SET007 Axioms: SET007+803.ax
%------------------------------------------------------------------------------
% File : SET007+803 : TPTP v9.0.0. 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 unt; 0 def)
% Number of atoms : 1427 ( 138 equ)
% Maximal formula atoms : 30 ( 14 avg)
% Number of connectives : 1476 ( 149 ~; 7 |;1003 &)
% ( 16 <=>; 301 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 12 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 60 ( 59 usr; 0 prp; 1-3 aty)
% Number of functors : 82 ( 82 usr; 7 con; 0-4 aty)
% Number of variables : 302 ( 292 !; 10 ?)
% 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) ) ) ) ) ).
%------------------------------------------------------------------------------