SET007 Axioms: SET007+679.ax
%------------------------------------------------------------------------------
% File : SET007+679 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Hilbert Basis 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 : hilbasis [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 57 ( 1 unt; 0 def)
% Number of atoms : 806 ( 66 equ)
% Maximal formula atoms : 35 ( 14 avg)
% Number of connectives : 846 ( 97 ~; 0 |; 558 &)
% ( 6 <=>; 185 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 12 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of predicates : 62 ( 60 usr; 1 prp; 0-4 aty)
% Number of functors : 67 ( 67 usr; 5 con; 0-5 aty)
% Number of variables : 191 ( 188 !; 3 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_hilbasis,axiom,
! [A,B,C] :
( ( v3_ordinal1(A)
& m1_subset_1(B,A)
& ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ( v1_relat_1(k3_hilbasis(A,B,C))
& v1_funct_1(k3_hilbasis(A,B,C))
& v1_funct_2(k3_hilbasis(A,B,C),k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(k3_hilbasis(A,B,C),k14_polynom1(A),C) ) ) ).
fof(fc2_hilbasis,axiom,
! [A,B] :
( ( ~ 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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k6_hilbasis(A,B))
& v1_funct_1(k6_hilbasis(A,B))
& v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
& v1_grcat_1(k6_hilbasis(A,B),k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,np__1),A)) ) ) ).
fof(fc3_hilbasis,axiom,
! [A,B] :
( ( ~ 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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k6_hilbasis(A,B))
& v1_funct_1(k6_hilbasis(A,B))
& v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
& v1_group_6(k6_hilbasis(A,B),k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,np__1),A)) ) ) ).
fof(fc4_hilbasis,axiom,
! [A,B] :
( ( ~ 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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k6_hilbasis(A,B))
& v1_funct_1(k6_hilbasis(A,B))
& v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
& v1_endalg(k6_hilbasis(A,B),k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,np__1),A)) ) ) ).
fof(fc5_hilbasis,axiom,
! [A,B] :
( ( ~ 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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k6_hilbasis(A,B))
& v1_funct_1(k6_hilbasis(A,B))
& v2_funct_1(k6_hilbasis(A,B))
& v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A))) ) ) ).
fof(fc6_hilbasis,axiom,
! [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)
& v7_vectsp_1(A)
& v7_ideal_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v2_group_1(k16_polynom3(A))
& v4_group_1(k16_polynom3(A))
& v7_group_1(k16_polynom3(A))
& v3_rlvect_1(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))
& v4_vectsp_1(k16_polynom3(A))
& v5_vectsp_1(k16_polynom3(A))
& v6_vectsp_1(k16_polynom3(A))
& v7_vectsp_1(k16_polynom3(A))
& v8_vectsp_1(k16_polynom3(A))
& v1_binom(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))
& v7_ideal_1(k16_polynom3(A)) ) ) ).
fof(t1_hilbasis,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ~ ( r1_tarski(k2_xboole_0(k2_relat_1(A),k2_relat_1(B)),k1_relat_1(C))
& ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v1_finseq_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_finseq_1(E) )
=> ~ ( D = k5_relat_1(A,C)
& E = k5_relat_1(B,C)
& k5_relat_1(k7_finseq_1(A,B),C) = k7_finseq_1(D,E) ) ) ) ) ) ) ) ).
fof(t2_hilbasis,axiom,
! [A] :
( ( v7_seqm_3(A)
& v1_polynom1(A)
& m1_pboole(A,np__0) )
=> k21_polynom1(np__0,A) = k9_finseq_1(k10_finseq_1(k1_xboole_0,k1_xboole_0)) ) ).
fof(t3_hilbasis,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,B) )
=> ( r1_xreal_0(A,B)
=> m1_polynom1(k7_relat_1(C,A),A,k14_polynom1(A)) ) ) ) ) ).
fof(t4_hilbasis,axiom,
! [A,B,C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,B) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,B) )
=> ! [E] :
( ( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,A) )
=> ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> ( ( E = k7_relat_1(C,A)
& F = k7_relat_1(D,A)
& r3_polynom1(B,C,D) )
=> r3_polynom1(A,E,F) ) ) ) ) ) ).
fof(t5_hilbasis,axiom,
! [A,B,C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,B) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,B) )
=> ! [E] :
( ( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,A) )
=> ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> ( ( E = k7_relat_1(C,A)
& F = k7_relat_1(D,A) )
=> ( k7_relat_1(k10_polynom1(B,C,D),A) = k10_polynom1(A,E,F)
& k7_relat_1(k9_polynom1(B,C,D),A) = k9_polynom1(A,E,F) ) ) ) ) ) ) ).
fof(d1_hilbasis,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ! [D] :
( m1_polynom1(D,k1_nat_1(A,np__1),k14_polynom1(k1_nat_1(A,np__1)))
=> ( D = k1_hilbasis(A,B,C)
<=> ( k7_relat_1(D,A) = C
& k8_polynom1(D,A) = B ) ) ) ) ) ) ).
fof(t6_hilbasis,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k16_polynom1(k1_nat_1(A,np__1)) = k1_hilbasis(A,np__0,k16_polynom1(A)) ) ).
fof(t7_hilbasis,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ( r2_hidden(C,k5_relset_1(k5_numbers,k13_polynom1(A),k20_polynom1(A,B)))
<=> r3_polynom1(A,C,B) ) ) ) ) ).
fof(d2_hilbasis,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> k2_hilbasis(A,B) = k2_polynom1(A,k16_polynom1(A),B,np__1) ) ).
fof(t8_hilbasis,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> k1_polynom2(A,k2_hilbasis(A,B)) = k15_cqc_sim1(A,B) ) ) ).
fof(t9_hilbasis,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( k8_polynom1(k2_hilbasis(A,B),B) = np__1
& ! [C] :
( m1_subset_1(C,A)
=> ( B != C
=> k8_polynom1(k2_hilbasis(A,B),C) = np__0 ) ) ) ) ) ).
fof(t10_hilbasis,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( k2_hilbasis(A,B) = k2_hilbasis(A,C)
=> B = C ) ) ) ) ).
fof(t11_hilbasis,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_ordinal1(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,u1_struct_0(C))
& m2_relset_1(D,A,u1_struct_0(C)) )
=> k3_polynom2(A,k2_hilbasis(A,B),C,D) = k8_funct_2(A,u1_struct_0(C),D,B) ) ) ) ) ).
fof(d3_hilbasis,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& l2_vectsp_1(C) )
=> k3_hilbasis(A,B,C) = k1_polynom1(k14_polynom1(A),u1_struct_0(C),k26_polynom1(A,C),k2_hilbasis(A,B),k2_group_1(C)) ) ) ).
fof(t12_hilbasis,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( k15_polynom1(A,B,k3_hilbasis(A,C,B),k2_hilbasis(A,C)) = k2_group_1(B)
& ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( D != k2_hilbasis(A,C)
=> k15_polynom1(A,B,k3_hilbasis(A,C,B),D) = k1_rlvect_1(B) ) ) ) ) ) ).
fof(t13_hilbasis,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> k12_polynom1(k14_polynom1(A),C,k3_hilbasis(A,B,C)) = k15_cqc_sim1(k14_polynom1(A),k2_hilbasis(A,B)) ) ) ).
fof(t14_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,B)
=> ! [D] :
( m1_subset_1(D,B)
=> ( k3_hilbasis(B,C,A) = k3_hilbasis(B,D,A)
=> C = D ) ) ) ) ) ).
fof(t15_hilbasis,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(k16_polynom3(A)))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( B = C
=> k5_rlvect_1(k16_polynom3(A),B) = k10_polynom3(A,C) ) ) ) ) ).
fof(t16_hilbasis,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(k16_polynom3(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k16_polynom3(A)))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
=> ( ( B = D
& C = E )
=> k6_rlvect_1(k16_polynom3(A),B,C) = k11_polynom3(A,D,E) ) ) ) ) ) ) ).
fof(t17_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k16_polynom3(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)) )
=> ( ( r2_hidden(C,k4_hilbasis(A,B))
& r2_hidden(D,B) )
=> ( r2_hidden(C,B)
& r1_xreal_0(k3_algseq_1(A,C),k3_algseq_1(A,D)) ) ) ) ) ) ) ).
fof(d5_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [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)) )
=> ( D = k5_hilbasis(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( E = B
=> k8_funct_2(k5_numbers,u1_struct_0(A),D,E) = C )
& ( E != B
=> k8_funct_2(k5_numbers,u1_struct_0(A),D,E) = k1_rlvect_1(A) ) ) ) ) ) ) ) ) ).
fof(t18_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( C != k1_rlvect_1(A)
=> k3_algseq_1(A,k5_hilbasis(A,B,C)) = k1_nat_1(B,np__1) )
& ( C = k1_rlvect_1(A)
=> k3_algseq_1(A,k5_hilbasis(A,B,C)) = np__0 )
& r1_xreal_0(k3_algseq_1(A,k5_hilbasis(A,B,C)),k1_nat_1(B,np__1)) ) ) ) ) ).
fof(t19_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_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] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& v1_algseq_1(E,A)
& m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
=> k8_funct_2(k5_numbers,u1_struct_0(A),k14_polynom3(A,k5_hilbasis(A,B,D),E),k1_nat_1(C,B)) = k1_group_1(A,D,k8_funct_2(k5_numbers,u1_struct_0(A),E,C)) ) ) ) ) ) ).
fof(t20_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_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] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& v1_algseq_1(E,A)
& m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
=> k8_funct_2(k5_numbers,u1_struct_0(A),k14_polynom3(A,E,k5_hilbasis(A,B,D)),k1_nat_1(C,B)) = k1_group_1(A,k8_funct_2(k5_numbers,u1_struct_0(A),E,C),D) ) ) ) ) ) ).
fof(t21_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_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)) )
=> r1_xreal_0(k3_algseq_1(A,k14_polynom3(A,B,C)),k5_binarith(k1_nat_1(k3_algseq_1(A,B),k3_algseq_1(A,C)),np__1)) ) ) ) ).
fof(t22_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,A)
& v2_ideal_1(C,A)
& v3_ideal_1(C,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ( v4_quofield(D,A,B)
=> ( ~ v1_xboole_0(k4_pre_topc(A,B,D,C))
& v1_ideal_1(k4_pre_topc(A,B,D,C),B)
& v2_ideal_1(k4_pre_topc(A,B,D,C),B)
& v3_ideal_1(k4_pre_topc(A,B,D,C),B)
& m1_subset_1(k4_pre_topc(A,B,D,C),k1_zfmisc_1(u1_struct_0(B))) ) ) ) ) ) ) ).
fof(t23_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_quofield(C,A,B)
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_rlvect_1(A)) = k1_rlvect_1(B) ) ) ) ) ).
fof(t24_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
=> ! [F] :
( m1_ideal_1(F,A,C)
=> ! [G] :
( m1_ideal_1(G,B,D)
=> ! [H] :
( m2_finseq_1(H,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)))
=> ( ( v1_quofield(E,A,B)
& k3_finseq_1(F) = k3_finseq_1(G)
& r1_ideal_1(A,C,F,H)
& ! [I] :
( r2_hidden(I,k4_finseq_1(G))
=> k1_funct_1(G,I) = k1_group_1(B,k1_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,k5_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),H,I))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,k6_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),H,I)))),k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,k7_mcart_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),k4_finseq_4(k5_numbers,k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),H,I)))) ) )
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),E,k9_rlvect_1(A,F)) = k9_rlvect_1(B,G) ) ) ) ) ) ) ) ) ) ).
fof(t25_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ~ ( v4_quofield(C,A,B)
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> ~ ( v4_quofield(D,B,A)
& D = k2_tops_2(A,B,C) ) ) ) ) ) ) ).
fof(t26_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(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] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ( v4_quofield(D,A,B)
=> k4_pre_topc(A,B,D,k7_ideal_1(A,C)) = k7_ideal_1(B,k4_pre_topc(A,B,D,C)) ) ) ) ) ) ).
fof(t27_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(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] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_group_1(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( ( v4_quofield(C,A,B)
& v7_ideal_1(A) )
=> v7_ideal_1(B) ) ) ) ) ).
fof(t28_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(k30_polynom1(np__0,A)))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k30_polynom1(np__0,A)))
& v4_quofield(B,A,k30_polynom1(np__0,A)) ) ) ).
fof(t29_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(B),u1_struct_0(A))
& v2_polynom1(D,k14_polynom1(B),A)
& m2_relset_1(D,k14_polynom1(B),u1_struct_0(A)) )
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(k30_polynom1(B,A)))
=> ~ ( D = k9_rlvect_1(k30_polynom1(B,A),E)
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k30_polynom1(B,A)),u1_struct_0(A))
& m2_relset_1(F,u1_struct_0(k30_polynom1(B,A)),u1_struct_0(A)) )
=> ~ ( ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(B),u1_struct_0(A))
& v2_polynom1(G,k14_polynom1(B),A)
& m2_relset_1(G,k14_polynom1(B),u1_struct_0(A)) )
=> k1_funct_1(F,G) = k15_polynom1(B,A,G,C) )
& k15_polynom1(B,A,D,C) = k9_rlvect_1(A,k5_finseqop(u1_struct_0(k30_polynom1(B,A)),u1_struct_0(A),E,F)) ) ) ) ) ) ) ) ) ).
fof(d6_hilbasis,axiom,
! [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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
& m2_relset_1(C,u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A))) )
=> ( C = k6_hilbasis(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(k30_polynom1(B,A)))
& v1_algseq_1(D,k30_polynom1(B,A))
& m2_relset_1(D,k5_numbers,u1_struct_0(k30_polynom1(B,A))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(B),u1_struct_0(A))
& v2_polynom1(E,k14_polynom1(B),A)
& m2_relset_1(E,k14_polynom1(B),u1_struct_0(A)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(k1_nat_1(B,np__1)),u1_struct_0(A))
& v2_polynom1(F,k14_polynom1(k1_nat_1(B,np__1)),A)
& m2_relset_1(F,k14_polynom1(k1_nat_1(B,np__1)),u1_struct_0(A)) )
=> ! [G] :
( ( v7_seqm_3(G)
& v1_polynom1(G)
& m1_pboole(G,k1_nat_1(B,np__1)) )
=> ( ( F = k1_funct_1(C,D)
& E = k8_funct_2(k5_numbers,u1_struct_0(k30_polynom1(B,A)),D,k8_polynom1(G,B)) )
=> k15_polynom1(k1_nat_1(B,np__1),A,F,G) = k1_funct_1(E,k7_relat_1(G,B)) ) ) ) ) ) ) ) ) ) ).
fof(d7_hilbasis,axiom,
! [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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A))))
& m2_relset_1(C,u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A)))) )
=> ( C = k7_hilbasis(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(k1_nat_1(B,np__1)),u1_struct_0(A))
& v2_polynom1(D,k14_polynom1(k1_nat_1(B,np__1)),A)
& m2_relset_1(D,k14_polynom1(k1_nat_1(B,np__1)),u1_struct_0(A)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(B),u1_struct_0(A))
& v2_polynom1(E,k14_polynom1(B),A)
& m2_relset_1(E,k14_polynom1(B),u1_struct_0(A)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,u1_struct_0(k30_polynom1(B,A)))
& v1_algseq_1(F,k30_polynom1(B,A))
& m2_relset_1(F,k5_numbers,u1_struct_0(k30_polynom1(B,A))) )
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( ( v7_seqm_3(H)
& v1_polynom1(H)
& m1_pboole(H,B) )
=> ( ( F = k1_funct_1(C,D)
& E = k8_funct_2(k5_numbers,u1_struct_0(k30_polynom1(B,A)),F,G) )
=> k15_polynom1(B,A,E,H) = k15_polynom1(k1_nat_1(B,np__1),A,D,k1_hilbasis(B,G,H)) ) ) ) ) ) ) ) ) ) ) ).
fof(t30_hilbasis,axiom,
! [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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
=> k8_funct_2(u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)),k6_hilbasis(A,B),k8_funct_2(u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),k7_hilbasis(A,B),C)) = C ) ) ) ).
fof(t31_hilbasis,axiom,
! [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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
& m2_relset_1(C,u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
& v4_quofield(C,k16_polynom3(k30_polynom1(B,A)),k30_polynom1(k1_nat_1(B,np__1),A)) ) ) ) ).
fof(t32_hilbasis,axiom,
$true ).
fof(t33_hilbasis,axiom,
! [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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ( v7_ideal_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> v7_ideal_1(k30_polynom1(B,A)) ) ) ) ).
fof(t34_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> v7_ideal_1(A) ) ).
fof(t35_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> v7_ideal_1(k30_polynom1(B,A)) ) ) ).
fof(t36_hilbasis,axiom,
! [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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& ~ v1_finset_1(B) )
=> ~ v7_ideal_1(k30_polynom1(B,A)) ) ) ).
fof(dt_k1_hilbasis,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers)
& v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> m1_polynom1(k1_hilbasis(A,B,C),k1_nat_1(A,np__1),k14_polynom1(k1_nat_1(A,np__1))) ) ).
fof(dt_k2_hilbasis,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> m1_polynom1(k2_hilbasis(A,B),A,k14_polynom1(A)) ) ).
fof(dt_k3_hilbasis,axiom,
! [A,B,C] :
( ( m1_subset_1(B,A)
& ~ v3_struct_0(C)
& v2_group_1(C)
& l2_vectsp_1(C) )
=> ( v1_funct_1(k3_hilbasis(A,B,C))
& v1_funct_2(k3_hilbasis(A,B,C),k14_polynom1(A),u1_struct_0(C))
& m2_relset_1(k3_hilbasis(A,B,C),k14_polynom1(A),u1_struct_0(C)) ) ) ).
fof(dt_k4_hilbasis,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k16_polynom3(A)))) )
=> ( ~ v1_xboole_0(k4_hilbasis(A,B))
& m1_subset_1(k4_hilbasis(A,B),k1_zfmisc_1(B)) ) ) ).
fof(dt_k5_hilbasis,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(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,k5_numbers)
& m1_subset_1(C,u1_struct_0(A)) )
=> ( v1_funct_1(k5_hilbasis(A,B,C))
& v1_funct_2(k5_hilbasis(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k5_hilbasis(A,B,C),A)
& m2_relset_1(k5_hilbasis(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k6_hilbasis,axiom,
! [A,B] :
( ( ~ 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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_funct_1(k6_hilbasis(A,B))
& v1_funct_2(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)))
& m2_relset_1(k6_hilbasis(A,B),u1_struct_0(k16_polynom3(k30_polynom1(B,A))),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A))) ) ) ).
fof(dt_k7_hilbasis,axiom,
! [A,B] :
( ( ~ 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)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_funct_1(k7_hilbasis(A,B))
& v1_funct_2(k7_hilbasis(A,B),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A))))
& m2_relset_1(k7_hilbasis(A,B),u1_struct_0(k30_polynom1(k1_nat_1(B,np__1),A)),u1_struct_0(k16_polynom3(k30_polynom1(B,A)))) ) ) ).
fof(d4_hilbasis,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k16_polynom3(A)))) )
=> k4_hilbasis(A,B) = a_2_0_hilbasis(A,B) ) ) ).
fof(fraenkel_a_2_0_hilbasis,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& l3_vectsp_1(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k16_polynom3(B)))) )
=> ( r2_hidden(A,a_2_0_hilbasis(B,C))
<=> ? [D] :
( m1_struct_0(D,k16_polynom3(B),C)
& A = D
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(B))
& v1_algseq_1(E,B)
& m2_relset_1(E,k5_numbers,u1_struct_0(B)) )
=> ! [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)) )
=> ( ( E = D
& r2_hidden(F,C) )
=> r1_xreal_0(k3_algseq_1(B,E),k3_algseq_1(B,F)) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------