SET007 Axioms: SET007+650.ax


%------------------------------------------------------------------------------
% File     : SET007+650 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Ring of 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    : polynom3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   94 (   0 unit)
%            Number of atoms       :  909 (  86 equality)
%            Maximal formula depth :   21 (  10 average)
%            Number of connectives :  916 ( 101 ~  ;   6  |; 562  &)
%                                         (  13 <=>; 234 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   51 (   0 propositional; 1-3 arity)
%            Number of functors    :   72 (  10 constant; 0-6 arity)
%            Number of variables   :  278 (   0 singleton; 273 !;   5 ?)
%            Maximal term depth    :    7 (   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(cc1_polynom3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers) )
     => ! [C] :
          ( m1_finseq_1(C,k4_finseq_2(B,A))
         => v1_matrlin(C) ) ) )).

fof(fc1_polynom3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_relat_1(k5_polynom3(A))
        & v1_relat_2(k5_polynom3(A))
        & v4_relat_2(k5_polynom3(A))
        & v8_relat_2(k5_polynom3(A))
        & v1_partfun1(k5_polynom3(A),k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers))
        & v3_orders_1(k5_polynom3(A)) ) ) )).

fof(fc2_polynom3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( ~ v1_xboole_0(k6_polynom3(A,B))
        & v1_relat_1(k6_polynom3(A,B))
        & v1_funct_1(k6_polynom3(A,B))
        & v2_funct_1(k6_polynom3(A,B))
        & v1_finset_1(k6_polynom3(A,B))
        & v1_finseq_1(k6_polynom3(A,B))
        & v1_funcop_1(k6_polynom3(A,B))
        & v1_matrlin(k6_polynom3(A,B))
        & v1_polynom1(k6_polynom3(A,B)) ) ) )).

fof(fc3_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(C,A)
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k8_polynom3(A,B,C))
        & v1_relat_1(k8_polynom3(A,B,C))
        & v1_funct_1(k8_polynom3(A,B,C))
        & v1_funct_2(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k8_polynom3(A,B,C),A) ) ) )).

fof(fc4_polynom3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k10_polynom3(A,B))
        & v1_relat_1(k10_polynom3(A,B))
        & v1_funct_1(k10_polynom3(A,B))
        & v1_funct_2(k10_polynom3(A,B),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k10_polynom3(A,B),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k10_polynom3(A,B),A) ) ) )).

fof(fc5_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(C,A)
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k11_polynom3(A,B,C))
        & v1_relat_1(k11_polynom3(A,B,C))
        & v1_funct_1(k11_polynom3(A,B,C))
        & v1_funct_2(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k11_polynom3(A,B,C),A) ) ) )).

fof(fc6_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ( ~ v1_xboole_0(k12_polynom3(A))
        & v1_relat_1(k12_polynom3(A))
        & v1_funct_1(k12_polynom3(A))
        & v1_funct_2(k12_polynom3(A),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k12_polynom3(A),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k12_polynom3(A),A) ) ) )).

fof(fc7_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( ~ v1_xboole_0(k13_polynom3(A))
        & v1_relat_1(k13_polynom3(A))
        & v1_funct_1(k13_polynom3(A))
        & v1_funct_2(k13_polynom3(A),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k13_polynom3(A),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k13_polynom3(A),A) ) ) )).

fof(fc8_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ 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_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(C,A)
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k14_polynom3(A,B,C))
        & v1_relat_1(k14_polynom3(A,B,C))
        & v1_funct_1(k14_polynom3(A,B,C))
        & v1_funct_2(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k14_polynom3(A,B,C),A) ) ) )).

fof(fc9_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v3_rlvect_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A)) ) ) )).

fof(fc10_polynom3,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) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v4_rlvect_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A)) ) ) )).

fof(fc11_polynom3,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) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v5_rlvect_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A)) ) ) )).

fof(fc12_polynom3,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) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v6_rlvect_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A)) ) ) )).

fof(fc13_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v3_rlvect_1(k16_polynom3(A))
        & v4_rlvect_1(k16_polynom3(A))
        & v5_rlvect_1(k16_polynom3(A))
        & v6_rlvect_1(k16_polynom3(A))
        & v7_group_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A)) ) ) )).

fof(fc14_polynom3,axiom,(
    ! [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_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v3_rlvect_1(k16_polynom3(A))
        & v4_rlvect_1(k16_polynom3(A))
        & v5_rlvect_1(k16_polynom3(A))
        & v6_rlvect_1(k16_polynom3(A))
        & v4_group_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A)) ) ) )).

fof(fc15_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v3_rlvect_1(k16_polynom3(A))
        & v4_rlvect_1(k16_polynom3(A))
        & v5_rlvect_1(k16_polynom3(A))
        & v6_rlvect_1(k16_polynom3(A))
        & v2_group_1(k16_polynom3(A))
        & v7_group_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A))
        & v6_vectsp_1(k16_polynom3(A))
        & v8_vectsp_1(k16_polynom3(A)) ) ) )).

fof(fc16_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v3_rlvect_1(k16_polynom3(A))
        & v4_rlvect_1(k16_polynom3(A))
        & v5_rlvect_1(k16_polynom3(A))
        & v6_rlvect_1(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A))
        & v4_vectsp_1(k16_polynom3(A))
        & v5_vectsp_1(k16_polynom3(A))
        & v7_vectsp_1(k16_polynom3(A)) ) ) )).

fof(t1_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(C,k5_finsop_1(B))
                 => k1_funct_1(B,C) = k1_rlvect_1(A) ) )
           => k9_rlvect_1(A,B) = k1_rlvect_1(A) ) ) ) )).

fof(t2_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => k9_rlvect_1(A,B) = k9_rlvect_1(A,k4_finseq_5(u1_struct_0(A),B)) ) ) )).

fof(t3_polynom3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => k15_rvsum_1(A) = k15_rvsum_1(k4_finseq_5(k1_numbers,A)) ) )).

fof(t4_polynom3,axiom,(
    ! [A] :
      ( m1_trees_4(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k5_finsop_1(A))
           => r1_xreal_0(k3_wsierp_1(A,B),k9_wsierp_1(A)) ) ) ) )).

fof(d1_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ( r1_polynom3(A,B,C)
              <=> ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & r2_hidden(D,k2_finseq_1(A))
                    & ~ r1_xreal_0(k3_wsierp_1(C,D),k3_wsierp_1(B,D))
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r1_xreal_0(np__1,E)
                         => ( r1_xreal_0(D,E)
                            | k3_wsierp_1(B,E) = k3_wsierp_1(C,E) ) ) ) ) ) ) ) ) )).

fof(d2_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ( r2_polynom3(A,B,C)
              <=> ( r1_polynom3(A,B,C)
                  | B = C ) ) ) ) ) )).

fof(t5_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k5_numbers,k4_finseq_2(A,k5_numbers))
                 => ( ( ( r1_polynom3(A,B,C)
                        & r1_polynom3(A,C,D) )
                     => r1_polynom3(A,B,D) )
                    & ( ~ ( ~ ( r1_polynom3(A,B,C)
                              & r2_polynom3(A,C,D) )
                          & ~ ( r2_polynom3(A,B,C)
                              & r1_polynom3(A,C,D) )
                          & ~ ( r2_polynom3(A,B,C)
                              & r2_polynom3(A,C,D) ) )
                     => r2_polynom3(A,B,D) ) ) ) ) ) ) )).

fof(t6_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ~ ( B != C
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( r2_hidden(D,k2_finseq_1(A))
                          & k3_wsierp_1(B,D) != k3_wsierp_1(C,D)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( r1_xreal_0(np__1,E)
                               => ( r1_xreal_0(D,E)
                                  | k3_wsierp_1(B,E) = k3_wsierp_1(C,E) ) ) ) ) ) ) ) ) ) )).

fof(t7_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ( r2_polynom3(A,B,C)
                | r1_polynom3(A,C,B) ) ) ) ) )).

fof(d3_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_2(B)
            & v4_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers))
            & m2_relset_1(B,k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers)) )
         => ( B = k5_polynom3(A)
          <=> ! [C] :
                ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
               => ! [D] :
                    ( m2_finseq_2(D,k5_numbers,k4_finseq_2(A,k5_numbers))
                   => ( r2_hidden(k4_tarski(C,D),B)
                    <=> r2_polynom3(A,C,D) ) ) ) ) ) ) )).

fof(d4_polynom3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k4_finseq_2(A,k5_numbers))
             => ( C = k6_polynom3(A,B)
              <=> ? [D] :
                    ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(k4_finseq_2(A,k5_numbers)))
                    & C = k2_triang_1(k4_finseq_2(A,k5_numbers),D,k5_polynom3(A))
                    & ! [E] :
                        ( m2_finseq_2(E,k5_numbers,k4_finseq_2(A,k5_numbers))
                       => ( r2_hidden(E,D)
                        <=> k9_wsierp_1(E) = B ) ) ) ) ) ) ) )).

fof(t8_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_finseq_1(k6_polynom3(np__1,A)) = np__1 ) )).

fof(t9_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_finseq_1(k6_polynom3(np__2,A)) = k1_nat_1(A,np__1) ) )).

fof(t10_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k6_polynom3(np__1,A) = k13_binarith(k4_finseq_2(np__1,k5_numbers),k13_binarith(k5_numbers,A)) ) )).

fof(t11_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( k1_funct_1(k6_polynom3(np__2,C),A) = k2_polynom3(k5_numbers,D,k5_binarith(C,D))
                          & k1_funct_1(k6_polynom3(np__2,C),B) = k2_polynom3(k5_numbers,E,k5_binarith(C,E)) )
                       => ( ~ ( ~ r1_xreal_0(B,A)
                              & r1_xreal_0(E,D) )
                          & ~ ( ~ r1_xreal_0(E,D)
                              & r1_xreal_0(B,A) ) ) ) ) ) ) ) ) )).

fof(t12_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( k1_funct_1(k6_polynom3(np__2,B),A) = k2_polynom3(k5_numbers,C,k5_binarith(B,C))
                      & k1_funct_1(k6_polynom3(np__2,B),k1_nat_1(A,np__1)) = k2_polynom3(k5_numbers,D,k5_binarith(B,D)) )
                   => D = k1_nat_1(C,np__1) ) ) ) ) ) )).

fof(t13_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_funct_1(k6_polynom3(np__2,A),np__1) = k2_polynom3(k5_numbers,np__0,A) ) )).

fof(t14_polynom3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k2_finseq_1(k1_nat_1(A,np__1)))
           => k1_funct_1(k6_polynom3(np__2,A),B) = k2_polynom3(k5_numbers,k5_binarith(B,np__1),k5_binarith(k1_nat_1(A,np__1),B)) ) ) ) )).

fof(d5_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ! [E] :
                      ( m2_finseq_1(E,k4_finseq_2(np__3,k5_numbers))
                     => ! [F] :
                          ( m2_finseq_2(F,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
                         => ( F = k7_polynom3(A,B,C,D,E)
                          <=> ( k3_finseq_1(F) = k3_finseq_1(E)
                              & ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ( r2_hidden(G,k5_finsop_1(E))
                                   => k1_funct_1(F,G) = k1_group_1(A,k1_group_1(A,k2_normsp_1(A,B,k4_finseq_4(k5_numbers,k5_numbers,k3_polynom1(k5_numbers,k5_numbers,k4_finseq_2(np__3,k5_numbers),E,G),np__1)),k2_normsp_1(A,C,k4_finseq_4(k5_numbers,k5_numbers,k3_polynom1(k5_numbers,k5_numbers,k4_finseq_2(np__3,k5_numbers),E,G),np__2))),k2_normsp_1(A,D,k4_finseq_4(k5_numbers,k5_numbers,k3_polynom1(k5_numbers,k5_numbers,k4_finseq_2(np__3,k5_numbers),E,G),np__3))) ) ) ) ) ) ) ) ) ) ) )).

fof(t15_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_group_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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ! [E] :
                      ( m2_finseq_1(E,k4_finseq_2(np__3,k5_numbers))
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,k5_finsop_1(E),k5_finsop_1(E))
                            & v3_funct_2(F,k5_finsop_1(E),k5_finsop_1(E))
                            & m2_relset_1(F,k5_finsop_1(E),k5_finsop_1(E)) )
                         => ! [G] :
                              ( m2_finseq_1(G,k4_finseq_2(np__3,k5_numbers))
                             => ( G = k1_partfun1(k5_finsop_1(E),k5_finsop_1(E),k5_numbers,k4_finseq_2(np__3,k5_numbers),F,E)
                               => k7_polynom3(A,B,C,D,G) = k1_partfun1(k5_finsop_1(E),k5_finsop_1(E),k5_numbers,u1_struct_0(A),F,k7_polynom3(A,B,C,D,E)) ) ) ) ) ) ) ) ) )).

fof(t16_polynom3,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k3_finseq_2(A))
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => k5_polynom1(A,k16_finseq_1(k3_finseq_2(A),B,C)) = k16_finseq_1(k5_numbers,k5_polynom1(A,B),C) ) ) )).

fof(t17_polynom3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m1_trees_4(B,k1_numbers,k5_numbers)
         => ( A = B
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k16_finseq_1(k1_numbers,A,C) = k16_finseq_1(k5_numbers,B,C) ) ) ) ) )).

fof(t18_polynom3,axiom,(
    ! [A] :
      ( m1_trees_4(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(B,C)
               => r1_xreal_0(k9_wsierp_1(k16_finseq_1(k5_numbers,A,B)),k9_wsierp_1(k16_finseq_1(k5_numbers,A,C))) ) ) ) ) )).

fof(t19_polynom3,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_finseq_1(B),C)
           => k16_finseq_1(A,B,k1_nat_1(C,np__1)) = k7_finseq_1(k16_finseq_1(A,B,C),k9_finseq_1(k1_funct_1(B,k1_nat_1(C,np__1)))) ) ) ) )).

fof(t20_polynom3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_finseq_1(A),B)
           => k15_rvsum_1(k16_finseq_1(k1_numbers,A,k1_nat_1(B,np__1))) = k2_xcmplx_0(k15_rvsum_1(k16_finseq_1(k1_numbers,A,B)),k1_wsierp_1(A,k1_nat_1(B,np__1))) ) ) ) )).

fof(t21_polynom3,axiom,(
    ! [A] :
      ( m1_trees_4(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(np__1,E)
                          & r1_xreal_0(D,k3_wsierp_1(A,k1_nat_1(B,np__1)))
                          & r1_xreal_0(E,k3_wsierp_1(A,k1_nat_1(C,np__1)))
                          & k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,A,B)),D) = k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,A,C)),E) )
                       => ( r1_xreal_0(k3_finseq_1(A),B)
                          | r1_xreal_0(k3_finseq_1(A),C)
                          | ( B = C
                            & D = E ) ) ) ) ) ) ) ) )).

fof(t22_polynom3,axiom,(
    ! [A,B,C] :
      ( m2_finseq_1(C,k3_finseq_2(A))
     => ! [D] :
          ( m2_finseq_1(D,k3_finseq_2(B))
         => ! [E] :
              ( m2_subset_1(E,k1_numbers,k5_numbers)
             => ! [F] :
                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                 => ! [G] :
                      ( m2_subset_1(G,k1_numbers,k5_numbers)
                     => ! [H] :
                          ( m2_subset_1(H,k1_numbers,k5_numbers)
                         => ( ( r2_hidden(E,k5_finsop_1(C))
                              & r2_hidden(F,k5_finsop_1(D))
                              & r2_hidden(G,k5_finsop_1(k1_funct_1(C,E)))
                              & r2_hidden(H,k5_finsop_1(k1_funct_1(D,F)))
                              & k5_polynom1(A,C) = k5_polynom1(B,D)
                              & k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,k5_polynom1(A,C),k5_binarith(E,np__1))),G) = k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,k5_polynom1(B,D),k5_binarith(F,np__1))),H) )
                           => ( E = F
                              & G = H ) ) ) ) ) ) ) ) )).

fof(t23_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(k3_algseq_1(A,B),C)
              <=> r1_algseq_1(A,B,C) ) ) ) ) )).

fof(d6_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k8_polynom3(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k2_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) )).

fof(t24_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r1_algseq_1(A,B,D)
                      & r1_algseq_1(A,C,D) )
                   => r1_algseq_1(A,k8_polynom3(A,B,C),D) ) ) ) ) ) )).

fof(t25_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => r1_tarski(k4_algseq_1(A,k8_polynom3(A,B,C)),k4_subset_1(k5_numbers,k4_algseq_1(A,B),k4_algseq_1(A,C))) ) ) ) )).

fof(t26_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & l1_rlvect_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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k8_polynom3(A,k8_polynom3(A,B,C),D) = k8_polynom3(A,B,k8_polynom3(A,C,D)) ) ) ) ) )).

fof(d7_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( C = k10_polynom3(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(A,C,D) = k5_rlvect_1(A,k2_normsp_1(A,B,D)) ) ) ) ) ) )).

fof(d8_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => k11_polynom3(A,B,C) = k8_polynom3(A,B,k10_polynom3(A,C)) ) ) ) )).

fof(t27_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_normsp_1(A,k11_polynom3(A,B,C),D) = k6_rlvect_1(A,k2_normsp_1(A,B,D),k2_normsp_1(A,C,D)) ) ) ) ) )).

fof(d9_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => k12_polynom3(A) = k3_finsop_1(u1_struct_0(A),k5_numbers,k1_rlvect_1(A)) ) )).

fof(t28_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_normsp_1(A,k12_polynom3(A),B) = k1_rlvect_1(A) ) ) )).

fof(t29_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v5_rlvect_1(A)
        & l1_rlvect_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)) )
         => k8_polynom3(A,B,k12_polynom3(A)) = B ) ) )).

fof(t30_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_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)) )
         => k11_polynom3(A,B,B) = k12_polynom3(A) ) ) )).

fof(d10_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => k13_polynom3(A) = k1_polynom1(k5_numbers,u1_struct_0(A),k12_polynom3(A),np__0,k2_group_1(A)) ) )).

fof(t31_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( k2_normsp_1(A,k13_polynom3(A),np__0) = k2_group_1(A)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( B != np__0
             => k2_normsp_1(A,k13_polynom3(A),B) = k1_rlvect_1(A) ) ) ) ) )).

fof(d11_polynom3,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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k14_polynom3(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ? [F] :
                            ( m2_finseq_1(F,u1_struct_0(A))
                            & k3_finseq_1(F) = k1_nat_1(E,np__1)
                            & k2_normsp_1(A,D,E) = k9_rlvect_1(A,F)
                            & ! [G] :
                                ( m2_subset_1(G,k1_numbers,k5_numbers)
                               => ( r2_hidden(G,k5_finsop_1(F))
                                 => k1_funct_1(F,G) = k1_group_1(A,k2_normsp_1(A,B,k5_binarith(G,np__1)),k2_normsp_1(A,C,k5_binarith(k1_nat_1(E,np__1),G))) ) ) ) ) ) ) ) ) ) )).

fof(t32_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_vectsp_1(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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k14_polynom3(A,B,k9_polynom3(A,C,D)) = k9_polynom3(A,k14_polynom3(A,B,C),k14_polynom3(A,B,D)) ) ) ) ) )).

fof(t33_polynom3,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)
        & 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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k14_polynom3(A,k9_polynom3(A,B,C),D) = k9_polynom3(A,k14_polynom3(A,B,D),k14_polynom3(A,C,D)) ) ) ) ) )).

fof(t34_polynom3,axiom,(
    ! [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_vectsp_1(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] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k14_polynom3(A,k14_polynom3(A,B,C),D) = k14_polynom3(A,B,k14_polynom3(A,C,D)) ) ) ) ) )).

fof(t35_polynom3,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) )
     => ! [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)) )
         => k14_polynom3(A,B,k12_polynom3(A)) = k12_polynom3(A) ) ) )).

fof(t36_polynom3,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_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k14_polynom3(A,B,k13_polynom3(A)) = B ) ) )).

fof(d12_polynom3,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) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v3_vectsp_1(B)
            & l3_vectsp_1(B) )
         => ( B = k16_polynom3(A)
          <=> ( ! [C] :
                  ( r2_hidden(C,u1_struct_0(B))
                <=> ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                    & v1_algseq_1(C,A)
                    & m2_relset_1(C,k5_numbers,u1_struct_0(A)) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,u1_struct_0(A))
                            & m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
                         => ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,k5_numbers,u1_struct_0(A))
                                & m2_relset_1(F,k5_numbers,u1_struct_0(A)) )
                             => ( ( C = E
                                  & D = F )
                               => k2_rlvect_1(B,C,D) = k8_polynom3(A,E,F) ) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] :
                      ( m1_subset_1(D,u1_struct_0(B))
                     => ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,u1_struct_0(A))
                            & m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
                         => ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,k5_numbers,u1_struct_0(A))
                                & m2_relset_1(F,k5_numbers,u1_struct_0(A)) )
                             => ( ( C = E
                                  & D = F )
                               => k1_group_1(B,C,D) = k14_polynom3(A,E,F) ) ) ) ) )
              & k1_rlvect_1(B) = k12_polynom3(A)
              & k2_vectsp_1(B) = k13_polynom3(A) ) ) ) ) )).

fof(t37_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A) )
     => k2_group_1(k16_polynom3(A)) = k13_polynom3(A) ) )).

fof(s1_polynom3,axiom,(
    ? [A] :
      ( m2_finseq_1(A,k3_finseq_2(f1_s1_polynom3))
      & k3_finseq_1(A) = f2_s1_polynom3
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k2_finseq_1(f2_s1_polynom3))
           => ( k3_finseq_1(k3_polynom1(f1_s1_polynom3,k5_numbers,k3_finseq_2(f1_s1_polynom3),A,B)) = f3_s1_polynom3(B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k5_finsop_1(k3_polynom1(f1_s1_polynom3,k5_numbers,k3_finseq_2(f1_s1_polynom3),A,B)))
                   => k1_funct_1(k3_polynom1(f1_s1_polynom3,k5_numbers,k3_finseq_2(f1_s1_polynom3),A,B),C) = f4_s1_polynom3(B,C) ) ) ) ) ) ) )).

fof(s2_polynom3,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,u1_struct_0(f1_s2_polynom3))
      & v1_algseq_1(A,f1_s2_polynom3)
      & m2_relset_1(A,k5_numbers,u1_struct_0(f1_s2_polynom3))
      & r1_xreal_0(k3_algseq_1(f1_s2_polynom3,A),f2_s2_polynom3)
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(f2_s2_polynom3,B)
           => k2_normsp_1(f1_s2_polynom3,A,B) = f3_s2_polynom3(B) ) ) ) )).

fof(antisymmetry_r1_polynom3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k5_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
     => ( r1_polynom3(A,B,C)
       => ~ r1_polynom3(A,C,B) ) ) )).

fof(reflexivity_r2_polynom3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k5_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
     => r2_polynom3(A,B,B) ) )).

fof(dt_k1_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_finseq_1(C,A) )
     => m2_finseq_1(k1_polynom3(A,B,C),A) ) )).

fof(redefinition_k1_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_finseq_1(C,A) )
     => k1_polynom3(A,B,C) = k2_finseq_3(B,C) ) )).

fof(dt_k2_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m2_finseq_2(k2_polynom3(A,B,C),A,k4_finseq_2(np__2,A)) ) )).

fof(redefinition_k2_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k2_polynom3(A,B,C) = k10_finseq_1(B,C) ) )).

fof(dt_k3_polynom3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k4_finseq_2(B,A))
        & m1_subset_1(E,k4_finseq_2(C,A)) )
     => m2_finseq_2(k3_polynom3(A,B,C,D,E),A,k4_finseq_2(k1_nat_1(B,C),A)) ) )).

fof(redefinition_k3_polynom3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k4_finseq_2(B,A))
        & m1_subset_1(E,k4_finseq_2(C,A)) )
     => k3_polynom3(A,B,C,D,E) = k7_finseq_1(D,E) ) )).

fof(dt_k4_polynom3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_finseq_1(D,k4_finseq_2(B,A))
        & m1_finseq_1(E,k4_finseq_2(C,A)) )
     => m2_finseq_2(k4_polynom3(A,B,C,D,E),k4_finseq_2(k1_nat_1(B,C),A),k3_finseq_2(k4_finseq_2(k1_nat_1(B,C),A))) ) )).

fof(redefinition_k4_polynom3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_finseq_1(D,k4_finseq_2(B,A))
        & m1_finseq_1(E,k4_finseq_2(C,A)) )
     => k4_polynom3(A,B,C,D,E) = k4_matrlin(D,E) ) )).

fof(dt_k5_polynom3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_relat_2(k5_polynom3(A))
        & v4_relat_2(k5_polynom3(A))
        & v8_relat_2(k5_polynom3(A))
        & v1_partfun1(k5_polynom3(A),k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers))
        & m2_relset_1(k5_polynom3(A),k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers)) ) ) )).

fof(dt_k6_polynom3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_finseq_1(k6_polynom3(A,B),k4_finseq_2(A,k5_numbers)) ) )).

fof(dt_k7_polynom3,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(A)
        & l1_group_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A))
        & v1_funct_1(D)
        & v1_funct_2(D,k5_numbers,u1_struct_0(A))
        & m1_relset_1(D,k5_numbers,u1_struct_0(A))
        & m1_finseq_1(E,k4_finseq_2(np__3,k5_numbers)) )
     => m2_finseq_2(k7_polynom3(A,B,C,D,E),u1_struct_0(A),k3_finseq_2(u1_struct_0(A))) ) )).

fof(dt_k8_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k8_polynom3(A,B,C))
        & v1_funct_2(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k9_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k9_polynom3(A,B,C))
        & v1_funct_2(k9_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k9_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(commutativity_k9_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => k9_polynom3(A,B,C) = k9_polynom3(A,C,B) ) )).

fof(redefinition_k9_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => k9_polynom3(A,B,C) = k8_polynom3(A,B,C) ) )).

fof(dt_k10_polynom3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k10_polynom3(A,B))
        & v1_funct_2(k10_polynom3(A,B),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k10_polynom3(A,B),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k11_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k11_polynom3(A,B,C))
        & v1_funct_2(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k12_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ( v1_funct_1(k12_polynom3(A))
        & v1_funct_2(k12_polynom3(A),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k12_polynom3(A),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k13_polynom3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => ( v1_funct_1(k13_polynom3(A))
        & v1_funct_2(k13_polynom3(A),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k13_polynom3(A),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k14_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l3_vectsp_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k14_polynom3(A,B,C))
        & v1_funct_2(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k15_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k15_polynom3(A,B,C))
        & v1_funct_2(k15_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k15_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(commutativity_k15_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => k15_polynom3(A,B,C) = k15_polynom3(A,C,B) ) )).

fof(redefinition_k15_polynom3,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v7_group_1(A)
        & l3_vectsp_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => k15_polynom3(A,B,C) = k14_polynom3(A,B,C) ) )).

fof(dt_k16_polynom3,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) )
     => ( ~ v3_struct_0(k16_polynom3(A))
        & v3_vectsp_1(k16_polynom3(A))
        & l3_vectsp_1(k16_polynom3(A)) ) ) )).
%------------------------------------------------------------------------------