  & v3_ideal_1(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ( C = k9_ideal_1(A,B)
                <=> ( r1_tarski(B,C)
                    & ! [D] :
                        ( ( ~ v1_xboole_0(D)
                          & v1_ideal_1(D,A)
                          & v3_ideal_1(D,A)
                          & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                       => ( r1_tarski(B,D)
                         => r1_tarski(C,D) ) ) ) ) ) ) ) ) )).

fof(t44_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k7_ideal_1(A,B) = B ) ) )).

fof(t45_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k8_ideal_1(A,B) = B ) ) )).

fof(t46_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k9_ideal_1(A,B) = B ) ) )).

fof(d18_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( m4_ideal_1(C,A,B)
              <=> k7_ideal_1(A,C) = B ) ) ) ) )).

fof(t47_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => k7_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) )).

fof(t48_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => k7_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) )).

fof(t49_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => k8_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) )).

fof(t50_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => k9_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A))) = k1_struct_0(A,k1_rlvect_1(A)) ) )).

fof(t51_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => k7_ideal_1(A,k1_struct_0(A,k2_group_1(A))) = u1_struct_0(A) ) )).

fof(t52_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_vectsp_1(A)
        & l3_vectsp_1(A) )
     => k8_ideal_1(A,k1_struct_0(A,k2_group_1(A))) = u1_struct_0(A) ) )).

fof(t53_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => k9_ideal_1(A,k1_struct_0(A,k2_group_1(A))) = u1_struct_0(A) ) )).

fof(t54_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => k7_ideal_1(A,k2_pre_topc(A)) = u1_struct_0(A) ) )).

fof(t55_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => k8_ideal_1(A,k2_pre_topc(A)) = u1_struct_0(A) ) )).

fof(t56_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => k9_ideal_1(A,k2_pre_topc(A)) = u1_struct_0(A) ) )).

fof(t57_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r1_tarski(B,C)
               => r1_tarski(k7_ideal_1(A,B),k7_ideal_1(A,C)) ) ) ) ) )).

fof(t58_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r1_tarski(B,C)
               => r1_tarski(k8_ideal_1(A,B),k8_ideal_1(A,C)) ) ) ) ) )).

fof(t59_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r1_tarski(B,C)
               => r1_tarski(k9_ideal_1(A,B),k9_ideal_1(A,C)) ) ) ) ) )).

fof(t60_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( r2_hidden(C,k7_ideal_1(A,B))
            <=> ? [D] :
                  ( m1_ideal_1(D,A,B)
                  & C = k9_rlvect_1(A,D) ) ) ) ) )).

fof(t61_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( r2_hidden(C,k8_ideal_1(A,B))
            <=> ? [D] :
                  ( m2_ideal_1(D,A,B)
                  & C = k9_rlvect_1(A,D) ) ) ) ) )).

fof(t62_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( r2_hidden(C,k9_ideal_1(A,B))
            <=> ? [D] :
                  ( m3_ideal_1(D,A,B)
                  & C = k9_rlvect_1(A,D) ) ) ) ) )).

fof(t63_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( k7_ideal_1(A,B) = k8_ideal_1(A,B)
            & k7_ideal_1(A,B) = k9_ideal_1(A,B) ) ) ) )).

fof(t66_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r2_hidden(B,k7_ideal_1(A,k1_struct_0(A,B))) ) ) )).

fof(t67_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,k7_ideal_1(A,B))
               => r1_tarski(k7_ideal_1(A,k1_struct_0(A,C)),k7_ideal_1(A,B)) ) ) ) ) )).

fof(t68_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(B,k7_ideal_1(A,k2_struct_0(A,B,C)))
                & r2_hidden(C,k7_ideal_1(A,k2_struct_0(A,B,C))) ) ) ) ) )).

fof(t69_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_tarski(k7_ideal_1(A,k1_struct_0(A,B)),k7_ideal_1(A,k2_struct_0(A,B,C)))
                & r1_tarski(k7_ideal_1(A,k1_struct_0(A,C)),k7_ideal_1(A,k2_struct_0(A,B,C))) ) ) ) ) )).

fof(t70_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k10_ideal_1(A,B,k1_rlvect_1(A)) = k1_struct_0(A,k1_rlvect_1(A)) ) ) )).

fof(t71_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k10_ideal_1(A,B,k2_group_1(A)) = B ) ) )).

fof(t72_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => k11_ideal_1(A,B,k11_ideal_1(A,C,D)) = k11_ideal_1(A,k11_ideal_1(A,B,C),D) ) ) ) ) )).

fof(t73_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v3_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => r1_tarski(B,k11_ideal_1(A,B,C)) ) ) ) )).

fof(t74_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v3_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => r1_tarski(C,k11_ideal_1(A,B,C)) ) ) ) )).

fof(t75_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( ( v1_ideal_1(D,A)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                 => ( ( r1_tarski(B,D)
                      & r1_tarski(C,D) )
                   => r1_tarski(k11_ideal_1(A,B,C),D) ) ) ) ) ) )).

fof(t76_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k7_ideal_1(A,k2_struct_0(A,B,C)) = k12_ideal_1(A,k7_ideal_1(A,k1_struct_0(A,B)),k7_ideal_1(A,k1_struct_0(A,C))) ) ) ) )).

fof(t77_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r1_tarski(k13_ideal_1(A,B,C),B)
                & r1_tarski(k13_ideal_1(A,B,C),C) ) ) ) ) )).

fof(t78_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => k13_ideal_1(A,B,k13_ideal_1(A,C,D)) = k13_ideal_1(A,k13_ideal_1(A,B,C),D) ) ) ) ) )).

fof(t79_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( r1_tarski(D,B)
                      & r1_tarski(D,C) )
                   => r1_tarski(D,k13_ideal_1(A,B,C)) ) ) ) ) ) )).

fof(t80_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                 => ( r1_tarski(C,B)
                   => k13_ideal_1(A,B,k12_ideal_1(A,C,D)) = k12_ideal_1(A,C,k13_ideal_1(A,B,D)) ) ) ) ) ) )).

fof(t81_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v1_algstr_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => k14_ideal_1(A,k1_struct_0(A,k1_rlvect_1(A)),B) = k1_struct_0(A,k1_rlvect_1(A)) ) ) )).

fof(t82_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_ideal_1(C,A)
                & v2_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => r1_tarski(k14_ideal_1(A,B,C),k13_ideal_1(A,B,C)) ) ) ) )).

fof(t83_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v3_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v3_ideal_1(D,A)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                 => k14_ideal_1(A,B,k12_ideal_1(A,C,D)) = k12_ideal_1(A,k14_ideal_1(A,B,C),k14_ideal_1(A,B,D)) ) ) ) ) )).

fof(t84_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v3_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => r1_tarski(k15_ideal_1(A,k12_ideal_1(A,B,C),k13_ideal_1(A,B,C)),k15_ideal_1(A,B,C)) ) ) ) )).

fof(t85_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_ideal_1(C,A)
                & v2_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => r1_tarski(k14_ideal_1(A,k11_ideal_1(A,B,C),k13_ideal_1(A,B,C)),k13_ideal_1(A,B,C)) ) ) ) )).

fof(d23_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r4_ideal_1(A,B,C)
              <=> k11_ideal_1(A,B,C) = u1_struct_0(A) ) ) ) ) )).

fof(t86_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r4_ideal_1(A,B,C)
               => r1_tarski(k13_ideal_1(A,B,C),k14_ideal_1(A,k11_ideal_1(A,B,C),k13_ideal_1(A,B,C))) ) ) ) ) )).

fof(t87_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_ideal_1(C,A)
                & v2_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r4_ideal_1(A,B,C)
               => k15_ideal_1(A,B,C) = k13_ideal_1(A,B,C) ) ) ) ) )).

fof(t88_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => r1_tarski(B,k16_ideal_1(A,B,C)) ) ) ) )).

fof(t89_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v5_vectsp_1(A)
        & v2_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => r1_tarski(k14_ideal_1(A,k16_ideal_1(A,B,C),C),B) ) ) ) )).

fof(t90_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => r1_tarski(k14_ideal_1(A,k16_ideal_1(A,B,C),C),B) ) ) ) )).

fof(t91_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => k16_ideal_1(A,k16_ideal_1(A,B,C),D) = k16_ideal_1(A,B,k15_ideal_1(A,C,D)) ) ) ) ) )).

fof(t92_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => k16_ideal_1(A,k13_ideal_1(A,C,D),B) = k13_ideal_1(A,k16_ideal_1(A,C,B),k16_ideal_1(A,D,B)) ) ) ) ) )).

fof(t93_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v4_vectsp_1(A)
        & v1_algstr_1(A)
        & v3_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v3_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & v3_ideal_1(D,A)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                 => k16_ideal_1(A,B,k11_ideal_1(A,C,D)) = k13_ideal_1(A,k16_ideal_1(A,B,C),k16_ideal_1(A,B,D)) ) ) ) ) )).

fof(t94_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,k17_ideal_1(A,B))
              <=> ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & r2_hidden(k2_binom(A,C,D),k17_ideal_1(A,B)) ) ) ) ) ) )).

fof(t95_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_ideal_1(C,A)
                & v2_ideal_1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => k17_ideal_1(A,k14_ideal_1(A,B,C)) = k17_ideal_1(A,k13_ideal_1(A,B,C)) ) ) ) )).

fof(d26_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v6_ideal_1(B,A)
          <=> ? [C] :
                ( ~ v1_xboole_0(C)
                & v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                & B = k7_ideal_1(A,C) ) ) ) ) )).

fof(d27_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v7_ideal_1(A)
      <=> ! [B] :
            ( ( ~ v1_xboole_0(B)
              & v1_ideal_1(B,A)
              & v2_ideal_1(B,A)
              & v3_ideal_1(B,A)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => v6_ideal_1(B,A) ) ) ) )).

fof(d28_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_ideal_1(B,A)
            & v2_ideal_1(B,A)
            & v3_ideal_1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( v8_ideal_1(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & B = k7_ideal_1(A,k1_struct_0(A,C)) ) ) ) ) )).

fof(d29_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v9_ideal_1(A)
      <=> ! [B] :
            ( ( ~ v1_xboole_0(B)
              & v1_ideal_1(B,A)
              & v2_ideal_1(B,A)
              & v3_ideal_1(B,A)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => v8_ideal_1(B,A) ) ) ) )).

fof(t96_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ~ ( B != k1_struct_0(A,k1_rlvect_1(A))
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( C != k1_rlvect_1(A)
                      & r2_hidden(C,B) ) ) ) ) ) )).

fof(t97_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v1_int_3(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => v9_ideal_1(A) ) )).

fof(t98_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( v9_ideal_1(A)
       => v7_ideal_1(A) ) ) )).

fof(t99_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & v7_ideal_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ? [C] :
              ( ~ v1_xboole_0(C)
              & v1_finset_1(C)
              & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
              & r1_tarski(C,B)
              & k7_ideal_1(A,C) = k7_ideal_1(A,B) ) ) ) )).

fof(t100_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ? [C] :
                ( ~ v1_xboole_0(C)
                & v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                & r1_tarski(C,B)
                & k7_ideal_1(A,C) = k7_ideal_1(A,B) ) )
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(A))
              & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
           => ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)),k7_ideal_1(A,k5_relset_1(k5_numbers,u1_struct_0(A),k2_partfun1(k5_numbers,u1_struct_0(A),B,k1_gr_cy_1(k1_nat_1(C,np__1)))))) ) ) ) ) )).

fof(t101_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(A))
              & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
           => ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & r2_hidden(k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)),k7_ideal_1(A,k5_relset_1(k5_numbers,u1_struct_0(A),k2_partfun1(k5_numbers,u1_struct_0(A),B,k1_gr_cy_1(k1_nat_1(C,np__1)))))) ) )
       => ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
              & m2_relset_1(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
           => ~ ( ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ( ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C))
                      & v1_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
                      & v2_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
                      & v3_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
                      & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k1_zfmisc_1(u1_struct_0(A))) ) )
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ( ~ r1_xreal_0(D,C)
                         => r2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,D)) ) ) ) ) ) ) ) )).

fof(t102_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ( ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A)))
              & m2_relset_1(B,k5_numbers,k1_zfmisc_1(u1_struct_0(A))) )
           => ~ ( ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ( ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C))
                      & v1_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
                      & v2_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
                      & v3_ideal_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),A)
                      & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k1_zfmisc_1(u1_struct_0(A))) ) )
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ( ~ r1_xreal_0(D,C)
                         => r2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,C),k8_funct_2(k5_numbers,k1_zfmisc_1(u1_struct_0(A)),B,D)) ) ) ) ) )
       => v7_ideal_1(A) ) ) )).

fof(dt_m1_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ! [C] :
          ( m1_ideal_1(C,A,B)
         => m2_finseq_1(C,u1_struct_0(A)) ) ) )).

fof(existence_m1_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] : m1_ideal_1(C,A,B) ) )).

fof(dt_m2_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ! [C] :
          ( m2_ideal_1(C,A,B)
         => m2_finseq_1(C,u1_struct_0(A)) ) ) )).

fof(existence_m2_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] : m2_ideal_1(C,A,B) ) )).

fof(dt_m3_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ! [C] :
          ( m3_ideal_1(C,A,B)
         => m2_finseq_1(C,u1_struct_0(A)) ) ) )).

fof(existence_m3_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] : m3_ideal_1(C,A,B) ) )).

fof(dt_m4_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & v2_ideal_1(B,A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ! [C] :
          ( m4_ideal_1(C,A,B)
         => ( ~ v1_xboole_0(C)
            & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ) )).

fof(existence_m4_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & v2_ideal_1(B,A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ? [C] : m4_ideal_1(C,A,B) ) )).

fof(redefinition_v4_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & v2_ideal_1(B,A)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v4_ideal_1(B,A)
      <=> v1_realset1(B) ) ) )).

fof(dt_k1_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k1_ideal_1(A,B))
        & v1_funct_2(k1_ideal_1(A,B),k2_zfmisc_1(B,B),B)
        & m2_relset_1(k1_ideal_1(A,B),k2_zfmisc_1(B,B),B) ) ) )).

fof(dt_k2_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & ~ v1_xboole_0(B)
        & v3_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k2_ideal_1(A,B))
        & v1_funct_2(k2_ideal_1(A,B),k2_zfmisc_1(B,B),B)
        & m2_relset_1(k2_ideal_1(A,B),k2_zfmisc_1(B,B),B) ) ) )).

fof(dt_k3_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & ~ v1_xboole_0(B)
        & v1_ideal_1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v3_struct_0(k3_ideal_1(A,B))
        & l1_rlvect_1(k3_ideal_1(A,B)) ) ) )).

fof(dt_k4_ideal_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
        & m1_ideal_1(D,A,B)
        & m1_ideal_1(E,A,C) )
     => m1_ideal_1(k4_ideal_1(A,B,C,D,E),A,k4_subset_1(u1_struct_0(A),B,C)) ) )).

fof(redefinition_k4_ideal_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
        & m1_ideal_1(D,A,B)
        & m1_ideal_1(E,A,C) )
     => k4_ideal_1(A,B,C,D,E) = k7_finseq_1(D,E) ) )).

fof(dt_k5_ideal_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
        & m2_ideal_1(D,A,B)
        & m2_ideal_1(E,A,C) )
     => m2_ideal_1(k5_ideal_1(A,B,C,D,E),A,k4_subset_1(u1_struct_0(A),B,C)) ) )).

fof(redefinition_k5_ideal_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
        & m2_ideal_1(D,A,B)
        & m2_ideal_1(E,A,C) )
     => k5_ideal_1(A,B,C,D,E) = k7_finseq_1(D,E) ) )).

fof(dt_k6_ideal_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
        & m3_ideal_1(D,A,B)
        & m3_ideal_1(E,A,C) )
     => m3_ideal_1(k6_ideal_1(A,B,C,D,E),A,k4_subset_1(u1_struct_0(A),B,C)) ) )).

fof(redefinition_k6_ideal_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & l1_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
        & m3_ideal_1(D,A,B)
        & m3_ideal_1(E,A,C) )
     => k6_ideal_1(A,B,C,D,E) = k7_finseq_1(D,E) ) )).

fof(dt_k7_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k7_ideal_1(A,B))
        & v1_ideal_1(k7_ideal_1(A,B),A)
        & v2_ideal_1(k7_ideal_1(A,B),A)
        & v3_ideal_1(k7_ideal_1(A,B),A)
        & m1_subset_1(k7_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k8_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k8_ideal_1(A,B))
        & v1_ideal_1(k8_ideal_1(A,B),A)
        & v2_ideal_1(k8_ideal_1(A,B),A)
        & m1_subset_1(k8_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k9_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( ~ v1_xboole_0(k9_ideal_1(A,B))
        & v1_ideal_1(k9_ideal_1(A,B),A)
        & v3_ideal_1(k9_ideal_1(A,B),A)
        & m1_subset_1(k9_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(dt_k10_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k10_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k11_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k11_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k12_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k12_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(commutativity_k12_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => k12_ideal_1(A,B,C) = k12_ideal_1(A,C,B) ) )).

fof(redefinition_k12_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => k12_ideal_1(A,B,C) = k11_ideal_1(A,B,C) ) )).

fof(dt_k13_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k13_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(commutativity_k13_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => k13_ideal_1(A,B,C) = k13_ideal_1(A,C,B) ) )).

fof(idempotence_k13_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => k13_ideal_1(A,B,B) = B ) )).

fof(redefinition_k13_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => k13_ideal_1(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k14_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k14_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k15_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k15_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(commutativity_k15_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => k15_ideal_1(A,B,C) = k15_ideal_1(A,C,B) ) )).

fof(redefinition_k15_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => k15_ideal_1(A,B,C) = k14_ideal_1(A,B,C) ) )).

fof(dt_k16_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k16_ideal_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k17_ideal_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k17_ideal_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(t64_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k7_ideal_1(A,k1_struct_0(A,B)) = a_2_0_ideal_1(A,B) ) ) )).

fof(t65_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_vectsp_1(A)
        & v1_binom(A)
        & v1_algstr_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k7_ideal_1(A,k2_struct_0(A,B,C)) = a_3_0_ideal_1(A,B,C) ) ) ) )).

fof(d19_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k10_ideal_1(A,B,C) = a_3_1_ideal_1(A,B,C) ) ) ) )).

fof(d20_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k11_ideal_1(A,B,C) = a_3_2_ideal_1(A,B,C) ) ) ) )).

fof(d21_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k13_ideal_1(A,B,C) = a_3_3_ideal_1(A,B,C) ) ) ) )).

fof(d22_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k14_ideal_1(A,B,C) = a_3_4_ideal_1(A,B,C) ) ) ) )).

fof(d24_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k16_ideal_1(A,B,C) = a_3_5_ideal_1(A,B,C) ) ) ) )).

fof(d25_ideal_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k17_ideal_1(A,B) = a_2_1_ideal_1(A,B) ) ) )).

fof(fraenkel_a_2_0_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v7_vectsp_1(B)
        & v1_binom(B)
        & v1_algstr_1(B)
        & l3_vectsp_1(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( r2_hidden(A,a_2_0_ideal_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = k10_group_1(B,C,D) ) ) ) )).

fof(fraenkel_a_3_0_ideal_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_group_1(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v7_vectsp_1(B)
        & v1_binom(B)
        & v1_algstr_1(B)
        & l3_vectsp_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_ideal_1(B,C,D))
      <=> ? [E,F] :
            ( m1_subset_1(E,u1_struct_0(B))
            & m1_subset_1(F,u1_struct_0(B))
            & A = k4_rlvect_1(B,k10_group_1(B,C,E),k10_group_1(B,D,F)) ) ) ) )).

fof(fraenkel_a_3_1_ideal_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_group_1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_1_ideal_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = k1_group_1(B,D,E)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_3_2_ideal_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_rlvect_1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_2_ideal_1(B,C,D))
      <=> ? [E,F] :
            ( m1_subset_1(E,u1_struct_0(B))
            & m1_subset_1(F,u1_struct_0(B))
            & A = k2_rlvect_1(B,E,F)
            & r2_hidden(E,C)
            & r2_hidden(F,D) ) ) ) )).

fof(fraenkel_a_3_3_ideal_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_struct_0(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_3_ideal_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r2_hidden(E,C)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_4_ideal_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l3_vectsp_1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_4_ideal_1(B,C,D))
      <=> ? [E] :
            ( m2_finseq_1(E,u1_struct_0(B))
            & A = k9_rlvect_1(B,E)
            & ! [F] :
                ( m2_subset_1(F,k1_numbers,k5_numbers)
               => ~ ( r1_xreal_0(np__1,F)
                    & r1_xreal_0(F,k3_finseq_1(E))
                    & ! [G] :
                        ( m1_subset_1(G,u1_struct_0(B))
                       => ! [H] :
                            ( m1_subset_1(H,u1_struct_0(B))
                           => ~ ( k1_funct_1(E,F) = k1_group_1(B,G,H)
                                & r2_hidden(G,C)
                                & r2_hidden(H,D) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_5_ideal_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_group_1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_5_ideal_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_tarski(k10_ideal_1(B,D,E),C) ) ) ) )).

fof(fraenkel_a_2_1_ideal_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_group_1(B)
        & l3_vectsp_1(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_2_1_ideal_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(B))
            & A = D
            & ? [E] :
                ( m2_subset_1(E,k1_numbers,k5_numbers)
                & r2_hidden(k2_binom(B,D,E),C) ) ) ) ) )).
%------------------------------------------------------------------------------