SET007 Axioms: SET007+777.ax
%------------------------------------------------------------------------------
% File : SET007+777 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Characterization and Existence of Grobner Bases
% 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 : groeb_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 55 ( 0 unt; 0 def)
% Number of atoms : 1342 ( 24 equ)
% Maximal formula atoms : 54 ( 24 avg)
% Number of connectives : 1408 ( 121 ~; 2 |; 988 &)
% ( 10 <=>; 287 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 17 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 68 ( 67 usr; 0 prp; 1-6 aty)
% Number of functors : 31 ( 31 usr; 3 con; 0-4 aty)
% Number of variables : 267 ( 261 !; 6 ?)
% 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_groeb_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( ~ v3_struct_0(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
& v1_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
& v2_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
& v3_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
& v4_orders_2(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))
& v4_dickson(g1_orders_2(k14_polynom1(A),k4_groeb_1(A))) ) ) ).
fof(t1_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( r4_polyred(A,B,C,D,E,F)
& ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(G,A,C)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
=> F != k25_polynom1(A,C,D,k28_polynom1(A,C,G,E)) ) ) ) ) ) ) ) ) ).
fof(t2_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( r4_polyred(A,B,C,D,E,F)
& ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(G,A,C)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( F = k25_polynom1(A,C,D,k29_polynom1(A,C,G,E))
& ~ r2_hidden(k3_termord(A,B,C,k29_polynom1(A,C,G,E)),k12_polynom1(k14_polynom1(A),C,F))
& r1_termord(A,B,k3_termord(A,B,C,k29_polynom1(A,C,G,E)),k3_termord(A,B,C,D)) ) ) ) ) ) ) ) ) ) ).
fof(t3_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ( r1_tarski(F,G)
& r5_polyred(A,B,C,D,E,F) )
=> r5_polyred(A,B,C,D,E,G) ) ) ) ) ) ) ) ) ).
fof(t4_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( r1_tarski(D,E)
=> r1_tarski(k3_polyred(A,B,C,D),k3_polyred(A,B,C,E)) ) ) ) ) ) ) ).
fof(t5_groeb_1,axiom,
! [A] :
( v3_ordinal1(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,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> k12_polynom1(k14_polynom1(A),B,k24_polynom1(A,B,C)) = k12_polynom1(k14_polynom1(A),B,C) ) ) ) ).
fof(t6_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> r6_pboole(A,k3_termord(A,B,C,k24_polynom1(A,C,D)),k3_termord(A,B,C,D)) ) ) ) ) ).
fof(t7_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> r1_termord(A,B,k3_termord(A,B,C,k25_polynom1(A,C,D,E)),k2_termord(A,B,k3_termord(A,B,C,D),k3_termord(A,B,C,E))) ) ) ) ) ) ).
fof(t8_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r1_tarski(k12_polynom1(k14_polynom1(A),C,E),k12_polynom1(k14_polynom1(A),C,D))
=> r1_polyred(A,B,C,E,D) ) ) ) ) ) ) ).
fof(t9_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& v1_polynom7(D,A,C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& v1_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r6_polyred(A,B,C,D,E)
=> r1_termord(A,B,k3_termord(A,B,C,E),k3_termord(A,B,C,D)) ) ) ) ) ) ) ).
fof(t10_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> v9_rewrite1(k3_polyred(A,B,C,k1_groeb_1(A,C,D))) ) ) ) ) ).
fof(t11_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C))
& r2_hidden(E,D)
& k7_ideal_1(k30_polynom1(A,C),D) = k7_ideal_1(k30_polynom1(A,C),k1_groeb_1(A,C,E)) )
=> v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ).
fof(t12_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( v9_rewrite1(k3_polyred(A,B,C,D))
=> v7_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ).
fof(t13_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( v7_rewrite1(k3_polyred(A,B,C,D))
=> v4_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ).
fof(t14_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( v4_rewrite1(k3_polyred(A,B,C,D))
=> v8_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ).
fof(t15_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( v8_rewrite1(k3_polyred(A,B,C,D))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
=> r1_rewrite1(k3_polyred(A,B,C,D),E,k26_polynom1(A,C)) ) ) ) ) ) ) ) ).
fof(t16_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
=> r1_rewrite1(k3_polyred(A,B,C,D),E,k26_polynom1(A,C)) ) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& v1_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
=> r7_polyred(A,B,C,E,D) ) ) ) ) ) ) ) ).
fof(t17_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& v1_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
=> r7_polyred(A,B,C,E,D) ) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& v1_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
=> r10_polyred(A,B,C,E,D) ) ) ) ) ) ) ) ).
fof(t18_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& v1_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,k7_ideal_1(k30_polynom1(A,C),D))
=> r10_polyred(A,B,C,E,D) ) )
=> ! [E] :
( ( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,A) )
=> ~ ( r2_hidden(E,k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)))
& ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> ~ ( r2_hidden(F,k2_groeb_1(A,B,C,D))
& r3_polynom1(A,F,E) ) ) ) ) ) ) ) ) ) ).
fof(t19_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [E] :
( ( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,A) )
=> ~ ( r2_hidden(E,k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)))
& ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> ~ ( r2_hidden(F,k2_groeb_1(A,B,C,D))
& r3_polynom1(A,F,E) ) ) ) )
=> r1_tarski(k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)),k3_groeb_1(A,k2_groeb_1(A,B,C,D))) ) ) ) ) ) ).
fof(t20_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( r1_tarski(k2_groeb_1(A,B,C,k7_ideal_1(k30_polynom1(A,C),D)),k3_groeb_1(A,k2_groeb_1(A,B,C,D)))
=> v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ).
fof(d3_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( r1_groeb_1(A,B,C,D)
<=> v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ).
fof(d4_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( r2_groeb_1(A,B,C,D,E)
<=> ( k7_ideal_1(k30_polynom1(A,C),D) = E
& v9_rewrite1(k3_polyred(A,B,C,D)) ) ) ) ) ) ) ) ).
fof(t21_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r2_groeb_1(A,B,C,D,E)
=> m2_rewrite1(k3_polyred(A,B,C,D),k3_polyred(A,B,C,E)) ) ) ) ) ) ) ).
fof(t22_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k30_polynom1(A,C)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k30_polynom1(A,C)))
=> ! [F] :
( ( ~ v1_xboole_0(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r1_groeb_1(A,B,C,F)
=> ( r11_polyred(k30_polynom1(A,C),k7_ideal_1(k30_polynom1(A,C),F),D,E)
<=> k2_rewrite1(k3_polyred(A,B,C,F),D) = k2_rewrite1(k3_polyred(A,B,C,F),E) ) ) ) ) ) ) ) ) ).
fof(t23_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r1_groeb_1(A,B,C,E)
=> ( r2_hidden(D,k7_ideal_1(k30_polynom1(A,C),E))
<=> r1_rewrite1(k3_polyred(A,B,C,E),D,k26_polynom1(A,C)) ) ) ) ) ) ) ) ).
fof(t24_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( r2_groeb_1(A,B,C,E,D)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(F,D)
=> r1_rewrite1(k3_polyred(A,B,C,E),F,k26_polynom1(A,C)) ) ) ) ) ) ) ) ) ).
fof(t25_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(F,E)
=> r1_rewrite1(k3_polyred(A,B,C,D),F,k26_polynom1(A,C)) ) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& v1_polynom7(F,A,C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(F,E)
=> r7_polyred(A,B,C,F,D) ) ) ) ) ) ) ) ) ).
fof(t26_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ( r1_tarski(E,D)
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& v1_polynom7(F,A,C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(F,D)
=> r7_polyred(A,B,C,F,E) ) ) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& v1_polynom7(F,A,C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(F,D)
=> r10_polyred(A,B,C,F,E) ) ) ) ) ) ) ) ) ).
fof(t27_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& v1_polynom7(F,A,C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(F,E)
=> r10_polyred(A,B,C,F,D) ) )
=> ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> ~ ( r2_hidden(F,k2_groeb_1(A,B,C,E))
& ! [G] :
( ( v7_seqm_3(G)
& v1_polynom1(G)
& m1_pboole(G,A) )
=> ~ ( r2_hidden(G,k2_groeb_1(A,B,C,D))
& r3_polynom1(A,G,F) ) ) ) ) ) ) ) ) ) ) ).
fof(t28_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> ~ ( r2_hidden(F,k2_groeb_1(A,B,C,E))
& ! [G] :
( ( v7_seqm_3(G)
& v1_polynom1(G)
& m1_pboole(G,A) )
=> ~ ( r2_hidden(G,k2_groeb_1(A,B,C,D))
& r3_polynom1(A,G,F) ) ) ) )
=> r1_tarski(k2_groeb_1(A,B,C,E),k3_groeb_1(A,k2_groeb_1(A,B,C,D))) ) ) ) ) ) ) ).
fof(t29_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( ( r1_tarski(E,D)
& r1_tarski(k2_groeb_1(A,B,C,D),k3_groeb_1(A,k2_groeb_1(A,B,C,E))) )
=> r2_groeb_1(A,B,C,E,D) ) ) ) ) ) ) ).
fof(t30_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> r2_groeb_1(A,B,C,k1_groeb_1(A,C,k26_polynom1(A,C)),k1_groeb_1(A,C,k26_polynom1(A,C))) ) ) ) ).
fof(t31_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> r2_groeb_1(A,B,C,k1_groeb_1(A,C,D),k7_ideal_1(k30_polynom1(A,C),k1_groeb_1(A,C,D))) ) ) ) ) ).
fof(t32_groeb_1,axiom,
! [A] :
( ( v1_partfun1(A,k14_polynom1(k1_xboole_0),k14_polynom1(k1_xboole_0))
& v1_relat_2(A)
& v4_relat_2(A)
& v6_relat_2(A)
& v8_relat_2(A)
& v2_bagorder(A,k1_xboole_0)
& m2_relset_1(A,k14_polynom1(k1_xboole_0),k14_polynom1(k1_xboole_0)) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_group_1(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)
& v7_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_ideal_1(C,k30_polynom1(k1_xboole_0,B))
& v2_ideal_1(C,k30_polynom1(k1_xboole_0,B))
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k30_polynom1(k1_xboole_0,B)))) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(k1_xboole_0,B)))) )
=> ( r1_tarski(D,C)
=> ( D = k1_groeb_1(k1_xboole_0,B,k26_polynom1(k1_xboole_0,B))
| r2_groeb_1(k1_xboole_0,A,B,D,C) ) ) ) ) ) ) ).
fof(t33_groeb_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_ordinal1(A) )
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ~ ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> r1_groeb_1(A,B,C,D) ) ) ) ) ).
fof(d5_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ( B = k4_groeb_1(A)
<=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( r2_hidden(k4_tarski(C,D),B)
<=> r3_polynom1(A,C,D) ) ) ) ) ) ) ).
fof(t34_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)))))
=> ? [C] :
( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k14_polynom1(A)))
& r1_dickson(g1_orders_2(k14_polynom1(A),k4_groeb_1(A)),B,C) ) ) ) ).
fof(t35_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ? [E] :
( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
& r2_groeb_1(A,B,C,E,D) ) ) ) ) ) ).
fof(t36_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ~ ( D != k1_groeb_1(A,C,k26_polynom1(A,C))
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ~ ( r2_groeb_1(A,B,C,E,D)
& ~ r2_hidden(k26_polynom1(A,C),E) ) ) ) ) ) ) ) ).
fof(d6_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l2_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(D,k14_polynom1(A),C)
& m2_relset_1(D,k14_polynom1(A),u1_struct_0(C)) )
=> ( r3_groeb_1(A,B,C,D)
<=> k4_termord(A,B,C,D) = k2_group_1(C) ) ) ) ) ) ).
fof(d7_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ( r4_groeb_1(A,B,C,D)
<=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(E,k14_polynom1(A),C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ( r2_hidden(E,D)
=> ( r3_groeb_1(A,B,C,E)
& ~ r7_polyred(A,B,C,E,k6_subset_1(u1_struct_0(k30_polynom1(A,C)),D,k1_groeb_1(A,C,E))) ) ) ) ) ) ) ) ) ).
fof(t37_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(C))
& v3_polynom7(E,A,C)
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(G,k14_polynom1(A),C)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(F,D)
& r2_hidden(G,D)
& k5_termord(A,B,C,F) = E
& k5_termord(A,B,C,G) = E
& ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(H,k14_polynom1(A),C)
& m2_relset_1(H,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( r2_hidden(H,D)
& r2_polyred(A,B,C,H,F)
& k5_termord(A,B,C,H) = E ) )
& ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(H,k14_polynom1(A),C)
& m2_relset_1(H,k14_polynom1(A),u1_struct_0(C)) )
=> ~ ( r2_hidden(H,D)
& r2_polyred(A,B,C,H,G)
& k5_termord(A,B,C,H) = E ) ) )
=> F = G ) ) ) ) ) ) ) ) ).
fof(t38_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(F,k14_polynom1(A),C)
& m2_relset_1(F,k14_polynom1(A),u1_struct_0(C)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k14_polynom1(A),u1_struct_0(C))
& v2_polynom1(G,k14_polynom1(A),C)
& v1_polynom7(G,A,C)
& m2_relset_1(G,k14_polynom1(A),u1_struct_0(C)) )
=> ( ( r2_hidden(F,E)
& r2_hidden(G,E)
& r3_polynom1(A,k3_termord(A,B,C,G),k3_termord(A,B,C,F))
& r2_groeb_1(A,B,C,E,D) )
=> ( F = G
| r2_groeb_1(A,B,C,k6_subset_1(u1_struct_0(k30_polynom1(A,C)),E,k1_groeb_1(A,C,F)),D) ) ) ) ) ) ) ) ) ) ).
fof(t39_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ~ ( D != k1_groeb_1(A,C,k26_polynom1(A,C))
& ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ~ ( r2_groeb_1(A,B,C,E,D)
& r4_groeb_1(A,B,C,E) ) ) ) ) ) ) ) ).
fof(t40_groeb_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& v2_bagorder(B,A)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v1_ideal_1(D,k30_polynom1(A,C))
& v2_ideal_1(D,k30_polynom1(A,C))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [E] :
( ( ~ v1_xboole_0(E)
& v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ! [F] :
( ( ~ v1_xboole_0(F)
& v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> ( ( r2_groeb_1(A,B,C,E,D)
& r4_groeb_1(A,B,C,E)
& r2_groeb_1(A,B,C,F,D)
& r4_groeb_1(A,B,C,F) )
=> E = F ) ) ) ) ) ) ) ).
fof(dt_k1_groeb_1,axiom,
! [A,B,C] :
( ( v3_ordinal1(A)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( ~ v1_xboole_0(k1_groeb_1(A,B,C))
& v1_finset_1(k1_groeb_1(A,B,C))
& m1_subset_1(k1_groeb_1(A,B,C),k1_zfmisc_1(u1_struct_0(k30_polynom1(A,B)))) ) ) ).
fof(redefinition_k1_groeb_1,axiom,
! [A,B,C] :
( ( v3_ordinal1(A)
& v2_group_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> k1_groeb_1(A,B,C) = k1_tarski(C) ) ).
fof(dt_k2_groeb_1,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(A)
& v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m1_relset_1(B,k14_polynom1(A),k14_polynom1(A))
& ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C)))) )
=> m1_subset_1(k2_groeb_1(A,B,C,D),k1_zfmisc_1(k14_polynom1(A))) ) ).
fof(dt_k3_groeb_1,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A))) )
=> m1_subset_1(k3_groeb_1(A,B),k1_zfmisc_1(k14_polynom1(A))) ) ).
fof(dt_k4_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v1_partfun1(k4_groeb_1(A),k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(k4_groeb_1(A))
& v4_relat_2(k4_groeb_1(A))
& v8_relat_2(k4_groeb_1(A))
& m2_relset_1(k4_groeb_1(A),k14_polynom1(A),k14_polynom1(A)) ) ) ).
fof(d1_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_partfun1(B,k14_polynom1(A),k14_polynom1(A))
& v1_relat_2(B)
& v4_relat_2(B)
& v6_relat_2(B)
& v8_relat_2(B)
& m2_relset_1(B,k14_polynom1(A),k14_polynom1(A)) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_group_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v7_vectsp_1(C)
& ~ v3_realset2(C)
& l3_vectsp_1(C) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k30_polynom1(A,C))))
=> k2_groeb_1(A,B,C,D) = a_4_0_groeb_1(A,B,C,D) ) ) ) ) ).
fof(d2_groeb_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k14_polynom1(A)))
=> k3_groeb_1(A,B) = a_2_0_groeb_1(A,B) ) ) ).
fof(fraenkel_a_4_0_groeb_1,axiom,
! [A,B,C,D,E] :
( ( v3_ordinal1(B)
& v1_partfun1(C,k14_polynom1(B),k14_polynom1(B))
& v1_relat_2(C)
& v4_relat_2(C)
& v6_relat_2(C)
& v8_relat_2(C)
& m2_relset_1(C,k14_polynom1(B),k14_polynom1(B))
& ~ v3_struct_0(D)
& v2_group_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v7_vectsp_1(D)
& ~ v3_realset2(D)
& l3_vectsp_1(D)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k30_polynom1(B,D)))) )
=> ( r2_hidden(A,a_4_0_groeb_1(B,C,D,E))
<=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k14_polynom1(B),u1_struct_0(D))
& v2_polynom1(F,k14_polynom1(B),D)
& m2_relset_1(F,k14_polynom1(B),u1_struct_0(D))
& A = k3_termord(B,C,D,F)
& r2_hidden(F,E)
& F != k26_polynom1(B,D) ) ) ) ).
fof(fraenkel_a_2_0_groeb_1,axiom,
! [A,B,C] :
( ( v3_ordinal1(B)
& m1_subset_1(C,k1_zfmisc_1(k14_polynom1(B))) )
=> ( r2_hidden(A,a_2_0_groeb_1(B,C))
<=> ? [D] :
( m1_polynom1(D,B,k14_polynom1(B))
& A = D
& ? [E] :
( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,B)
& r2_hidden(E,C)
& r3_polynom1(B,E,D) ) ) ) ) ).
%------------------------------------------------------------------------------