SET007 Axioms: SET007+710.ax


%------------------------------------------------------------------------------
% File     : SET007+710 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Multivariate Polynomials: Monomials and Constant Polynomials
% 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    : polynom7 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   68 (   1 unit)
%            Number of atoms       :  691 (  63 equality)
%            Maximal formula depth :   25 (  10 average)
%            Number of connectives :  705 (  82 ~  ;   4  |; 444  &)
%                                         (  12 <=>; 163 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   47 (   1 propositional; 0-3 arity)
%            Number of functors    :   26 (   1 constant; 0-5 arity)
%            Number of variables   :  227 (   0 singleton; 216 !;  11 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_polynom7,axiom,(
    ? [A] :
      ( l2_struct_0(A)
      & ~ v3_struct_0(A)
      & ~ v3_realset2(A) ) )).

fof(cc1_polynom7,axiom,(
    ! [A] :
      ( l2_struct_0(A)
     => ( ~ v3_realset2(A)
       => ~ v3_struct_0(A) ) ) )).

fof(rc2_polynom7,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(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)
      & v2_vectsp_2(A)
      & v1_binom(A)
      & ~ v3_realset2(A) ) )).

fof(rc3_polynom7,axiom,(
    ! [A] :
      ( ( ~ v3_realset2(A)
        & l2_struct_0(A) )
     => ? [B] :
          ( m1_subset_1(B,u1_struct_0(A))
          & v8_rlvect_1(B,A) ) ) )).

fof(rc4_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_realset2(B)
        & l2_struct_0(B) )
     => ? [C] :
          ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
          & v1_polynom7(C,A,B) ) ) )).

fof(rc5_polynom7,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & ~ v3_realset2(B)
        & l2_struct_0(B) )
     => ? [C] :
          ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
          & v2_polynom1(C,k14_polynom1(A),B)
          & v1_polynom7(C,A,B) ) ) )).

fof(rc6_polynom7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_pboole(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_seq_1(B)
          & v7_seqm_3(B)
          & v1_polynom1(B)
          & v2_polynom7(B,A) ) ) )).

fof(cc2_polynom7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ( ( v7_seqm_3(B)
              & v1_polynom1(B)
              & v2_polynom7(B,A) )
           => ( v7_seqm_3(B)
              & v1_polynom1(B)
              & ~ v1_polynom2(B,A) ) ) ) ) )).

fof(rc7_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ? [C] :
          ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
          & v3_polynom7(C,A,B) ) ) )).

fof(cc3_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
              & v3_polynom7(C,A,B) )
           => ( v1_funct_1(C)
              & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
              & v2_polynom1(C,k14_polynom1(A),B) ) ) ) ) )).

fof(rc8_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ? [C] :
          ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
          & v4_polynom7(C,A,B) ) ) )).

fof(cc4_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
              & v4_polynom7(C,A,B) )
           => ( v1_funct_1(C)
              & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
              & v2_polynom1(C,k14_polynom1(A),B)
              & v3_polynom7(C,A,B) ) ) ) ) )).

fof(fc1_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ( v1_relat_1(k26_polynom1(A,B))
        & v1_funct_1(k26_polynom1(A,B))
        & v1_funct_2(k26_polynom1(A,B),k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(k26_polynom1(A,B),k14_polynom1(A),B)
        & v3_polynom7(k26_polynom1(A,B),A,B)
        & v4_polynom7(k26_polynom1(A,B),A,B) ) ) )).

fof(fc2_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_group_1(B)
        & l3_vectsp_1(B) )
     => ( v1_relat_1(k27_polynom1(A,B))
        & v1_funct_1(k27_polynom1(A,B))
        & v1_funct_2(k27_polynom1(A,B),k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(k27_polynom1(A,B),k14_polynom1(A),B)
        & v3_polynom7(k27_polynom1(A,B),A,B)
        & v4_polynom7(k27_polynom1(A,B),A,B) ) ) )).

fof(fc3_polynom7,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( v1_relat_1(k4_polynom7(A,B,C))
        & v1_funct_1(k4_polynom7(A,B,C))
        & v1_funct_2(k4_polynom7(A,B,C),k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(k4_polynom7(A,B,C),k14_polynom1(A),B)
        & v3_polynom7(k4_polynom7(A,B,C),A,B)
        & v4_polynom7(k4_polynom7(A,B,C),A,B) ) ) )).

fof(fc4_polynom7,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v5_rlvect_1(B)
        & v7_vectsp_1(B)
        & v1_algstr_1(B)
        & v1_binom(B)
        & l3_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(C,k14_polynom1(A),B)
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( v1_relat_1(k5_polynom7(A,B,C,D))
        & v1_funct_1(k5_polynom7(A,B,C,D))
        & v1_funct_2(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(k5_polynom7(A,B,C,D),k14_polynom1(A),B) ) ) )).

fof(fc5_polynom7,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v5_rlvect_1(B)
        & v7_vectsp_1(B)
        & v1_algstr_1(B)
        & v1_binom(B)
        & l3_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(C,k14_polynom1(A),B)
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( v1_relat_1(k6_polynom7(A,B,C,D))
        & v1_funct_1(k6_polynom7(A,B,C,D))
        & v1_funct_2(k6_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
        & v2_polynom1(k6_polynom7(A,B,C,D),k14_polynom1(A),B) ) ) )).

fof(d1_polynom7,axiom,(
    $true )).

fof(d2_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( v1_polynom7(C,A,B)
          <=> C != k26_polynom1(A,B) ) ) ) )).

fof(t1_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( C = k26_polynom1(A,B)
          <=> k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0 ) ) ) )).

fof(t2_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ( ~ ( ~ v3_realset2(B)
            & ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                  & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
               => k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0 ) )
        & ~ ( ~ ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                    & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
                 => k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0 )
            & v3_realset2(B) ) ) ) )).

fof(d3_polynom7,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ( v2_polynom7(B,A)
      <=> ? [C] :
            ( m1_subset_1(C,A)
            & k1_polynom2(A,B) = k1_tarski(C) ) ) ) )).

fof(t3_polynom7,axiom,(
    ! [A] :
      ( ( v7_seqm_3(A)
        & v1_polynom1(A)
        & m1_pboole(A,k1_xboole_0) )
     => A = k16_polynom1(k1_xboole_0) ) )).

fof(t4_polynom7,axiom,(
    ! [A] :
      ( ( v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v2_group_1(A)
        & v7_vectsp_1(A)
        & ~ v3_realset2(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k14_polynom1(k1_xboole_0),u1_struct_0(A))
            & v2_polynom1(B,k14_polynom1(k1_xboole_0),A)
            & m2_relset_1(B,k14_polynom1(k1_xboole_0),u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k1_xboole_0,u1_struct_0(A))
                & m2_relset_1(C,k1_xboole_0,u1_struct_0(A)) )
             => k5_polynom2(k1_xboole_0,A,B,C) = k15_polynom1(k1_xboole_0,A,B,k16_polynom1(k1_xboole_0)) ) ) ) )).

fof(t5_polynom7,axiom,(
    ! [A] :
      ( ( v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_vectsp_1(A)
        & ~ v3_realset2(A)
        & l3_vectsp_1(A) )
     => r2_quofield(k30_polynom1(k1_xboole_0,A),A) ) )).

fof(d4_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( v3_polynom7(C,A,B)
          <=> ? [D] :
                ( v7_seqm_3(D)
                & v1_polynom1(D)
                & m1_pboole(D,A)
                & ! [E] :
                    ( ( v7_seqm_3(E)
                      & v1_polynom1(E)
                      & m1_pboole(E,A) )
                   => ( E != D
                     => k15_polynom1(A,B,C,E) = k1_rlvect_1(B) ) ) ) ) ) ) )).

fof(t6_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
              & v3_polynom7(C,A,B)
              & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
          <=> ~ ( k12_polynom1(k14_polynom1(A),B,C) != k1_xboole_0
                & ! [D] :
                    ( ( v7_seqm_3(D)
                      & v1_polynom1(D)
                      & m1_pboole(D,A) )
                   => k12_polynom1(k14_polynom1(A),B,C) != k1_tarski(D) ) ) ) ) ) )).

fof(d5_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( ( v7_seqm_3(D)
                & v1_polynom1(D)
                & m1_pboole(D,A) )
             => k1_polynom7(A,B,C,D) = k1_polynom1(k14_polynom1(A),u1_struct_0(B),k26_polynom1(A,B),D,C) ) ) ) )).

fof(d6_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & v3_polynom7(C,A,B)
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ! [D] :
              ( ( v7_seqm_3(D)
                & v1_polynom1(D)
                & m1_pboole(D,A) )
             => ( D = k2_polynom7(A,B,C)
              <=> ~ ( k15_polynom1(A,B,C,D) = k1_rlvect_1(B)
                    & ~ ( k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0
                        & D = k16_polynom1(A) ) ) ) ) ) ) )).

fof(d7_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & v3_polynom7(C,A,B)
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => k3_polynom7(A,B,C) = k15_polynom1(A,B,C,k2_polynom7(A,B,C)) ) ) )).

fof(t7_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & v3_polynom7(C,A,B)
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0
            | k12_polynom1(k14_polynom1(A),B,C) = k1_tarski(k2_polynom7(A,B,C)) ) ) ) )).

fof(t8_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v7_seqm_3(C)
            & v1_polynom1(C)
            & m1_pboole(C,A) )
         => ( k3_polynom7(A,B,k1_polynom7(A,B,k1_rlvect_1(B),C)) = k1_rlvect_1(B)
            & k2_polynom7(A,B,k1_polynom7(A,B,k1_rlvect_1(B),C)) = k16_polynom1(A) ) ) ) )).

fof(t9_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( ( v7_seqm_3(D)
                & v1_polynom1(D)
                & m1_pboole(D,A) )
             => k3_polynom7(A,B,k1_polynom7(A,B,C,D)) = C ) ) ) )).

fof(t10_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_realset2(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v8_rlvect_1(C,B)
            & m1_subset_1(C,u1_struct_0(B)) )
         => ! [D] :
              ( ( v7_seqm_3(D)
                & v1_polynom1(D)
                & m1_pboole(D,A) )
             => k2_polynom7(A,B,k1_polynom7(A,B,C,D)) = D ) ) ) )).

fof(t11_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & v3_polynom7(C,A,B)
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => k1_polynom7(A,B,k3_polynom7(A,B,C),k2_polynom7(A,B,C)) = C ) ) )).

fof(t12_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v7_vectsp_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v3_polynom7(C,A,B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,u1_struct_0(B))
                    & m2_relset_1(D,A,u1_struct_0(B)) )
                 => k5_polynom2(A,B,C,D) = k1_group_1(B,k3_polynom7(A,B,C),k3_polynom2(A,k2_polynom7(A,B,C),B,D)) ) ) ) ) )).

fof(t13_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v7_vectsp_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( ( v7_seqm_3(D)
                    & v1_polynom1(D)
                    & m1_pboole(D,A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,u1_struct_0(B))
                        & m2_relset_1(E,A,u1_struct_0(B)) )
                     => k5_polynom2(A,B,k1_polynom7(A,B,C,D),E) = k1_group_1(B,C,k3_polynom2(A,D,B,E)) ) ) ) ) ) )).

fof(d8_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( v4_polynom7(C,A,B)
          <=> ! [D] :
                ( ( v7_seqm_3(D)
                  & v1_polynom1(D)
                  & m1_pboole(D,A) )
               => ( D != k16_polynom1(A)
                 => k15_polynom1(A,B,C,D) = k1_rlvect_1(B) ) ) ) ) ) )).

fof(t14_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
              & v4_polynom7(C,A,B)
              & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
          <=> ( C = k26_polynom1(A,B)
              | k12_polynom1(k14_polynom1(A),B,C) = k1_tarski(k16_polynom1(A)) ) ) ) ) )).

fof(t15_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & v4_polynom7(C,A,B)
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0
            | k12_polynom1(k14_polynom1(A),B,C) = k1_tarski(k16_polynom1(A)) ) ) ) )).

fof(t16_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & v4_polynom7(C,A,B)
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( k2_polynom7(A,B,C) = k16_polynom1(A)
            & k3_polynom7(A,B,C) = k15_polynom1(A,B,C,k16_polynom1(A)) ) ) ) )).

fof(d9_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => k4_polynom7(A,B,C) = k1_polynom1(k14_polynom1(A),u1_struct_0(B),k26_polynom1(A,B),k16_polynom1(A),C) ) ) )).

fof(t17_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
              & v4_polynom7(C,A,B)
              & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
          <=> ? [D] :
                ( m1_subset_1(D,u1_struct_0(B))
                & C = k4_polynom7(A,B,D) ) ) ) ) )).

fof(t18_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_vectsp_1(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ( k15_polynom1(A,B,k4_polynom7(A,B,C),k16_polynom1(A)) = C
            & ! [D] :
                ( ( v7_seqm_3(D)
                  & v1_polynom1(D)
                  & m1_pboole(D,A) )
               => ( D != k16_polynom1(A)
                 => k15_polynom1(A,B,k4_polynom7(A,B,C),D) = k1_rlvect_1(B) ) ) ) ) ) )).

fof(t19_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => k4_polynom7(A,B,k1_rlvect_1(B)) = k26_polynom1(A,B) ) )).

fof(t20_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_group_1(B)
        & l2_vectsp_1(B) )
     => k4_polynom7(A,B,k2_group_1(B)) = k27_polynom1(A,B) ) )).

fof(t21_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => ( k4_polynom7(A,B,C) = k4_polynom7(A,B,D)
              <=> C = D ) ) ) ) )).

fof(t22_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ( k12_polynom1(k14_polynom1(A),B,k4_polynom7(A,B,C)) = k1_xboole_0
            | k12_polynom1(k14_polynom1(A),B,k4_polynom7(A,B,C)) = k1_tarski(k16_polynom1(A)) ) ) ) )).

fof(t23_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ( k2_polynom7(A,B,k4_polynom7(A,B,C)) = k16_polynom1(A)
            & k3_polynom7(A,B,k4_polynom7(A,B,C)) = C ) ) ) )).

fof(t24_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v7_vectsp_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v4_polynom7(C,A,B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,u1_struct_0(B))
                    & m2_relset_1(D,A,u1_struct_0(B)) )
                 => k5_polynom2(A,B,C,D) = k3_polynom7(A,B,C) ) ) ) ) )).

fof(t25_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v7_vectsp_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,u1_struct_0(B))
                    & m2_relset_1(D,A,u1_struct_0(B)) )
                 => k5_polynom2(A,B,k4_polynom7(A,B,C),D) = C ) ) ) ) )).

fof(d10_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_vectsp_1(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,k14_polynom1(A),u1_struct_0(B))
                    & m2_relset_1(E,k14_polynom1(A),u1_struct_0(B)) )
                 => ( E = k5_polynom7(A,B,C,D)
                  <=> ! [F] :
                        ( ( v7_seqm_3(F)
                          & v1_polynom1(F)
                          & m1_pboole(F,A) )
                       => k15_polynom1(A,B,E,F) = k1_group_1(B,D,k15_polynom1(A,B,C,F)) ) ) ) ) ) ) )).

fof(d11_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & l2_vectsp_1(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & v1_funct_2(E,k14_polynom1(A),u1_struct_0(B))
                    & m2_relset_1(E,k14_polynom1(A),u1_struct_0(B)) )
                 => ( E = k6_polynom7(A,B,C,D)
                  <=> ! [F] :
                        ( ( v7_seqm_3(F)
                          & v1_polynom1(F)
                          & m1_pboole(F,A) )
                       => k15_polynom1(A,B,E,F) = k1_group_1(B,k15_polynom1(A,B,C,F),D) ) ) ) ) ) ) )).

fof(t26_polynom7,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v7_group_1(B)
        & l2_vectsp_1(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
            & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => k5_polynom7(A,B,C,D) = k6_polynom7(A,B,C,D) ) ) ) )).

fof(t27_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v5_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k5_polynom7(A,B,C,D) = k28_polynom1(A,B,k4_polynom7(A,B,D),C) ) ) ) ) )).

fof(t28_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v4_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k6_polynom7(A,B,C,D) = k28_polynom1(A,B,C,k4_polynom7(A,B,D)) ) ) ) ) )).

fof(t29_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v1_algstr_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v2_polynom1(C,k14_polynom1(A),B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,u1_struct_0(B))
                        & m2_relset_1(E,A,u1_struct_0(B)) )
                     => k5_polynom2(A,B,k5_polynom7(A,B,C,D),E) = k10_group_1(B,D,k5_polynom2(A,B,C,E)) ) ) ) ) ) )).

fof(t30_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v7_vectsp_1(B)
            & v1_algstr_1(B)
            & v2_algstr_1(B)
            & v2_vectsp_2(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v2_polynom1(C,k14_polynom1(A),B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,u1_struct_0(B))
                        & m2_relset_1(E,A,u1_struct_0(B)) )
                     => k5_polynom2(A,B,k5_polynom7(A,B,C,D),E) = k1_group_1(B,D,k5_polynom2(A,B,C,E)) ) ) ) ) ) )).

fof(t31_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v1_algstr_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v2_polynom1(C,k14_polynom1(A),B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,u1_struct_0(B))
                        & m2_relset_1(E,A,u1_struct_0(B)) )
                     => k5_polynom2(A,B,k6_polynom7(A,B,C,D),E) = k10_group_1(B,k5_polynom2(A,B,C,E),D) ) ) ) ) ) )).

fof(t32_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v1_algstr_1(B)
            & v2_algstr_1(B)
            & v2_vectsp_2(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v2_polynom1(C,k14_polynom1(A),B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,u1_struct_0(B))
                        & m2_relset_1(E,A,u1_struct_0(B)) )
                     => k5_polynom2(A,B,k6_polynom7(A,B,C,D),E) = k10_group_1(B,k5_polynom2(A,B,C,E),D) ) ) ) ) ) )).

fof(t33_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v1_algstr_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v2_polynom1(C,k14_polynom1(A),B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,u1_struct_0(B))
                        & m2_relset_1(E,A,u1_struct_0(B)) )
                     => k5_polynom2(A,B,k29_polynom1(A,B,k4_polynom7(A,B,D),C),E) = k10_group_1(B,D,k5_polynom2(A,B,C,E)) ) ) ) ) ) )).

fof(t34_polynom7,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( ( v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v2_group_1(B)
            & v4_group_1(B)
            & v7_group_1(B)
            & v7_vectsp_1(B)
            & v1_algstr_1(B)
            & ~ v3_realset2(B)
            & l3_vectsp_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
                & v2_polynom1(C,k14_polynom1(A),B)
                & m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,A,u1_struct_0(B))
                        & m2_relset_1(E,A,u1_struct_0(B)) )
                     => k5_polynom2(A,B,k29_polynom1(A,B,C,k4_polynom7(A,B,D)),E) = k10_group_1(B,k5_polynom2(A,B,C,E),D) ) ) ) ) ) )).

fof(dt_k1_polynom7,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B)
        & m1_subset_1(C,u1_struct_0(B))
        & v7_seqm_3(D)
        & v1_polynom1(D)
        & m1_pboole(D,A) )
     => ( v1_funct_1(k1_polynom7(A,B,C,D))
        & v1_funct_2(k1_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
        & v3_polynom7(k1_polynom7(A,B,C,D),A,B)
        & m2_relset_1(k1_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) )).

fof(dt_k2_polynom7,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & v3_polynom7(C,A,B)
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
     => ( v7_seqm_3(k2_polynom7(A,B,C))
        & v1_polynom1(k2_polynom7(A,B,C))
        & m1_pboole(k2_polynom7(A,B,C),A) ) ) )).

fof(dt_k3_polynom7,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & v3_polynom7(C,A,B)
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
     => m1_subset_1(k3_polynom7(A,B,C),u1_struct_0(B)) ) )).

fof(dt_k4_polynom7,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & l2_struct_0(B)
        & m1_subset_1(C,u1_struct_0(B)) )
     => ( v1_funct_1(k4_polynom7(A,B,C))
        & v1_funct_2(k4_polynom7(A,B,C),k14_polynom1(A),u1_struct_0(B))
        & m2_relset_1(k4_polynom7(A,B,C),k14_polynom1(A),u1_struct_0(B)) ) ) )).

fof(dt_k5_polynom7,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l2_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( v1_funct_1(k5_polynom7(A,B,C,D))
        & v1_funct_2(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
        & m2_relset_1(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) )).

fof(dt_k6_polynom7,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l2_vectsp_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
        & m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( v1_funct_1(k6_polynom7(A,B,C,D))
        & v1_funct_2(k6_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
        & m2_relset_1(k6_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) )).
%------------------------------------------------------------------------------