SET007 Axioms: SET007+804.ax


%------------------------------------------------------------------------------
% File     : SET007+804 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Primitive Roots of Unity and Cyclotomic Polynomials
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : uniroots [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   88 (   3 unit)
%            Number of atoms       :  698 ( 120 equality)
%            Maximal formula depth :   25 (   9 average)
%            Number of connectives :  759 ( 149 ~  ;  10  |; 362  &)
%                                         (  12 <=>; 226 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   46 (   1 propositional; 0-3 arity)
%            Number of functors    :   83 (   8 constant; 0-6 arity)
%            Number of variables   :  200 (   0 singleton; 186 !;  14 ?)
%            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_uniroots,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v6_group_1(A)
      & v7_group_1(A)
      & v4_vectsp_1(A)
      & v5_vectsp_1(A)
      & v6_vectsp_1(A)
      & v7_vectsp_1(A)
      & v8_vectsp_1(A)
      & v9_vectsp_1(A)
      & ~ v10_vectsp_1(A)
      & ~ v3_realset2(A) ) )).

fof(rc2_uniroots,axiom,(
    ? [A] :
      ( l3_vectsp_1(A)
      & ~ v3_struct_0(A)
      & v3_rlvect_1(A)
      & v4_rlvect_1(A)
      & v5_rlvect_1(A)
      & v6_rlvect_1(A)
      & v2_group_1(A)
      & v4_group_1(A)
      & v6_group_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)
      & ~ v3_realset2(A) ) )).

fof(fc1_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_uniroots(A))
        & v1_group_1(k1_uniroots(A))
        & v2_group_1(k1_uniroots(A))
        & v3_group_1(k1_uniroots(A))
        & v4_group_1(k1_uniroots(A))
        & v6_group_1(k1_uniroots(A)) ) ) )).

fof(fc2_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => ~ v1_xboole_0(k2_uniroots(A)) ) )).

fof(fc3_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => v1_finset_1(k2_uniroots(A)) ) )).

fof(fc4_uniroots,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_relat_1(k4_uniroots(B,A))
        & v1_funct_1(k4_uniroots(B,A))
        & v1_funct_2(k4_uniroots(B,A),k5_numbers,u1_struct_0(A))
        & v1_algseq_1(k4_uniroots(B,A),A)
        & v1_uproots(k4_uniroots(B,A),A) ) ) )).

fof(t1_uniroots,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( A != np__0
          & A != np__1
          & ~ r1_xreal_0(np__2,A) ) ) )).

fof(t2_uniroots,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( r1_xreal_0(np__1,k3_finseq_1(A))
       => k7_relat_1(A,k2_finseq_1(np__1)) = k9_finseq_1(k1_funct_1(A,np__1)) ) ) )).

fof(t3_uniroots,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k1_complfld))
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k5_finsop_1(A))
                   => k3_complfld(k4_finseq_4(k5_numbers,u1_struct_0(k1_complfld),A,C)) = k1_funct_1(B,C) ) ) )
           => k3_complfld(k13_fvsum_1(k1_complfld,A)) = k16_rvsum_1(B) ) ) ) )).

fof(t4_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k1_complfld))) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ! [C] :
              ( m2_finseq_1(C,k1_numbers)
             => ( ( k3_finseq_1(C) = k4_card_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(k1_complfld))
                         => ( ( r2_hidden(D,k5_finsop_1(C))
                              & E = k1_funct_1(k1_uproots(A),D) )
                           => k1_funct_1(C,D) = k3_complfld(k6_rlvect_1(k1_complfld,B,E)) ) ) ) )
               => k3_complfld(k2_polynom4(k1_complfld,k10_uproots(k1_complfld,k2_uproots(u1_struct_0(k1_complfld),A,np__1)),B)) = k16_rvsum_1(C) ) ) ) ) )).

fof(t5_uniroots,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k1_complfld))
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,k5_finsop_1(A))
             => v1_int_1(k1_funct_1(A,B)) ) )
       => v1_int_1(k9_rlvect_1(k1_complfld,A)) ) ) )).

fof(t6_uniroots,axiom,(
    $true )).

fof(t7_uniroots,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,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( C = A
                      & D = B )
                   => ( k11_binop_2(C,D) = k10_group_1(k1_complfld,A,B)
                      & k9_binop_2(C,D) = k4_rlvect_1(k1_complfld,A,B) ) ) ) ) ) ) )).

fof(t8_uniroots,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( v1_int_1(A)
       => ( r1_xreal_0(A,np__0)
          | ! [B] :
              ( m1_subset_1(B,u1_struct_0(k1_complfld))
             => ~ ( k3_complfld(B) = np__1
                  & B != k1_hahnban1(np__1,np__0)
                  & r1_xreal_0(k3_complfld(k6_rlvect_1(k1_complfld,k1_hahnban1(A,np__0),B)),k10_binop_2(A,np__1)) ) ) ) ) ) )).

fof(t9_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_finseq_1(A,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( r1_xreal_0(np__1,B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( r2_hidden(C,k5_finsop_1(A))
                      & r1_xreal_0(k1_funct_1(A,C),B) ) )
              & r1_xreal_0(k16_rvsum_1(A),B) ) ) ) )).

fof(t10_uniroots,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_group_1(k1_complfld) = k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),k2_group_1(k1_complfld),A) ) )).

fof(t11_uniroots,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)) = k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),k4_nat_1(B,A)),A))
            & k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)) = k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),k4_nat_1(B,A)),A)) ) ) ) )).

fof(t12_uniroots,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A))) = k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),k4_nat_1(B,A)),A)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),k4_nat_1(B,A)),A))) ) ) )).

fof(t13_uniroots,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k10_group_1(k1_complfld,k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A))),k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),C),A)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),C),A)))) = k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),k4_nat_1(k23_binop_2(B,C),A)),A)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),k4_nat_1(k23_binop_2(B,C),A)),A))) ) ) ) )).

fof(t14_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_group_1(A)
        & v4_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] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,k24_binop_2(C,D)) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,C),D) ) ) ) ) )).

fof(t15_uniroots,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ( v1_int_1(B)
           => v1_int_1(k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),B,A)) ) ) ) )).

fof(t16_uniroots,axiom,(
    ! [A] :
      ( m2_finseq_1(A,u1_struct_0(k1_complfld))
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,k5_finsop_1(A))
             => v1_int_1(k1_funct_1(A,B)) ) )
       => v1_int_1(k9_rlvect_1(k1_complfld,A)) ) ) )).

fof(t17_uniroots,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( ( r1_xreal_0(np__0,A)
          & k22_sin_cos(A) = np__1 )
       => ( r1_xreal_0(k11_binop_2(np__2,k32_sin_cos),A)
          | A = np__0 ) ) ) )).

fof(d1_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ( B = k1_uniroots(A)
          <=> ( u1_struct_0(B) = k4_xboole_0(u1_struct_0(A),k7_rlvect_2(A,k1_rlvect_1(A)))
              & u1_group_1(B) = k1_realset1(u1_group_1(A),u1_struct_0(B)) ) ) ) ) )).

fof(t18_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => u1_struct_0(A) = k2_xboole_0(u1_struct_0(k1_uniroots(A)),k7_rlvect_2(A,k1_rlvect_1(A))) ) )).

fof(t19_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_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(k1_uniroots(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k1_uniroots(A)))
                     => ( ( B = D
                          & C = E )
                       => k1_group_1(k1_uniroots(A),D,E) = k1_group_1(A,B,C) ) ) ) ) ) ) )).

fof(t20_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => k2_group_1(A) = k2_group_1(k1_uniroots(A)) ) )).

fof(t21_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => k9_group_1(k1_uniroots(A)) = k10_binop_2(k4_card_1(u1_struct_0(A)),np__1) ) )).

fof(t22_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( r2_hidden(B,u1_struct_0(k1_uniroots(A)))
         => r2_hidden(B,u1_struct_0(A)) ) ) )).

fof(t23_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => r1_tarski(u1_struct_0(k1_uniroots(A)),u1_struct_0(A)) ) )).

fof(t24_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ( r2_hidden(B,k2_uniroots(A))
          <=> m1_comptrig(B,k2_group_1(k1_complfld),A) ) ) ) )).

fof(t25_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => r2_hidden(k2_group_1(k1_complfld),k2_uniroots(A)) ) )).

fof(t26_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ( r2_hidden(B,k2_uniroots(A))
           => k3_complfld(B) = np__1 ) ) ) )).

fof(t27_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ( r2_hidden(B,k2_uniroots(A))
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & B = k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),C),A)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),C),A))) ) ) ) ) )).

fof(t28_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k2_numbers)
         => ! [C] :
              ( m1_subset_1(C,k2_numbers)
             => ( ( r2_hidden(B,k2_uniroots(A))
                  & r2_hidden(C,k2_uniroots(A)) )
               => r2_hidden(k5_binop_2(B,C),k2_uniroots(A)) ) ) ) ) )).

fof(t30_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k1_card_1(k2_uniroots(A)) = A ) )).

fof(t31_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( r1_nat_1(B,A)
           => r1_tarski(k2_uniroots(B),k2_uniroots(A)) ) ) ) )).

fof(t32_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_uniroots(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( C = B
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_binop_1(u1_struct_0(k1_uniroots(A)),k5_numbers,u1_struct_0(k1_uniroots(A)),k5_group_1(k1_uniroots(A)),B,D) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),C,D) ) ) ) ) ) )).

fof(t33_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_uniroots(k1_complfld)))
         => ~ ( r2_hidden(B,k2_uniroots(A))
              & v5_group_1(B,k1_uniroots(k1_complfld)) ) ) ) )).

fof(t34_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_uniroots(k1_complfld)))
             => ( C = k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),B),A)))
               => k7_group_1(k1_uniroots(k1_complfld),C) = k5_int_1(A,k3_int_2(B,A)) ) ) ) ) )).

fof(t35_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => r1_tarski(k2_uniroots(A),u1_struct_0(k1_uniroots(k1_complfld))) ) )).

fof(t36_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ? [B] :
          ( m1_subset_1(B,u1_struct_0(k1_uniroots(k1_complfld)))
          & k7_group_1(k1_uniroots(k1_complfld),B) = A ) ) )).

fof(t37_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_uniroots(k1_complfld)))
         => ( r1_nat_1(k7_group_1(k1_uniroots(k1_complfld),B),A)
          <=> r2_hidden(B,k2_uniroots(A)) ) ) ) )).

fof(t39_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( r2_hidden(B,k2_uniroots(A))
        <=> ? [C] :
              ( m1_subset_1(C,u1_struct_0(k1_uniroots(k1_complfld)))
              & B = C
              & r1_nat_1(k7_group_1(k1_uniroots(k1_complfld),C),A) ) ) ) )).

fof(d3_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_group_1(B)
            & v3_group_1(B)
            & v4_group_1(B)
            & l1_group_1(B) )
         => ( B = k3_uniroots(A)
          <=> ( u1_struct_0(B) = k2_uniroots(A)
              & u1_group_1(B) = k1_realset1(u1_group_1(k1_complfld),k2_uniroots(A)) ) ) ) ) )).

fof(t40_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => m1_group_2(k3_uniroots(A),k1_uniroots(k1_complfld)) ) )).

fof(d4_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v8_vectsp_1(B)
            & l3_vectsp_1(B) )
         => k4_uniroots(A,B) = k1_polynom1(k5_numbers,u1_struct_0(B),k1_polynom1(k5_numbers,u1_struct_0(B),k12_polynom3(B),np__0,k5_rlvect_1(B,k2_group_1(B))),A,k2_group_1(B)) ) ) )).

fof(t41_uniroots,axiom,(
    k4_uniroots(np__1,k1_complfld) = k4_polynom5(k1_complfld,k5_rlvect_1(k1_complfld,k2_group_1(k1_complfld)),k2_group_1(k1_complfld)) )).

fof(t42_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ( k2_normsp_1(A,k4_uniroots(B,A),np__0) = k5_rlvect_1(A,k2_group_1(A))
            & k2_normsp_1(A,k4_uniroots(B,A),B) = k2_group_1(A) ) ) ) )).

fof(t43_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( C != np__0
                  & C != B
                  & k2_normsp_1(A,k4_uniroots(B,A),C) != k1_rlvect_1(A) ) ) ) ) )).

fof(t44_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v8_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k3_algseq_1(A,k4_uniroots(B,A)) = k23_binop_2(B,np__1) ) ) )).

fof(t45_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => k2_polynom4(k1_complfld,k4_uniroots(A,k1_complfld),B) = k6_xcmplx_0(k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),B,A),np__1) ) ) )).

fof(t46_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k6_polynom5(k1_complfld,k4_uniroots(A,k1_complfld)) = k2_uniroots(A) ) )).

fof(t47_uniroots,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ~ ( m1_subset_1(B,k1_numbers)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( C = B
                      & k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),B,A) = k3_newton(C,A) ) ) ) ) ) )).

fof(t48_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ? [C] :
              ( m1_subset_1(C,u1_struct_0(k1_complfld))
              & C = B
              & k2_polynom4(k1_complfld,k4_uniroots(A,k1_complfld),C) = k10_binop_2(k3_newton(B,A),np__1) ) ) ) )).

fof(t49_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k8_uproots(k1_complfld,k4_uniroots(A,k1_complfld)) = k2_uproots(u1_struct_0(k1_complfld),k2_uniroots(A),np__1) ) )).

fof(t50_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k4_uniroots(A,k1_complfld) = k10_uproots(k1_complfld,k2_uproots(u1_struct_0(k1_complfld),k2_uniroots(A),np__1)) ) )).

fof(t51_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ( v1_int_1(B)
           => v1_int_1(k2_polynom4(k1_complfld,k4_uniroots(A,k1_complfld),B)) ) ) ) )).

fof(t52_uniroots,axiom,(
    k6_uniroots(np__1) = k4_polynom5(k1_complfld,k5_rlvect_1(k1_complfld,k2_group_1(k1_complfld)),k2_group_1(k1_complfld)) )).

fof(t53_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_1(B,u1_struct_0(k16_polynom3(k1_complfld)))
         => ( ( k3_finseq_1(B) = A
              & ! [C] :
                  ( ( ~ v1_xboole_0(C)
                    & m2_subset_1(C,k1_numbers,k5_numbers) )
                 => ( r2_hidden(C,k5_finsop_1(B))
                   => ( ( ~ r1_nat_1(C,A)
                       => k1_funct_1(B,C) = k5_algseq_1(k1_complfld,k2_group_1(k1_complfld)) )
                      & ( r1_nat_1(C,A)
                       => k1_funct_1(B,C) = k6_uniroots(C) ) ) ) ) )
           => k4_uniroots(A,k1_complfld) = k13_fvsum_1(k16_polynom3(k1_complfld),B) ) ) ) )).

fof(t54_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ? [B] :
          ( m2_finseq_1(B,u1_struct_0(k16_polynom3(k1_complfld)))
          & ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,k5_numbers,u1_struct_0(k1_complfld))
              & v1_algseq_1(C,k1_complfld)
              & m2_relset_1(C,k5_numbers,u1_struct_0(k1_complfld))
              & C = k13_fvsum_1(k16_polynom3(k1_complfld),B)
              & k5_finsop_1(B) = k2_finseq_1(A)
              & ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m2_subset_1(D,k1_numbers,k5_numbers) )
                 => ( r2_hidden(D,k2_finseq_1(A))
                   => ( ( ~ ( r1_nat_1(D,A)
                            & D != A )
                       => k1_funct_1(B,D) = k5_algseq_1(k1_complfld,k2_group_1(k1_complfld)) )
                      & ( r1_nat_1(D,A)
                       => ( D = A
                          | k1_funct_1(B,D) = k6_uniroots(D) ) ) ) ) )
              & k4_uniroots(A,k1_complfld) = k15_polynom3(k1_complfld,k6_uniroots(A),C) ) ) ) )).

fof(t55_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( k2_normsp_1(k1_complfld,k6_uniroots(A),np__0) = np__1
              | k2_normsp_1(k1_complfld,k6_uniroots(A),np__0) = k7_binop_2(np__1) )
            & v1_int_1(k2_normsp_1(k1_complfld,k6_uniroots(A),B)) ) ) ) )).

fof(t56_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ( v1_int_1(B)
           => v1_int_1(k2_polynom4(k1_complfld,k6_uniroots(A),B)) ) ) ) )).

fof(t58_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ~ ( ~ r1_xreal_0(A,B)
              & r1_nat_1(B,A)
              & ! [C] :
                  ( m2_finseq_1(C,u1_struct_0(k16_polynom3(k1_complfld)))
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,u1_struct_0(k1_complfld))
                        & v1_algseq_1(D,k1_complfld)
                        & m2_relset_1(D,k5_numbers,u1_struct_0(k1_complfld)) )
                     => ~ ( D = k13_fvsum_1(k16_polynom3(k1_complfld),C)
                          & k5_finsop_1(C) = k2_finseq_1(A)
                          & ! [E] :
                              ( ( ~ v1_xboole_0(E)
                                & m2_subset_1(E,k1_numbers,k5_numbers) )
                             => ( r2_hidden(E,k2_finseq_1(A))
                               => ( ( ~ ( r1_nat_1(E,A)
                                        & ~ r1_nat_1(E,B)
                                        & E != A )
                                   => k1_funct_1(C,E) = k5_algseq_1(k1_complfld,k2_group_1(k1_complfld)) )
                                  & ( r1_nat_1(E,A)
                                   => ( r1_nat_1(E,B)
                                      | E = A
                                      | k1_funct_1(C,E) = k6_uniroots(E) ) ) ) ) )
                          & k4_uniroots(A,k1_complfld) = k15_polynom3(k1_complfld,k15_polynom3(k1_complfld,k4_uniroots(B,k1_complfld),k6_uniroots(A)),D) ) ) ) ) ) ) )).

fof(t59_uniroots,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k1_complfld))
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k16_polynom3(k1_complfld)))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k1_complfld))
                    & v1_algseq_1(D,k1_complfld)
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k1_complfld)) )
                 => ( ( D = k13_fvsum_1(k16_polynom3(k1_complfld),C)
                      & B = A
                      & ! [E] :
                          ( ( ~ v1_xboole_0(E)
                            & m2_subset_1(E,k1_numbers,k5_numbers) )
                         => ~ ( r2_hidden(E,k5_finsop_1(C))
                              & k1_funct_1(C,E) != k5_algseq_1(k1_complfld,k2_group_1(k1_complfld))
                              & k1_funct_1(C,E) != k6_uniroots(E) ) ) )
                   => v1_int_1(k2_polynom4(k1_complfld,D,B)) ) ) ) ) ) )).

fof(t60_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k1_complfld))
                     => ( ( E = D
                          & B = k2_polynom4(k1_complfld,k6_uniroots(A),E)
                          & C = k2_polynom4(k1_complfld,k4_uniroots(A,k1_complfld),E) )
                       => r2_int_1(B,C) ) ) ) ) ) ) )).

fof(t61_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_nat_1(B,A)
               => ( r1_xreal_0(A,B)
                  | ! [D] :
                      ( m1_subset_1(D,u1_struct_0(k1_complfld))
                     => ( D = C
                       => ! [E] :
                            ( v1_int_1(E)
                           => ! [F] :
                                ( v1_int_1(F)
                               => ! [G] :
                                    ( v1_int_1(G)
                                   => ( ( E = k2_polynom4(k1_complfld,k6_uniroots(A),D)
                                        & F = k2_polynom4(k1_complfld,k4_uniroots(A,k1_complfld),D)
                                        & G = k2_polynom4(k1_complfld,k4_uniroots(B,k1_complfld),D) )
                                     => r2_int_1(E,k5_int_1(F,G)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t62_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_complfld))
             => ( C = B
               => ! [D] :
                    ( v1_int_1(D)
                   => ( D = k2_polynom4(k1_complfld,k6_uniroots(A),C)
                     => r2_int_1(D,k6_xcmplx_0(k5_uniroots(B,A),np__1)) ) ) ) ) ) ) )).

fof(t63_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( r1_nat_1(B,A)
               => ( r1_xreal_0(A,B)
                  | ! [D] :
                      ( m1_subset_1(D,u1_struct_0(k1_complfld))
                     => ( D = C
                       => ! [E] :
                            ( v1_int_1(E)
                           => ( E = k2_polynom4(k1_complfld,k6_uniroots(A),D)
                             => r2_int_1(E,k5_int_1(k6_xcmplx_0(k5_uniroots(C,A),np__1),k6_xcmplx_0(k5_uniroots(C,B),np__1))) ) ) ) ) ) ) ) ) ) )).

fof(t64_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ( ~ r1_xreal_0(A,np__1)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( ~ r1_xreal_0(B,np__1)
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k1_complfld))
                 => ( C = B
                   => ! [D] :
                        ( v1_int_1(D)
                       => ~ ( D = k2_polynom4(k1_complfld,k6_uniroots(A),C)
                            & r1_xreal_0(k1_prepower(D),k10_binop_2(B,np__1)) ) ) ) ) ) ) ) ) )).

fof(s1_uniroots,axiom,
    ( ! [A] :
        ( ( ~ v1_xboole_0(A)
          & m2_subset_1(A,k1_numbers,k5_numbers) )
       => ( ! [B] :
              ( ( ~ v1_xboole_0(B)
                & m2_subset_1(B,k1_numbers,k5_numbers) )
             => ( ~ r1_xreal_0(A,B)
               => p1_s1_uniroots(B) ) )
         => p1_s1_uniroots(A) ) )
   => ! [A] :
        ( ( ~ v1_xboole_0(A)
          & m2_subset_1(A,k1_numbers,k5_numbers) )
       => p1_s1_uniroots(A) ) )).

fof(dt_k1_uniroots,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v4_group_1(A)
        & v6_vectsp_1(A)
        & v7_vectsp_1(A)
        & v8_vectsp_1(A)
        & v9_vectsp_1(A)
        & ~ v10_vectsp_1(A)
        & l3_vectsp_1(A) )
     => ( ~ v3_struct_0(k1_uniroots(A))
        & v1_group_1(k1_uniroots(A))
        & v3_group_1(k1_uniroots(A))
        & v4_group_1(k1_uniroots(A))
        & l1_group_1(k1_uniroots(A)) ) ) )).

fof(dt_k2_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => m1_subset_1(k2_uniroots(A),k1_zfmisc_1(u1_struct_0(k1_complfld))) ) )).

fof(dt_k3_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => ( ~ v3_struct_0(k3_uniroots(A))
        & v1_group_1(k3_uniroots(A))
        & v3_group_1(k3_uniroots(A))
        & v4_group_1(k3_uniroots(A))
        & l1_group_1(k3_uniroots(A)) ) ) )).

fof(dt_k4_uniroots,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & ~ v3_struct_0(B)
        & v8_vectsp_1(B)
        & l3_vectsp_1(B) )
     => ( v1_funct_1(k4_uniroots(A,B))
        & v1_funct_2(k4_uniroots(A,B),k5_numbers,u1_struct_0(B))
        & v1_algseq_1(k4_uniroots(A,B),B)
        & m2_relset_1(k4_uniroots(A,B),k5_numbers,u1_struct_0(B)) ) ) )).

fof(dt_k5_uniroots,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & m1_subset_1(B,k5_numbers) )
     => v1_int_1(k5_uniroots(A,B)) ) )).

fof(redefinition_k5_uniroots,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & m1_subset_1(B,k5_numbers) )
     => k5_uniroots(A,B) = k2_newton(A,B) ) )).

fof(dt_k6_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => ( v1_funct_1(k6_uniroots(A))
        & v1_funct_2(k6_uniroots(A),k5_numbers,u1_struct_0(k1_complfld))
        & v1_algseq_1(k6_uniroots(A),k1_complfld)
        & m2_relset_1(k6_uniroots(A),k5_numbers,u1_struct_0(k1_complfld)) ) ) )).

fof(d2_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k2_uniroots(A) = a_1_0_uniroots(A) ) )).

fof(t29_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k2_uniroots(A) = a_1_1_uniroots(A) ) )).

fof(t38_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k2_uniroots(A) = a_1_2_uniroots(A) ) )).

fof(d5_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k1_complfld))
            & v1_algseq_1(B,k1_complfld)
            & m2_relset_1(B,k5_numbers,u1_struct_0(k1_complfld)) )
         => ( B = k6_uniroots(A)
          <=> ? [C] :
                ( ~ v1_xboole_0(C)
                & v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k1_complfld)))
                & C = a_1_3_uniroots(A)
                & B = k10_uproots(k1_complfld,k2_uproots(u1_struct_0(k1_complfld),C,np__1)) ) ) ) ) )).

fof(t57_uniroots,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(k16_polynom3(k1_complfld)))
             => ! [D] :
                  ( ( v1_finset_1(D)
                    & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_complfld))) )
                 => ( ( D = a_2_0_uniroots(A,B)
                      & k5_finsop_1(C) = k2_finseq_1(A)
                      & ! [E] :
                          ( ( ~ v1_xboole_0(E)
                            & m2_subset_1(E,k1_numbers,k5_numbers) )
                         => ( r2_hidden(E,k5_finsop_1(C))
                           => ( ( ~ ( r1_nat_1(E,A)
                                    & ~ r1_nat_1(E,B)
                                    & E != A )
                               => k1_funct_1(C,E) = k5_algseq_1(k1_complfld,k2_group_1(k1_complfld)) )
                              & ( r1_nat_1(E,A)
                               => ( r1_nat_1(E,B)
                                  | E = A
                                  | k1_funct_1(C,E) = k6_uniroots(E) ) ) ) ) ) )
                   => k13_fvsum_1(k16_polynom3(k1_complfld),C) = k10_uproots(k1_complfld,k2_uproots(u1_struct_0(k1_complfld),D,np__1)) ) ) ) ) ) )).

fof(fraenkel_a_1_0_uniroots,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_1_0_uniroots(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k1_complfld))
            & A = C
            & m1_comptrig(C,k2_group_1(k1_complfld),B) ) ) ) )).

fof(fraenkel_a_1_1_uniroots,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_1_1_uniroots(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k1_hahnban1(k23_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),C),B)),k20_sin_cos(k12_binop_2(k11_binop_2(k11_binop_2(np__2,k32_sin_cos),C),B)))
            & ~ r1_xreal_0(B,C) ) ) ) )).

fof(fraenkel_a_1_2_uniroots,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_1_2_uniroots(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k1_uniroots(k1_complfld)))
            & A = C
            & r1_nat_1(k7_group_1(k1_uniroots(k1_complfld),C),B) ) ) ) )).

fof(fraenkel_a_1_3_uniroots,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_1_3_uniroots(B))
      <=> ? [C] :
            ( m1_subset_1(C,u1_struct_0(k1_uniroots(k1_complfld)))
            & A = C
            & k7_group_1(k1_uniroots(k1_complfld),C) = B ) ) ) )).

fof(fraenkel_a_2_0_uniroots,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers)
        & ~ v1_xboole_0(C)
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_0_uniroots(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(k1_uniroots(k1_complfld)))
            & A = D
            & r1_nat_1(k7_group_1(k1_uniroots(k1_complfld),D),B)
            & ~ r1_nat_1(k7_group_1(k1_uniroots(k1_complfld),D),C)
            & k7_group_1(k1_uniroots(k1_complfld),D) != B ) ) ) )).
%------------------------------------------------------------------------------