SET007 Axioms: SET007+803.ax


%------------------------------------------------------------------------------
% File     : SET007+803 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Little Bezout Theorem (Factor Theorem)
% 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    : uproots [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  100 (   1 unit)
%            Number of atoms       : 1427 ( 138 equality)
%            Maximal formula depth :   27 (  12 average)
%            Number of connectives : 1476 ( 149 ~  ;   7  |;1003  &)
%                                         (  16 <=>; 301 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   60 (   0 propositional; 1-3 arity)
%            Number of functors    :   82 (   7 constant; 0-4 arity)
%            Number of variables   :  302 (   0 singleton; 292 !;  10 ?)
%            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(rc1_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & l2_struct_0(A) )
     => ? [B] :
          ( m1_relset_1(B,k5_numbers,u1_struct_0(A))
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_funct_2(B,k5_numbers,u1_struct_0(A))
          & v1_algseq_1(B,A)
          & v1_uproots(B,A) ) ) )).

fof(fc1_uproots,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v10_vectsp_1(A)
        & l2_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ( ~ v1_xboole_0(k4_polynom5(A,B,k2_group_1(A)))
        & v1_relat_1(k4_polynom5(A,B,k2_group_1(A)))
        & v1_funct_1(k4_polynom5(A,B,k2_group_1(A)))
        & v1_funct_2(k4_polynom5(A,B,k2_group_1(A)),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k4_polynom5(A,B,k2_group_1(A)),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k4_polynom5(A,B,k2_group_1(A)),A)
        & v1_uproots(k4_polynom5(A,B,k2_group_1(A)),A) ) ) )).

fof(rc2_uproots,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v7_group_1(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & v2_vectsp_2(A)
      & v2_polynom5(A)
      & v1_algstr_1(A)
      & v2_algstr_1(A)
      & v3_algstr_1(A)
      & v4_algstr_1(A)
      & v5_algstr_1(A)
      & v6_algstr_1(A)
      & ~ v3_realset2(A) ) )).

fof(fc2_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & v2_vectsp_2(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(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))
        & v2_vectsp_2(k16_polynom3(A))
        & v1_algstr_1(k16_polynom3(A))
        & v2_algstr_1(k16_polynom3(A))
        & v3_algstr_1(k16_polynom3(A))
        & v4_algstr_1(k16_polynom3(A))
        & v5_algstr_1(k16_polynom3(A))
        & v6_algstr_1(k16_polynom3(A)) ) ) )).

fof(fc3_uproots,axiom,(
    ! [A,B,C] :
      ( ( ~ 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)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & v1_uproots(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)
        & v1_uproots(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)
        & v1_uproots(k14_polynom3(A,B,C),A) ) ) )).

fof(fc4_uproots,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => ( ~ v1_xboole_0(k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C))
        & v1_relat_1(k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C))
        & v1_funct_1(k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C))
        & v1_funct_2(k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C),k5_numbers,u1_struct_0(A))
        & v1_partfun1(k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C),A)
        & v1_uproots(k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C),A) ) ) )).

fof(fc5_uproots,axiom,(
    ! [A,B] :
      ( ( ~ 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)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & v1_uproots(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => v1_finset_1(k6_polynom5(A,B)) ) )).

fof(t1_uproots,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ ( ~ v1_xboole_0(A)
            & A != np__1
            & r1_xreal_0(A,np__1) )
        & ~ ( ~ ( A != np__1
                & r1_xreal_0(A,np__1) )
            & v1_xboole_0(A) ) ) ) )).

fof(t2_uproots,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k5_numbers)
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( r2_hidden(B,k5_finsop_1(A))
                & k3_wsierp_1(A,B) = np__0 ) )
       => ( k9_wsierp_1(A) = k3_finseq_1(A)
        <=> A = k1_finsop_1(k5_numbers,k3_finseq_1(A),np__1) ) ) ) )).

fof(t3_uproots,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))
         => ( ( r1_xreal_0(np__2,k3_finseq_1(B))
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k5_finsop_1(B))
                   => ( r1_xreal_0(C,np__2)
                      | k1_funct_1(B,C) = k1_rlvect_1(A) ) ) ) )
           => k9_rlvect_1(A,B) = k2_rlvect_1(A,k4_finseq_4(k5_numbers,u1_struct_0(A),B,np__1),k4_finseq_4(k5_numbers,u1_struct_0(A),B,np__2)) ) ) ) )).

fof(d1_uproots,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ( B = k1_uproots(A)
          <=> ( k3_finseq_1(B) = k4_card_1(A)
              & ? [C] :
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v1_finseq_1(C)
                  & k3_finseq_1(C) = k4_card_1(A)
                  & ( k1_funct_1(C,np__1) = k4_tarski(k8_subset_1(A),k4_xboole_0(A,k1_tarski(k8_subset_1(A))))
                    | k4_card_1(A) = np__0 )
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(np__1,D)
                       => ( r1_xreal_0(k4_card_1(A),D)
                          | ! [E] :
                              ( k1_funct_1(C,D) = E
                             => k1_funct_1(C,k23_binop_2(D,np__1)) = k4_tarski(k8_subset_1(k2_mcart_1(E)),k4_xboole_0(k2_mcart_1(E),k1_tarski(k8_subset_1(k2_mcart_1(E))))) ) ) ) )
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k5_finsop_1(B))
                       => k1_funct_1(B,D) = k1_mcart_1(k1_funct_1(C,D)) ) ) ) ) ) ) ) )).

fof(t4_uproots,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => v2_funct_1(k1_uproots(A)) ) )).

fof(t5_uproots,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => k2_relat_1(k1_uproots(A)) = A ) )).

fof(t6_uproots,axiom,(
    ! [A] : k1_uproots(k1_tarski(A)) = k9_finseq_1(A) )).

fof(t7_uproots,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ( v1_funct_1(k2_funct_1(k1_uproots(A)))
        & v1_funct_2(k2_funct_1(k1_uproots(A)),A,k2_finseq_1(k4_card_1(A)))
        & m2_relset_1(k2_funct_1(k1_uproots(A)),A,k2_finseq_1(k4_card_1(A))) ) ) )).

fof(d2_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => k2_uproots(A,B,C) = k1_funct_4(k16_polynom1(A),k10_pboole(B,C)) ) ) )).

fof(t8_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] :
              ( ~ r2_hidden(D,B)
             => k8_polynom1(k2_uproots(A,B,C),D) = np__0 ) ) ) )).

fof(t9_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] :
              ( r2_hidden(D,B)
             => k8_polynom1(k2_uproots(A,B,C),D) = C ) ) ) )).

fof(t10_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( C != np__0
           => k11_polynom1(k2_uproots(A,B,C)) = B ) ) ) )).

fof(t11_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ( v1_xboole_0(B)
              | C = np__0 )
           => r6_pboole(A,k2_uproots(A,B,C),k16_polynom1(A)) ) ) ) )).

fof(t12_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v1_finset_1(C)
            & m1_subset_1(C,k1_zfmisc_1(A)) )
         => ! [D] :
              ( m2_subset_1(D,k1_numbers,k5_numbers)
             => ( r1_xboole_0(B,C)
               => r6_pboole(A,k2_uproots(A,k4_subset_1(A,B,C),D),k9_polynom1(A,k2_uproots(A,B,D),k2_uproots(A,C,D))) ) ) ) ) )).

fof(d3_uproots,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( v1_xreal_0(C)
         => ( C = k3_uproots(A,B)
          <=> ? [D] :
                ( m2_finseq_1(D,k1_numbers)
                & C = k15_rvsum_1(D)
                & D = k5_relat_1(k1_uproots(k11_polynom1(B)),B) ) ) ) ) )).

fof(d4_uproots,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( C = k4_uproots(A,B)
          <=> ? [D] :
                ( m2_finseq_1(D,k5_numbers)
                & C = k9_wsierp_1(D)
                & D = k5_relat_1(k1_uproots(k11_polynom1(B)),B) ) ) ) ) )).

fof(t13_uproots,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ( r6_pboole(A,B,k16_polynom1(A))
       => k3_uproots(A,B) = np__0 ) ) )).

fof(t14_uproots,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ( k3_uproots(A,B) = np__0
       => r6_pboole(A,B,k16_polynom1(A)) ) ) )).

fof(t15_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v7_seqm_3(C)
            & v1_polynom1(C)
            & m1_pboole(C,A) )
         => ( ( B = k11_polynom1(C)
              & k4_uproots(A,C) = k4_card_1(B) )
          <=> r6_pboole(A,C,k2_uproots(A,B,np__1)) ) ) ) )).

fof(t16_uproots,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] :
          ( ( v1_seq_1(C)
            & v1_polynom1(C)
            & m1_pboole(C,A) )
         => ~ ( r1_tarski(k11_polynom1(C),B)
              & ! [D] :
                  ( m2_finseq_1(D,k1_numbers)
                 => ~ ( D = k5_relat_1(k1_uproots(B),C)
                      & k3_uproots(A,C) = k15_rvsum_1(D) ) ) ) ) ) )).

fof(t17_uproots,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => ! [C] :
          ( ( v1_seq_1(C)
            & v1_polynom1(C)
            & m1_pboole(C,A) )
         => ! [D] :
              ( ( v1_seq_1(D)
                & v1_polynom1(D)
                & m1_pboole(D,A) )
             => ( r6_pboole(A,B,k9_polynom1(A,C,D))
               => k3_uproots(A,B) = k2_xcmplx_0(k3_uproots(A,C),k3_uproots(A,D)) ) ) ) ) )).

fof(t18_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & l1_group_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_finsop_1(B),k5_finsop_1(B))
                    & v3_funct_2(D,k5_finsop_1(B),k5_finsop_1(B))
                    & m2_relset_1(D,k5_finsop_1(B),k5_finsop_1(B)) )
                 => ( C = k5_relat_1(D,B)
                   => k13_fvsum_1(A,C) = k13_fvsum_1(A,B) ) ) ) ) ) )).

fof(d5_uproots,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)) )
         => ( v1_uproots(B,A)
          <=> B != k12_polynom3(A) ) ) ) )).

fof(t19_uproots,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)) )
         => ( v1_uproots(B,A)
          <=> ~ r1_xreal_0(k3_algseq_1(A,B),np__0) ) ) ) )).

fof(t20_uproots,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)) )
         => ~ ( ~ r1_xreal_0(k3_algseq_1(A,B),np__0)
              & k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1)) = k1_rlvect_1(A) ) ) ) )).

fof(t21_uproots,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)) )
         => ( k3_algseq_1(A,B) = np__1
           => ( B = k5_algseq_1(A,k2_normsp_1(A,B,np__0))
              & k2_normsp_1(A,B,np__0) != k1_rlvect_1(A) ) ) ) ) )).

fof(t22_uproots,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))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k14_polynom3(A,B,k12_polynom3(A)) = k12_polynom3(A) ) ) )).

fof(t23_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_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)) )
         => ! [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)) )
             => ~ ( k14_polynom3(A,B,C) = k12_polynom3(A)
                  & B != k12_polynom3(A)
                  & C != k12_polynom3(A) ) ) ) ) )).

fof(t24_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [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_subset_1(u1_struct_0(A),k6_polynom5(A,B),k6_polynom5(A,C)),k6_polynom5(A,k15_polynom3(A,B,C))) ) ) ) )).

fof(t25_uproots,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)) )
         => ! [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)) )
             => k6_polynom5(A,k15_polynom3(A,B,C)) = k4_subset_1(u1_struct_0(A),k6_polynom5(A,B),k6_polynom5(A,C)) ) ) ) )).

fof(t26_uproots,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)) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k16_polynom3(A)))
             => ( B = C
               => k10_polynom3(A,B) = k5_rlvect_1(k16_polynom3(A),C) ) ) ) ) )).

fof(t27_uproots,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)) )
         => ! [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(k16_polynom3(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k16_polynom3(A)))
                     => ( ( B = D
                          & C = E )
                       => k11_polynom3(A,B,C) = k6_rlvect_1(k16_polynom3(A),D,E) ) ) ) ) ) ) )).

fof(t28_uproots,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) )
     => ! [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)) )
                 => k11_polynom3(A,k14_polynom3(A,B,C),k14_polynom3(A,B,D)) = k14_polynom3(A,B,k11_polynom3(A,C,D)) ) ) ) ) )).

fof(t29_uproots,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)) )
         => ! [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)) )
             => ( k11_polynom3(A,B,C) = k12_polynom3(A)
               => B = C ) ) ) ) )).

fof(t30_uproots,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)
        & 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)) )
         => ! [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)) )
                 => ( k14_polynom3(A,B,C) = k14_polynom3(A,B,D)
                   => ( B = k12_polynom3(A)
                      | C = D ) ) ) ) ) ) )).

fof(t31_uproots,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] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [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)) )
             => ~ ( C != k12_polynom3(A)
                  & k2_polynom5(A,C,B) = k12_polynom3(A) ) ) ) ) )).

fof(t32_uproots,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)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [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)) )
                 => k15_polynom3(A,k2_polynom5(A,D,B),k2_polynom5(A,D,C)) = k2_polynom5(A,D,k23_binop_2(B,C)) ) ) ) ) )).

fof(t33_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_vectsp_1(A) )
     => k13_polynom3(A) = k5_algseq_1(A,k2_group_1(A)) ) )).

fof(t34_uproots,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))
            & v1_algseq_1(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k14_polynom3(A,B,k5_algseq_1(A,k2_group_1(A))) = B ) ) )).

fof(t35_uproots,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)) )
         => ! [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
                  | k3_algseq_1(A,C) = np__0 )
               => k3_algseq_1(A,k14_polynom3(A,B,C)) = np__0 ) ) ) ) )).

fof(t36_uproots,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)) )
         => ! [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)) )
             => ( v1_uproots(k14_polynom3(A,B,C),A)
               => ( v1_uproots(B,A)
                  & v1_uproots(C,A) ) ) ) ) ) )).

fof(t37_uproots,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)
        & 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)) )
             => ~ ( k10_group_1(A,k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1)),k2_normsp_1(A,C,k5_binarith(k3_algseq_1(A,C),np__1))) != k1_rlvect_1(A)
                  & r1_xreal_0(k3_algseq_1(A,k14_polynom3(A,B,C)),np__0) ) ) ) ) )).

fof(t38_uproots,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] :
          ( ( 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_xreal_0(k3_algseq_1(A,B),np__1)
                  & ~ r1_xreal_0(k3_algseq_1(A,C),np__1)
                  & r1_xreal_0(k3_algseq_1(A,k14_polynom3(A,B,C)),k3_algseq_1(A,B)) ) ) ) ) )).

fof(t39_uproots,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] :
          ( m1_subset_1(B,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))
                    & v1_algseq_1(D,A)
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( k2_normsp_1(A,k14_polynom3(A,k4_polynom5(A,B,C),D),np__0) = k1_group_1(A,B,k2_normsp_1(A,D,np__0))
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,k14_polynom3(A,k4_polynom5(A,B,C),D),k23_binop_2(E,np__1)) = k2_rlvect_1(A,k1_group_1(A,B,k2_normsp_1(A,D,k23_binop_2(E,np__1))),k1_group_1(A,C,k2_normsp_1(A,D,E))) ) ) ) ) ) ) )).

fof(t40_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & v1_uproots(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => k3_algseq_1(A,k14_polynom3(A,k4_polynom5(A,B,k2_group_1(A)),C)) = k23_binop_2(k3_algseq_1(A,C),np__1) ) ) ) )).

fof(t41_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_algseq_1(A,k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),C)) = k23_binop_2(C,np__1) ) ) ) )).

fof(t42_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & v1_uproots(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k3_algseq_1(A,k15_polynom3(A,k2_polynom5(A,k4_polynom5(A,B,k2_group_1(A)),D),C)) = k23_binop_2(D,k3_algseq_1(A,C)) ) ) ) ) )).

fof(t43_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,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)) )
                 => ( ( C = k14_polynom3(A,k4_polynom5(A,B,k2_group_1(A)),D)
                      & k2_normsp_1(A,C,k5_binarith(k3_algseq_1(A,C),np__1)) = k2_group_1(A) )
                   => k2_normsp_1(A,D,k5_binarith(k3_algseq_1(A,D),np__1)) = k2_group_1(A) ) ) ) ) ) )).

fof(d6_uproots,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)
             => ! [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_uproots(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k2_normsp_1(A,B,k23_binop_2(C,E)) ) ) ) ) ) ) )).

fof(t44_uproots,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)) )
         => k5_uproots(A,B,np__0) = B ) ) )).

fof(t45_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [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_xreal_0(k3_algseq_1(A,C),B)
               => k5_uproots(A,C,B) = k12_polynom3(A) ) ) ) ) )).

fof(t46_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v10_vectsp_1(A)
        & l2_vectsp_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [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_xreal_0(B,k3_algseq_1(A,C))
               => k23_binop_2(k3_algseq_1(A,k5_uproots(A,C,B)),B) = k3_algseq_1(A,C) ) ) ) ) )).

fof(t47_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [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,k5_numbers,u1_struct_0(A))
                    & v1_algseq_1(D,A)
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( ~ r1_xreal_0(k3_algseq_1(A,D),C)
                   => k2_polynom4(A,k5_uproots(A,D,C),B) = k4_rlvect_1(A,k10_group_1(A,B,k2_polynom4(A,k5_uproots(A,D,k23_binop_2(C,np__1)),B)),k2_normsp_1(A,D,C)) ) ) ) ) ) )).

fof(t48_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [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__1
           => k6_polynom5(A,B) = k1_xboole_0 ) ) ) )).

fof(d7_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,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_polynom5(A,C,B)
               => ! [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)) )
                   => ( ( ~ r1_xreal_0(k3_algseq_1(A,C),np__0)
                       => ( D = k6_uproots(A,B,C)
                        <=> ( k23_binop_2(k3_algseq_1(A,D),np__1) = k3_algseq_1(A,C)
                            & ! [E] :
                                ( m2_subset_1(E,k1_numbers,k5_numbers)
                               => k2_normsp_1(A,D,E) = k2_polynom4(A,k5_uproots(A,C,k23_binop_2(E,np__1)),B) ) ) ) )
                      & ( r1_xreal_0(k3_algseq_1(A,C),np__0)
                       => ( D = k6_uproots(A,B,C)
                        <=> D = k12_polynom3(A) ) ) ) ) ) ) ) ) )).

fof(t49_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & v1_uproots(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ~ ( r1_polynom5(A,C,B)
                  & r1_xreal_0(k3_algseq_1(A,k6_uproots(A,B,C)),np__0) ) ) ) ) )).

fof(t50_uproots,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))
         => k6_polynom5(A,k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A))) = k7_rlvect_2(A,B) ) ) )).

fof(t51_uproots,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)
        & ~ v3_realset2(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,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)) )
                 => ( C = k15_polynom3(A,k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A)),D)
                   => r1_polynom5(A,C,B) ) ) ) ) ) )).

fof(t52_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,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_polynom5(A,C,B)
               => C = k15_polynom3(A,k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A)),k6_uproots(A,B,C)) ) ) ) ) )).

fof(t53_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,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)) )
                 => ( C = k15_polynom3(A,k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A)),D)
                   => r1_polynom5(A,C,B) ) ) ) ) ) )).

fof(t54_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & v1_algseq_1(B,A)
            & v1_uproots(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)
              <=> r1_xreal_0(np__1,k7_uproots(A,C,B)) ) ) ) ) )).

fof(t55_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k7_uproots(A,B,k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A))) = np__1 ) ) )).

fof(d9_uproots,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)
            & v1_uproots(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v7_seqm_3(C)
                & v1_polynom1(C)
                & m1_pboole(C,u1_struct_0(A)) )
             => ( C = k8_uproots(A,B)
              <=> ( k11_polynom1(C) = k6_polynom5(A,B)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k8_polynom1(C,D) = k7_uproots(A,D,B) ) ) ) ) ) ) )).

fof(t56_uproots,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r6_pboole(u1_struct_0(A),k8_uproots(A,k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A))),k2_uproots(u1_struct_0(A),k7_rlvect_2(A,B),np__1)) ) ) )).

fof(t57_uproots,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & v1_uproots(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)
                    & v1_uproots(D,A)
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => k7_uproots(A,B,k15_polynom3(A,C,D)) = k23_binop_2(k7_uproots(A,B,C),k7_uproots(A,B,D)) ) ) ) ) )).

fof(t58_uproots,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)
            & v1_uproots(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)
                & v1_uproots(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => r6_pboole(u1_struct_0(A),k8_uproots(A,k15_polynom3(A,B,C)),k9_polynom1(u1_struct_0(A),k8_uproots(A,B),k8_uproots(A,C))) ) ) ) )).

fof(t59_uproots,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)
            & v1_uproots(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( k3_algseq_1(A,B) = np__1
           => k4_uproots(u1_struct_0(A),k8_uproots(A,B)) = np__0 ) ) ) )).

fof(t60_uproots,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k4_uproots(u1_struct_0(A),k8_uproots(A,k2_polynom5(A,k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A)),C))) = C ) ) ) )).

fof(t61_uproots,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)
        & v2_polynom5(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)
            & v1_uproots(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k4_uproots(u1_struct_0(A),k8_uproots(A,B)) = k5_binarith(k3_algseq_1(A,B),np__1) ) ) )).

fof(d10_uproots,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(k16_polynom3(A)))
                 => ( D = k9_uproots(A,B,C)
                  <=> ( k3_finseq_1(D) = C
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k5_finsop_1(D))
                           => k1_funct_1(D,E) = k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A)) ) ) ) ) ) ) ) ) )).

fof(d11_uproots,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] :
          ( ( v7_seqm_3(B)
            & v1_polynom1(B)
            & m1_pboole(B,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)) )
             => ( C = k10_uproots(A,B)
              <=> ? [D] :
                    ( m2_finseq_1(D,k3_finseq_2(u1_struct_0(k16_polynom3(A))))
                    & ? [E] :
                        ( m2_finseq_1(E,u1_struct_0(A))
                        & k3_finseq_1(D) = k4_card_1(k11_polynom1(B))
                        & E = k1_uproots(k11_polynom1(B))
                        & ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( r2_hidden(F,k5_finsop_1(D))
                             => k1_funct_1(D,F) = k9_uproots(A,k4_finseq_4(k5_numbers,u1_struct_0(A),E,F),k8_polynom1(B,k4_finseq_4(k5_numbers,u1_struct_0(A),E,F))) ) )
                        & C = k13_fvsum_1(k16_polynom3(A),k15_dtconstr(u1_struct_0(k16_polynom3(A)),D)) ) ) ) ) ) ) )).

fof(t62_uproots,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) )
     => k10_uproots(A,k16_polynom1(u1_struct_0(A))) = k5_algseq_1(A,k2_group_1(A)) ) )).

fof(t63_uproots,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] :
          ( m1_subset_1(B,u1_struct_0(A))
         => k10_uproots(A,k2_uproots(u1_struct_0(A),k7_rlvect_2(A,B),np__1)) = k4_polynom5(A,k5_rlvect_1(A,B),k2_group_1(A)) ) ) )).

fof(t64_uproots,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] :
          ( ( v7_seqm_3(B)
            & v1_polynom1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m2_finseq_1(C,k3_finseq_2(u1_struct_0(k16_polynom3(A))))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(A))
                 => ( ( k3_finseq_1(C) = k4_card_1(k11_polynom1(B))
                      & D = k1_uproots(k11_polynom1(B))
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k5_finsop_1(C))
                           => k1_funct_1(C,E) = k9_uproots(A,k4_finseq_4(k5_numbers,u1_struct_0(A),D,E),k8_polynom1(B,k4_finseq_4(k5_numbers,u1_struct_0(A),D,E))) ) ) )
                   => k3_finseq_1(k15_dtconstr(u1_struct_0(k16_polynom3(A)),C)) = k4_uproots(u1_struct_0(A),B) ) ) ) ) ) )).

fof(t65_uproots,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] :
          ( ( v7_seqm_3(B)
            & v1_polynom1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( m2_finseq_1(C,k3_finseq_2(u1_struct_0(k16_polynom3(A))))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ( ( k3_finseq_1(C) = k4_card_1(k11_polynom1(B))
                          & D = k1_uproots(k11_polynom1(B))
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ( r2_hidden(F,k5_finsop_1(C))
                               => k1_funct_1(C,F) = k9_uproots(A,k4_finseq_4(k5_numbers,u1_struct_0(A),D,F),k8_polynom1(B,k4_finseq_4(k5_numbers,u1_struct_0(A),D,F))) ) ) )
                       => ( ( r2_hidden(E,k11_polynom1(B))
                           => k4_card_1(k3_funct_2(k5_numbers,u1_struct_0(k16_polynom3(A)),k15_dtconstr(u1_struct_0(k16_polynom3(A)),C),k2_setwiseo(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,u1_struct_0(A))),k4_polynom5(A,k5_rlvect_1(A,E),k2_group_1(A))))) = k8_polynom1(B,E) )
                          & ( ~ r2_hidden(E,k11_polynom1(B))
                           => k4_card_1(k3_funct_2(k5_numbers,u1_struct_0(k16_polynom3(A)),k15_dtconstr(u1_struct_0(k16_polynom3(A)),C),k2_setwiseo(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,u1_struct_0(A))),k4_polynom5(A,k5_rlvect_1(A,E),k2_group_1(A))))) = np__0 ) ) ) ) ) ) ) ) )).

fof(t66_uproots,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)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( v7_seqm_3(B)
            & v1_polynom1(B)
            & m1_pboole(B,u1_struct_0(A)) )
         => ! [C] :
              ( ( v7_seqm_3(C)
                & v1_polynom1(C)
                & m1_pboole(C,u1_struct_0(A)) )
             => k10_uproots(A,k9_polynom1(u1_struct_0(A),B,C)) = k15_polynom3(A,k10_uproots(A,B),k10_uproots(A,C)) ) ) ) )).

fof(t67_uproots,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)
        & v2_polynom5(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)
            & v1_uproots(B,A)
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1)) = k2_group_1(A)
           => B = k10_uproots(A,k8_uproots(A,B)) ) ) ) )).

fof(t68_uproots,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)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k16_polynom3(A)))
             => ( ( k3_finseq_1(C) = k4_card_1(B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ( ( r2_hidden(D,k5_finsop_1(C))
                              & E = k1_funct_1(k1_uproots(B),D) )
                           => k1_funct_1(C,D) = k4_polynom5(A,k5_rlvect_1(A,E),k2_group_1(A)) ) ) ) )
               => k10_uproots(A,k2_uproots(u1_struct_0(A),B,np__1)) = k13_fvsum_1(k16_polynom3(A),C) ) ) ) ) )).

fof(t69_uproots,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)
        & ~ v3_realset2(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_finseq_1(D,u1_struct_0(A))
                 => ( ( k3_finseq_1(D) = k4_card_1(B)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(A))
                             => ( ( r2_hidden(E,k5_finsop_1(D))
                                  & F = k1_funct_1(k1_uproots(B),E) )
                               => k1_funct_1(D,E) = k2_polynom4(A,k4_polynom5(A,k5_rlvect_1(A,F),k2_group_1(A)),C) ) ) ) )
                   => k2_polynom4(A,k10_uproots(A,k2_uproots(u1_struct_0(A),B,np__1)),C) = k13_fvsum_1(A,D) ) ) ) ) ) )).

fof(s1_uproots,axiom,
    ( ( p1_s1_uproots(np__1,k1_funct_1(f1_s1_uproots,np__1))
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,A)
              & p1_s1_uproots(A,k1_funct_1(f1_s1_uproots,A)) )
           => ( r1_xreal_0(k3_finseq_1(f1_s1_uproots),A)
              | p1_s1_uproots(k23_binop_2(A,np__1),k1_funct_1(f1_s1_uproots,k23_binop_2(A,np__1))) ) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(np__1,A)
            & r1_xreal_0(A,k3_finseq_1(f1_s1_uproots)) )
         => p1_s1_uproots(A,k1_funct_1(f1_s1_uproots,A)) ) ) )).

fof(dt_k1_uproots,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => m2_finseq_1(k1_uproots(A),A) ) )).

fof(dt_k2_uproots,axiom,(
    ! [A,B,C] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_polynom1(k2_uproots(A,B,C),A,k14_polynom1(A)) ) )).

fof(dt_k3_uproots,axiom,(
    ! [A,B] :
      ( ( v1_seq_1(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => v1_xreal_0(k3_uproots(A,B)) ) )).

fof(dt_k4_uproots,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => m2_subset_1(k4_uproots(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k4_uproots,axiom,(
    ! [A,B] :
      ( ( v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,A) )
     => k4_uproots(A,B) = k3_uproots(A,B) ) )).

fof(dt_k5_uproots,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l2_struct_0(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(k5_uproots(A,B,C))
        & v1_funct_2(k5_uproots(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k5_uproots(A,B,C),A)
        & m2_relset_1(k5_uproots(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k6_uproots,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,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(k6_uproots(A,B,C))
        & v1_funct_2(k6_uproots(A,B,C),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k6_uproots(A,B,C),A)
        & m2_relset_1(k6_uproots(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k7_uproots,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(C,A)
        & v1_uproots(C,A)
        & m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
     => m2_subset_1(k7_uproots(A,B,C),k1_numbers,k5_numbers) ) )).

fof(dt_k8_uproots,axiom,(
    ! [A,B] :
      ( ( ~ 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)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & v1_algseq_1(B,A)
        & v1_uproots(B,A)
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => ( v7_seqm_3(k8_uproots(A,B))
        & v1_polynom1(k8_uproots(A,B))
        & m1_pboole(k8_uproots(A,B),u1_struct_0(A)) ) ) )).

fof(dt_k9_uproots,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)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_1(k9_uproots(A,B,C),u1_struct_0(k16_polynom3(A))) ) )).

fof(dt_k10_uproots,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_vectsp_1(A)
        & l3_vectsp_1(A)
        & v7_seqm_3(B)
        & v1_polynom1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => ( v1_funct_1(k10_uproots(A,B))
        & v1_funct_2(k10_uproots(A,B),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k10_uproots(A,B),A)
        & m2_relset_1(k10_uproots(A,B),k5_numbers,u1_struct_0(A)) ) ) )).

fof(d8_uproots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_group_1(A)
        & v7_group_1(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & v1_algseq_1(C,A)
                & v1_uproots(C,A)
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( D = k7_uproots(A,B,C)
                  <=> ? [E] :
                        ( ~ v1_xboole_0(E)
                        & v1_finset_1(E)
                        & m1_subset_1(E,k1_zfmisc_1(k5_numbers))
                        & E = a_3_0_uproots(A,B,C)
                        & D = k1_pre_circ(E) ) ) ) ) ) ) )).

fof(fraenkel_a_3_0_uproots,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v4_group_1(B)
        & v7_group_1(B)
        & v3_rlvect_1(B)
        & v4_rlvect_1(B)
        & v5_rlvect_1(B)
        & v6_rlvect_1(B)
        & v6_vectsp_1(B)
        & v7_vectsp_1(B)
        & v8_vectsp_1(B)
        & ~ v10_vectsp_1(B)
        & l3_vectsp_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & v1_funct_1(D)
        & v1_funct_2(D,k5_numbers,u1_struct_0(B))
        & v1_algseq_1(D,B)
        & v1_uproots(D,B)
        & m2_relset_1(D,k5_numbers,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_uproots(B,C,D))
      <=> ? [E] :
            ( m2_subset_1(E,k1_numbers,k5_numbers)
            & A = E
            & ? [F] :
                ( v1_funct_1(F)
                & v1_funct_2(F,k5_numbers,u1_struct_0(B))
                & v1_algseq_1(F,B)
                & m2_relset_1(F,k5_numbers,u1_struct_0(B))
                & D = k15_polynom3(B,k2_polynom5(B,k4_polynom5(B,k5_rlvect_1(B,C),k2_group_1(B)),E),F) ) ) ) ) )).
%------------------------------------------------------------------------------