SET007 Axioms: SET007+667.ax


%------------------------------------------------------------------------------
% File     : SET007+667 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Fundamental Theorem of Algebra
% 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    : polynom5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  104 (   4 unit)
%            Number of atoms       : 1078 ( 102 equality)
%            Maximal formula depth :   27 (  10 average)
%            Number of connectives : 1101 ( 127 ~  ;   1  |; 706  &)
%                                         (  13 <=>; 254 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   49 (   1 propositional; 0-3 arity)
%            Number of functors    :   70 (   8 constant; 0-6 arity)
%            Number of variables   :  257 (   0 singleton; 248 !;   9 ?)
%            Maximal term depth    :    6 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(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)
        & 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))
        & m1_subset_1(C,k5_numbers) )
     => ( ~ v1_xboole_0(k2_polynom5(A,B,C))
        & v1_relat_1(k2_polynom5(A,B,C))
        & v1_funct_1(k2_polynom5(A,B,C))
        & v1_funct_2(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k2_polynom5(A,B,C),A) ) ) )).

fof(fc2_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ 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_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))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k3_polynom5(A,B,C))
        & v1_relat_1(k3_polynom5(A,B,C))
        & v1_funct_1(k3_polynom5(A,B,C))
        & v1_funct_2(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k3_polynom5(A,B,C),A) ) ) )).

fof(fc3_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k4_polynom5(A,B,C))
        & v1_relat_1(k4_polynom5(A,B,C))
        & v1_funct_1(k4_polynom5(A,B,C))
        & v1_funct_2(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k4_polynom5(A,B,C),A) ) ) )).

fof(fc4_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(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)
        & v1_polynom5(k12_polynom3(A),A) ) ) )).

fof(rc1_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ? [B] :
          ( m1_relset_1(B,k5_numbers,u1_struct_0(A))
          & ~ v1_xboole_0(B)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,u1_struct_0(A))
          & v1_partfun1(B,k5_numbers,u1_struct_0(A))
          & v1_algseq_1(B,A)
          & v1_polynom5(B,A) ) ) )).

fof(fc5_polynom5,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_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)
        & 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_xboole_0(k7_polynom5(A,B))
        & v1_relat_1(k7_polynom5(A,B))
        & v1_funct_1(k7_polynom5(A,B))
        & v1_funct_2(k7_polynom5(A,B),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k7_polynom5(A,B),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k7_polynom5(A,B),A) ) ) )).

fof(fc6_polynom5,axiom,
    ( ~ v3_struct_0(k1_complfld)
    & v2_group_1(k1_complfld)
    & v4_group_1(k1_complfld)
    & v7_group_1(k1_complfld)
    & v3_rlvect_1(k1_complfld)
    & v4_rlvect_1(k1_complfld)
    & v5_rlvect_1(k1_complfld)
    & v6_rlvect_1(k1_complfld)
    & v3_vectsp_1(k1_complfld)
    & v4_vectsp_1(k1_complfld)
    & v5_vectsp_1(k1_complfld)
    & v6_vectsp_1(k1_complfld)
    & v7_vectsp_1(k1_complfld)
    & v8_vectsp_1(k1_complfld)
    & v9_vectsp_1(k1_complfld)
    & ~ v10_vectsp_1(k1_complfld)
    & v2_vectsp_2(k1_complfld)
    & v1_algstr_1(k1_complfld)
    & v2_algstr_1(k1_complfld)
    & v3_algstr_1(k1_complfld)
    & v4_algstr_1(k1_complfld)
    & v5_algstr_1(k1_complfld)
    & v6_algstr_1(k1_complfld)
    & ~ v3_realset2(k1_complfld)
    & v2_polynom5(k1_complfld) )).

fof(rc2_polynom5,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)
      & v9_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v2_vectsp_2(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)
      & v2_polynom5(A) ) )).

fof(t1_polynom5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( A != np__0
              & B != np__0
              & ~ r1_xreal_0(np__0,k3_real_1(k5_real_1(k5_real_1(k2_nat_1(A,B),A),B),np__1)) ) ) ) )).

fof(t2_polynom5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ~ r1_xreal_0(B,np__0)
           => r1_xreal_0(k7_xcmplx_0(k1_square_1(A,B),k2_square_1(A,B)),np__1) ) ) ) )).

fof(t3_polynom5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ! [C] :
                ( v1_xreal_0(C)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ~ r1_xreal_0(np__1,C)
                    & ~ r1_xreal_0(B,k3_xcmplx_0(C,A)) ) )
           => r1_xreal_0(B,np__0) ) ) ) )).

fof(t4_polynom5,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,k5_finsop_1(A))
             => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,k5_finsop_1(A))
             => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k15_rvsum_1(A)) ) ) ) ) )).

fof(t5_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k5_rlvect_1(k1_complfld,k1_hahnban1(A,B)) = k1_hahnban1(k1_real_1(A),k1_real_1(B)) ) ) )).

fof(t6_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => k6_rlvect_1(k1_complfld,k1_hahnban1(A,B),k1_hahnban1(C,D)) = k1_hahnban1(k5_real_1(A,C),k5_real_1(B,D)) ) ) ) ) )).

fof(t7_polynom5,axiom,(
    $true )).

fof(t8_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => ( A != k1_rlvect_1(k1_complfld)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k3_complfld(k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),A,B)) = k4_power(k3_complfld(A),B) ) ) ) )).

fof(d1_polynom5,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( B = k1_polynom5(A)
          <=> ( k3_finseq_1(B) = k3_finseq_1(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k5_finsop_1(A))
                   => k4_finseq_4(k5_numbers,k1_numbers,B,C) = k3_complfld(k4_finseq_4(k5_numbers,u1_struct_0(k1_complfld),A,C)) ) ) ) ) ) ) )).

fof(t9_polynom5,axiom,(
    k1_polynom5(k6_finseq_1(u1_struct_0(k1_complfld))) = k6_finseq_1(k1_numbers) )).

fof(t10_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => k1_polynom5(k13_binarith(u1_struct_0(k1_complfld),A)) = k13_binarith(k1_numbers,k3_complfld(A)) ) )).

fof(t11_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => k1_polynom5(k2_polynom3(u1_struct_0(k1_complfld),A,B)) = k2_polynom3(k1_numbers,k3_complfld(A),k3_complfld(B)) ) ) )).

fof(t12_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_complfld))
             => k1_polynom5(k3_finseq_4(u1_struct_0(k1_complfld),A,B,C)) = k3_finseq_4(k1_numbers,k3_complfld(A),k3_complfld(B),k3_complfld(C)) ) ) ) )).

fof(t13_polynom5,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k1_complfld))
         => k1_polynom5(k8_finseq_1(u1_struct_0(k1_complfld),A,B)) = k8_finseq_1(k1_numbers,k1_polynom5(A),k1_polynom5(B)) ) ) )).

fof(t14_polynom5,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ( k1_polynom5(k8_finseq_1(u1_struct_0(k1_complfld),A,k13_binarith(u1_struct_0(k1_complfld),B))) = k8_finseq_1(k1_numbers,k1_polynom5(A),k13_binarith(k1_numbers,k3_complfld(B)))
            & k1_polynom5(k8_finseq_1(u1_struct_0(k1_complfld),k13_binarith(u1_struct_0(k1_complfld),B),A)) = k8_finseq_1(k1_numbers,k13_binarith(k1_numbers,k3_complfld(B)),k1_polynom5(A)) ) ) ) )).

fof(t15_polynom5,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k1_complfld))
     => r1_xreal_0(k3_complfld(k9_rlvect_1(k1_complfld,A)),k15_rvsum_1(k1_polynom5(A))) ) )).

fof(d2_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_polynom5(A,B,C) = k1_binop_1(k5_group_1(k16_polynom3(A)),B,C) ) ) ) )).

fof(t16_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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)) )
         => k2_polynom5(A,B,np__0) = k13_polynom3(A) ) ) )).

fof(t17_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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)) )
         => k2_polynom5(A,B,np__1) = B ) ) )).

fof(t18_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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)) )
         => k2_polynom5(A,B,np__2) = k15_polynom3(A,B,B) ) ) )).

fof(t19_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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)) )
         => k2_polynom5(A,B,np__3) = k15_polynom3(A,k15_polynom3(A,B,B),B) ) ) )).

fof(t20_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_polynom5(A,B,k1_nat_1(C,np__1)) = k15_polynom3(A,k2_polynom5(A,B,C),B) ) ) ) )).

fof(t21_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_polynom5(A,k12_polynom3(A),k1_nat_1(B,np__1)) = k12_polynom3(A) ) ) )).

fof(t22_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_polynom5(A,k13_polynom3(A),B) = k13_polynom3(A) ) ) )).

fof(t23_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_polynom4(A,k2_polynom5(A,B,D),C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),k2_polynom4(A,B,C),D) ) ) ) ) )).

fof(t24_polynom5,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)
        & v2_vectsp_2(A)
        & l3_vectsp_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)) )
         => ( k3_algseq_1(A,B) != np__0
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k3_algseq_1(A,k2_polynom5(A,B,C)) = k3_real_1(k5_real_1(k2_nat_1(C,k3_algseq_1(A,B)),C),np__1) ) ) ) ) )).

fof(d3_polynom5,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] :
              ( m1_subset_1(C,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 = k3_polynom5(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k1_group_1(A,C,k2_normsp_1(A,B,E)) ) ) ) ) ) ) )).

fof(t25_polynom5,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] :
          ( ( 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)) )
         => k3_algseq_1(A,k3_polynom5(A,B,k1_rlvect_1(A))) = np__0 ) ) )).

fof(t26_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_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)
        & l3_vectsp_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] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C != k1_rlvect_1(A)
               => k3_algseq_1(A,k3_polynom5(A,B,C)) = k3_algseq_1(A,B) ) ) ) ) )).

fof(t27_polynom5,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) )
     => ! [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)) )
         => k3_polynom5(A,B,k1_rlvect_1(A)) = k12_polynom3(A) ) ) )).

fof(t28_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l1_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)) )
         => k3_polynom5(A,B,k2_group_1(A)) = B ) ) )).

fof(t29_polynom5,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k3_polynom5(A,k12_polynom3(A),B) = k12_polynom3(A) ) ) )).

fof(t30_polynom5,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k3_polynom5(A,k13_polynom3(A),B) = k5_algseq_1(A,B) ) ) )).

fof(t31_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_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)
        & l3_vectsp_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] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k2_polynom4(A,k3_polynom5(A,B,C),D) = k10_group_1(A,C,k2_polynom4(A,B,D)) ) ) ) ) )).

fof(t32_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_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))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k2_polynom4(A,B,k1_rlvect_1(A)) = k2_normsp_1(A,B,np__0) ) ) )).

fof(d4_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k4_polynom5(A,B,C) = k1_polynom1(k5_numbers,u1_struct_0(A),k1_polynom1(k5_numbers,u1_struct_0(A),k12_polynom3(A),np__0,B),np__1,C) ) ) ) )).

fof(t33_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( k2_normsp_1(A,k5_algseq_1(A,B),np__0) = B
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(np__1,C)
                 => k2_normsp_1(A,k5_algseq_1(A,B),C) = k1_rlvect_1(A) ) ) ) ) ) )).

fof(t34_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( B != k1_rlvect_1(A)
           => k3_algseq_1(A,k5_algseq_1(A,B)) = np__1 ) ) ) )).

fof(t35_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => k5_algseq_1(A,k1_rlvect_1(A)) = k12_polynom3(A) ) )).

fof(t36_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v2_vectsp_2(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k14_polynom3(A,k5_algseq_1(A,B),k5_algseq_1(A,C)) = k5_algseq_1(A,k10_group_1(A,B,C)) ) ) ) )).

fof(t37_polynom5,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)
        & v9_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_polynom5(A,k5_algseq_1(A,B),C) = k5_algseq_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,C)) ) ) ) )).

fof(t38_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k2_polynom4(A,k5_algseq_1(A,B),C) = B ) ) ) )).

fof(t39_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( k2_normsp_1(A,k4_polynom5(A,B,C),np__0) = B
                & k2_normsp_1(A,k4_polynom5(A,B,C),np__1) = C
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(np__2,D)
                     => k2_normsp_1(A,k4_polynom5(A,B,C),D) = k1_rlvect_1(A) ) ) ) ) ) ) )).

fof(t40_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k3_algseq_1(A,k4_polynom5(A,B,C)),np__2) ) ) ) )).

fof(t41_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C != k1_rlvect_1(A)
               => k3_algseq_1(A,k4_polynom5(A,B,C)) = np__2 ) ) ) ) )).

fof(t42_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( B != k1_rlvect_1(A)
           => k3_algseq_1(A,k4_polynom5(A,B,k1_rlvect_1(A))) = np__1 ) ) ) )).

fof(t43_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => k4_polynom5(A,k1_rlvect_1(A),k1_rlvect_1(A)) = k12_polynom3(A) ) )).

fof(t44_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k4_polynom5(A,B,k1_rlvect_1(A)) = k5_algseq_1(A,B) ) ) )).

fof(t45_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k2_polynom4(A,k4_polynom5(A,B,C),D) = k2_rlvect_1(A,B,k1_group_1(A,C,D)) ) ) ) ) )).

fof(t46_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k2_polynom4(A,k4_polynom5(A,B,k1_rlvect_1(A)),D) = B ) ) ) ) )).

fof(t47_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k2_polynom4(A,k4_polynom5(A,k1_rlvect_1(A),C),D) = k1_group_1(A,C,D) ) ) ) ) )).

fof(t48_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k2_polynom4(A,k4_polynom5(A,B,k2_group_1(A)),D) = k2_rlvect_1(A,B,D) ) ) ) ) )).

fof(t49_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v5_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k2_polynom4(A,k4_polynom5(A,k1_rlvect_1(A),k2_group_1(A)),D) = D ) ) ) ) )).

fof(d5_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & v1_algseq_1(D,A)
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k5_polynom5(A,B,C)
                  <=> ? [E] :
                        ( m2_finseq_1(E,u1_struct_0(k16_polynom3(A)))
                        & D = k9_rlvect_1(k16_polynom3(A),E)
                        & k3_finseq_1(E) = k3_algseq_1(A,B)
                        & ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( r2_hidden(F,k5_finsop_1(E))
                             => k1_funct_1(E,F) = k3_polynom5(A,k2_polynom5(A,C,k5_binarith(F,np__1)),k2_normsp_1(A,B,k5_binarith(F,np__1))) ) ) ) ) ) ) ) ) )).

fof(t50_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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)) )
         => k5_polynom5(A,k12_polynom3(A),B) = k12_polynom3(A) ) ) )).

fof(t51_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(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)
        & l3_vectsp_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)) )
         => k5_polynom5(A,B,k12_polynom3(A)) = k5_algseq_1(A,k2_normsp_1(A,B,np__0)) ) ) )).

fof(t52_polynom5,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)
        & v9_vectsp_1(A)
        & l3_vectsp_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] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(k3_algseq_1(A,k5_polynom5(A,B,k5_algseq_1(A,C))),np__1) ) ) ) )).

fof(t53_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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)) )
             => ~ ( k3_algseq_1(A,B) != np__0
                  & ~ r1_xreal_0(k3_algseq_1(A,C),np__1)
                  & k3_algseq_1(A,k5_polynom5(A,B,C)) != k3_real_1(k5_real_1(k5_real_1(k2_nat_1(k3_algseq_1(A,B),k3_algseq_1(A,C)),k3_algseq_1(A,B)),k3_algseq_1(A,C)),np__2) ) ) ) ) )).

fof(t54_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k2_polynom4(A,k5_polynom5(A,B,C),D) = k2_polynom4(A,B,k2_polynom4(A,C,D)) ) ) ) ) )).

fof(d6_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_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] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_polynom5(A,B,C)
              <=> k2_polynom4(A,B,C) = k1_rlvect_1(A) ) ) ) ) )).

fof(d7_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_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)) )
         => ( v1_polynom5(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & r1_polynom5(A,B,C) ) ) ) ) )).

fof(t55_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => v1_polynom5(k12_polynom3(A),A) ) )).

fof(t56_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_polynom5(A,k12_polynom3(A),B) ) ) )).

fof(d8_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_1(A) )
     => ( v2_polynom5(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)) )
           => ( ~ r1_xreal_0(k3_algseq_1(A,B),np__1)
             => v1_polynom5(B,A) ) ) ) ) )).

fof(d9_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_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] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( C = k6_polynom5(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r2_hidden(D,C)
                    <=> r1_polynom5(A,B,D) ) ) ) ) ) ) )).

fof(d10_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & l3_vectsp_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))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( C = k7_polynom5(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(A,C,D) = k5_vectsp_1(A,k2_normsp_1(A,B,D),k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1))) ) ) ) ) ) )).

fof(t57_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & l3_vectsp_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)) )
         => ( k3_algseq_1(A,B) != np__0
           => k2_normsp_1(A,k7_polynom5(A,B),k5_binarith(k3_algseq_1(A,B),np__1)) = k2_group_1(A) ) ) ) )).

fof(t58_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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)) )
         => ( k3_algseq_1(A,B) != np__0
           => k3_algseq_1(A,k7_polynom5(A,B)) = k3_algseq_1(A,B) ) ) ) )).

fof(t59_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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)) )
         => ( k3_algseq_1(A,B) != np__0
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => k2_polynom4(A,k7_polynom5(A,B),C) = k5_vectsp_1(A,k2_polynom4(A,B,C),k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1))) ) ) ) ) )).

fof(t60_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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)) )
         => ( k3_algseq_1(A,B) != np__0
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( r1_polynom5(A,B,C)
                <=> r1_polynom5(A,k7_polynom5(A,B),C) ) ) ) ) ) )).

fof(t61_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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)) )
         => ( k3_algseq_1(A,B) != np__0
           => ( v1_polynom5(B,A)
            <=> v1_polynom5(k7_polynom5(A,B),A) ) ) ) ) )).

fof(t62_polynom5,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)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_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)) )
         => ( k3_algseq_1(A,B) != np__0
           => k6_polynom5(A,B) = k6_polynom5(A,k7_polynom5(A,B)) ) ) ) )).

fof(t63_polynom5,axiom,(
    r2_cfcont_1(k6_partfun1(k2_numbers),k2_numbers) )).

fof(t64_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_numbers)
     => r2_cfcont_1(k3_finsop_1(k2_numbers,k2_numbers,A),k2_numbers) ) )).

fof(d11_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
                 => ( D = k8_polynom5(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,E) = k1_group_1(A,B,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),E,C)) ) ) ) ) ) ) )).

fof(t65_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => k8_polynom5(A,k2_group_1(A),np__1) = k6_partfun1(u1_struct_0(A)) ) )).

fof(t66_polynom5,axiom,(
    k8_polynom5(k1_complfld,k2_group_1(k1_complfld),np__2) = k3_comseq_1(k2_numbers,k6_partfun1(k2_numbers),k6_partfun1(k2_numbers)) )).

fof(t67_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k8_polynom5(A,B,np__0) = k3_finsop_1(u1_struct_0(A),u1_struct_0(A),B) ) ) )).

fof(t68_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => ? [B] :
          ( m1_subset_1(B,k2_numbers)
          & A = B
          & k8_polynom5(k1_complfld,A,np__1) = k4_comseq_1(k2_numbers,k6_partfun1(k2_numbers),B) ) ) )).

fof(t69_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => ? [B] :
          ( m1_subset_1(B,k2_numbers)
          & A = B
          & k8_polynom5(k1_complfld,A,np__2) = k4_comseq_1(k2_numbers,k3_comseq_1(k2_numbers,k6_partfun1(k2_numbers),k6_partfun1(k2_numbers)),B) ) ) )).

fof(t70_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,k2_numbers,k2_numbers)
              & m2_relset_1(C,k2_numbers,k2_numbers)
              & C = k8_polynom5(k1_complfld,A,B)
              & k8_polynom5(k1_complfld,A,k1_nat_1(B,np__1)) = k3_comseq_1(k2_numbers,C,k6_partfun1(k2_numbers)) ) ) ) )).

fof(t71_polynom5,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,k2_numbers,k2_numbers)
              & m2_relset_1(C,k2_numbers,k2_numbers)
              & C = k8_polynom5(k1_complfld,A,B)
              & r2_cfcont_1(C,k2_numbers) ) ) ) )).

fof(d12_polynom5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l3_vectsp_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,u1_struct_0(A),u1_struct_0(A))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
             => ( C = k9_polynom5(A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k2_polynom4(A,B,D) ) ) ) ) ) )).

fof(t72_polynom5,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
        & v1_algseq_1(A,k1_complfld)
        & m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k2_numbers,k2_numbers)
          & m2_relset_1(B,k2_numbers,k2_numbers)
          & B = k9_polynom5(k1_complfld,A)
          & r2_cfcont_1(B,k2_numbers) ) ) )).

fof(t73_polynom5,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
        & v1_algseq_1(A,k1_complfld)
        & m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
     => ( k3_complfld(k2_normsp_1(k1_complfld,A,k5_binarith(k3_algseq_1(k1_complfld,A),np__1))) = np__1
       => ( r1_xreal_0(k3_algseq_1(k1_complfld,A),np__2)
          | ! [B] :
              ( m2_finseq_1(B,k1_numbers)
             => ( ( k3_finseq_1(B) = k3_algseq_1(k1_complfld,A)
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ( r2_hidden(C,k5_finsop_1(B))
                       => k2_seq_1(k5_numbers,k1_numbers,B,C) = k3_complfld(k2_normsp_1(k1_complfld,A,k5_binarith(C,np__1))) ) ) )
               => ! [C] :
                    ( m1_subset_1(C,u1_struct_0(k1_complfld))
                   => ~ ( ~ r1_xreal_0(k3_complfld(C),k15_rvsum_1(B))
                        & r1_xreal_0(k3_complfld(k2_polynom4(k1_complfld,A,C)),k3_real_1(k3_complfld(k2_normsp_1(k1_complfld,A,np__0)),np__1)) ) ) ) ) ) ) ) )).

fof(t74_polynom5,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
        & v1_algseq_1(A,k1_complfld)
        & m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
     => ~ ( ~ r1_xreal_0(k3_algseq_1(k1_complfld,A),np__2)
          & ! [B] :
              ( m1_subset_1(B,u1_struct_0(k1_complfld))
             => ~ ! [C] :
                    ( m1_subset_1(C,u1_struct_0(k1_complfld))
                   => r1_xreal_0(k3_complfld(k2_polynom4(k1_complfld,A,B)),k3_complfld(k2_polynom4(k1_complfld,A,C))) ) ) ) ) )).

fof(t75_polynom5,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
        & v1_algseq_1(A,k1_complfld)
        & m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
     => ( ~ r1_xreal_0(k3_algseq_1(k1_complfld,A),np__1)
       => v1_polynom5(A,k1_complfld) ) ) )).

fof(dt_k1_polynom5,axiom,(
    ! [A] :
      ( m1_finseq_1(A,u1_struct_0(k1_complfld))
     => m2_finseq_1(k1_polynom5(A),k1_numbers) ) )).

fof(dt_k2_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(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)
        & 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))
        & m1_subset_1(C,k5_numbers) )
     => ( v1_funct_1(k2_polynom5(A,B,C))
        & v1_funct_2(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k3_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ 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))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( v1_funct_1(k3_polynom5(A,B,C))
        & v1_funct_2(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k4_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => ( v1_funct_1(k4_polynom5(A,B,C))
        & v1_funct_2(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k5_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(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)
        & 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_funct_1(k5_polynom5(A,B,C))
        & v1_funct_2(k5_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k5_polynom5(A,B,C),A)
        & m2_relset_1(k5_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k6_polynom5,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_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)) )
     => m1_subset_1(k6_polynom5(A,B),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k7_polynom5,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_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(k7_polynom5(A,B))
        & v1_funct_2(k7_polynom5(A,B),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k7_polynom5(A,B),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k8_polynom5,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & l1_group_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => ( v1_funct_1(k8_polynom5(A,B,C))
        & v1_funct_2(k8_polynom5(A,B,C),u1_struct_0(A),u1_struct_0(A))
        & m2_relset_1(k8_polynom5(A,B,C),u1_struct_0(A),u1_struct_0(A)) ) ) )).

fof(dt_k9_polynom5,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_group_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(k9_polynom5(A,B))
        & v1_funct_2(k9_polynom5(A,B),u1_struct_0(A),u1_struct_0(A))
        & m2_relset_1(k9_polynom5(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) )).
%------------------------------------------------------------------------------