SET007 Axioms: SET007+804.ax
%------------------------------------------------------------------------------
% File : SET007+804 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 698 ( 120 equ)
% Maximal formula atoms : 25 ( 7 avg)
% Number of connectives : 759 ( 149 ~; 10 |; 362 &)
% ( 12 <=>; 226 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 9 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 46 ( 44 usr; 1 prp; 0-3 aty)
% Number of functors : 83 ( 83 usr; 8 con; 0-6 aty)
% Number of variables : 200 ( 186 !; 14 ?)
% 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 ) ) ) ).
%------------------------------------------------------------------------------