SET007 Axioms: SET007+710.ax
%------------------------------------------------------------------------------
% File : SET007+710 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Multivariate Polynomials: Monomials and Constant Polynomials
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : polynom7 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 68 ( 1 unt; 0 def)
% Number of atoms : 691 ( 63 equ)
% Maximal formula atoms : 23 ( 10 avg)
% Number of connectives : 705 ( 82 ~; 4 |; 444 &)
% ( 12 <=>; 163 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 47 ( 45 usr; 1 prp; 0-3 aty)
% Number of functors : 26 ( 26 usr; 1 con; 0-5 aty)
% Number of variables : 227 ( 216 !; 11 ?)
% 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_polynom7,axiom,
? [A] :
( l2_struct_0(A)
& ~ v3_struct_0(A)
& ~ v3_realset2(A) ) ).
fof(cc1_polynom7,axiom,
! [A] :
( l2_struct_0(A)
=> ( ~ v3_realset2(A)
=> ~ v3_struct_0(A) ) ) ).
fof(rc2_polynom7,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(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)
& v2_vectsp_2(A)
& v1_binom(A)
& ~ v3_realset2(A) ) ).
fof(rc3_polynom7,axiom,
! [A] :
( ( ~ v3_realset2(A)
& l2_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& v8_rlvect_1(B,A) ) ) ).
fof(rc4_polynom7,axiom,
! [A,B] :
( ( ~ v3_realset2(B)
& l2_struct_0(B) )
=> ? [C] :
( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v1_polynom7(C,A,B) ) ) ).
fof(rc5_polynom7,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& ~ v3_realset2(B)
& l2_struct_0(B) )
=> ? [C] :
( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& v1_polynom7(C,A,B) ) ) ).
fof(rc6_polynom7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B)
& v7_seqm_3(B)
& v1_polynom1(B)
& v2_polynom7(B,A) ) ) ).
fof(cc2_polynom7,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( ( v7_seqm_3(B)
& v1_polynom1(B)
& v2_polynom7(B,A) )
=> ( v7_seqm_3(B)
& v1_polynom1(B)
& ~ v1_polynom2(B,A) ) ) ) ) ).
fof(rc7_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ? [C] :
( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B) ) ) ).
fof(cc3_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B) ) ) ) ) ).
fof(rc8_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ? [C] :
( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v4_polynom7(C,A,B) ) ) ).
fof(cc4_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v4_polynom7(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(C,k14_polynom1(A),B)
& v3_polynom7(C,A,B) ) ) ) ) ).
fof(fc1_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ( v1_relat_1(k26_polynom1(A,B))
& v1_funct_1(k26_polynom1(A,B))
& v1_funct_2(k26_polynom1(A,B),k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(k26_polynom1(A,B),k14_polynom1(A),B)
& v3_polynom7(k26_polynom1(A,B),A,B)
& v4_polynom7(k26_polynom1(A,B),A,B) ) ) ).
fof(fc2_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& l3_vectsp_1(B) )
=> ( v1_relat_1(k27_polynom1(A,B))
& v1_funct_1(k27_polynom1(A,B))
& v1_funct_2(k27_polynom1(A,B),k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(k27_polynom1(A,B),k14_polynom1(A),B)
& v3_polynom7(k27_polynom1(A,B),A,B)
& v4_polynom7(k27_polynom1(A,B),A,B) ) ) ).
fof(fc3_polynom7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( v1_relat_1(k4_polynom7(A,B,C))
& v1_funct_1(k4_polynom7(A,B,C))
& v1_funct_2(k4_polynom7(A,B,C),k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(k4_polynom7(A,B,C),k14_polynom1(A),B)
& v3_polynom7(k4_polynom7(A,B,C),A,B)
& v4_polynom7(k4_polynom7(A,B,C),A,B) ) ) ).
fof(fc4_polynom7,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v5_rlvect_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& v1_binom(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))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_relat_1(k5_polynom7(A,B,C,D))
& v1_funct_1(k5_polynom7(A,B,C,D))
& v1_funct_2(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(k5_polynom7(A,B,C,D),k14_polynom1(A),B) ) ) ).
fof(fc5_polynom7,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& v5_rlvect_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& v1_binom(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))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_relat_1(k6_polynom7(A,B,C,D))
& v1_funct_1(k6_polynom7(A,B,C,D))
& v1_funct_2(k6_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
& v2_polynom1(k6_polynom7(A,B,C,D),k14_polynom1(A),B) ) ) ).
fof(d1_polynom7,axiom,
$true ).
fof(d2_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( v1_polynom7(C,A,B)
<=> C != k26_polynom1(A,B) ) ) ) ).
fof(t1_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( C = k26_polynom1(A,B)
<=> k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0 ) ) ) ).
fof(t2_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ( ~ ( ~ v3_realset2(B)
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0 ) )
& ~ ( ~ ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0 )
& v3_realset2(B) ) ) ) ).
fof(d3_polynom7,axiom,
! [A,B] :
( ( v7_seqm_3(B)
& v1_polynom1(B)
& m1_pboole(B,A) )
=> ( v2_polynom7(B,A)
<=> ? [C] :
( m1_subset_1(C,A)
& k1_polynom2(A,B) = k1_tarski(C) ) ) ) ).
fof(t3_polynom7,axiom,
! [A] :
( ( v7_seqm_3(A)
& v1_polynom1(A)
& m1_pboole(A,k1_xboole_0) )
=> A = k16_polynom1(k1_xboole_0) ) ).
fof(t4_polynom7,axiom,
! [A] :
( ( v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k14_polynom1(k1_xboole_0),u1_struct_0(A))
& v2_polynom1(B,k14_polynom1(k1_xboole_0),A)
& m2_relset_1(B,k14_polynom1(k1_xboole_0),u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_xboole_0,u1_struct_0(A))
& m2_relset_1(C,k1_xboole_0,u1_struct_0(A)) )
=> k5_polynom2(k1_xboole_0,A,B,C) = k15_polynom1(k1_xboole_0,A,B,k16_polynom1(k1_xboole_0)) ) ) ) ).
fof(t5_polynom7,axiom,
! [A] :
( ( v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> r2_quofield(k30_polynom1(k1_xboole_0,A),A) ) ).
fof(d4_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( v3_polynom7(C,A,B)
<=> ? [D] :
( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A)
& ! [E] :
( ( v7_seqm_3(E)
& v1_polynom1(E)
& m1_pboole(E,A) )
=> ( E != D
=> k15_polynom1(A,B,C,E) = k1_rlvect_1(B) ) ) ) ) ) ) ).
fof(t6_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
<=> ~ ( k12_polynom1(k14_polynom1(A),B,C) != k1_xboole_0
& ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> k12_polynom1(k14_polynom1(A),B,C) != k1_tarski(D) ) ) ) ) ) ).
fof(d5_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> k1_polynom7(A,B,C,D) = k1_polynom1(k14_polynom1(A),u1_struct_0(B),k26_polynom1(A,B),D,C) ) ) ) ).
fof(d6_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( D = k2_polynom7(A,B,C)
<=> ~ ( k15_polynom1(A,B,C,D) = k1_rlvect_1(B)
& ~ ( k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0
& D = k16_polynom1(A) ) ) ) ) ) ) ).
fof(d7_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> k3_polynom7(A,B,C) = k15_polynom1(A,B,C,k2_polynom7(A,B,C)) ) ) ).
fof(t7_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0
| k12_polynom1(k14_polynom1(A),B,C) = k1_tarski(k2_polynom7(A,B,C)) ) ) ) ).
fof(t8_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v7_seqm_3(C)
& v1_polynom1(C)
& m1_pboole(C,A) )
=> ( k3_polynom7(A,B,k1_polynom7(A,B,k1_rlvect_1(B),C)) = k1_rlvect_1(B)
& k2_polynom7(A,B,k1_polynom7(A,B,k1_rlvect_1(B),C)) = k16_polynom1(A) ) ) ) ).
fof(t9_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> k3_polynom7(A,B,k1_polynom7(A,B,C,D)) = C ) ) ) ).
fof(t10_polynom7,axiom,
! [A,B] :
( ( ~ v3_realset2(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v8_rlvect_1(C,B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> k2_polynom7(A,B,k1_polynom7(A,B,C,D)) = D ) ) ) ).
fof(t11_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> k1_polynom7(A,B,k3_polynom7(A,B,C),k2_polynom7(A,B,C)) = C ) ) ).
fof(t12_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,u1_struct_0(B))
& m2_relset_1(D,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,C,D) = k1_group_1(B,k3_polynom7(A,B,C),k3_polynom2(A,k2_polynom7(A,B,C),B,D)) ) ) ) ) ).
fof(t13_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& m2_relset_1(E,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k1_polynom7(A,B,C,D),E) = k1_group_1(B,C,k3_polynom2(A,D,B,E)) ) ) ) ) ) ).
fof(d8_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( v4_polynom7(C,A,B)
<=> ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( D != k16_polynom1(A)
=> k15_polynom1(A,B,C,D) = k1_rlvect_1(B) ) ) ) ) ) ).
fof(t14_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v4_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
<=> ( C = k26_polynom1(A,B)
| k12_polynom1(k14_polynom1(A),B,C) = k1_tarski(k16_polynom1(A)) ) ) ) ) ).
fof(t15_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v4_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( k12_polynom1(k14_polynom1(A),B,C) = k1_xboole_0
| k12_polynom1(k14_polynom1(A),B,C) = k1_tarski(k16_polynom1(A)) ) ) ) ).
fof(t16_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v4_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( k2_polynom7(A,B,C) = k16_polynom1(A)
& k3_polynom7(A,B,C) = k15_polynom1(A,B,C,k16_polynom1(A)) ) ) ) ).
fof(d9_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k4_polynom7(A,B,C) = k1_polynom1(k14_polynom1(A),u1_struct_0(B),k26_polynom1(A,B),k16_polynom1(A),C) ) ) ).
fof(t17_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v4_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& C = k4_polynom7(A,B,D) ) ) ) ) ).
fof(t18_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k15_polynom1(A,B,k4_polynom7(A,B,C),k16_polynom1(A)) = C
& ! [D] :
( ( v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( D != k16_polynom1(A)
=> k15_polynom1(A,B,k4_polynom7(A,B,C),D) = k1_rlvect_1(B) ) ) ) ) ) ).
fof(t19_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> k4_polynom7(A,B,k1_rlvect_1(B)) = k26_polynom1(A,B) ) ).
fof(t20_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v2_group_1(B)
& l2_vectsp_1(B) )
=> k4_polynom7(A,B,k2_group_1(B)) = k27_polynom1(A,B) ) ).
fof(t21_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( k4_polynom7(A,B,C) = k4_polynom7(A,B,D)
<=> C = D ) ) ) ) ).
fof(t22_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k12_polynom1(k14_polynom1(A),B,k4_polynom7(A,B,C)) = k1_xboole_0
| k12_polynom1(k14_polynom1(A),B,k4_polynom7(A,B,C)) = k1_tarski(k16_polynom1(A)) ) ) ) ).
fof(t23_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ( k2_polynom7(A,B,k4_polynom7(A,B,C)) = k16_polynom1(A)
& k3_polynom7(A,B,k4_polynom7(A,B,C)) = C ) ) ) ).
fof(t24_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v4_polynom7(C,A,B)
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,u1_struct_0(B))
& m2_relset_1(D,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,C,D) = k3_polynom7(A,B,C) ) ) ) ) ).
fof(t25_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v7_vectsp_1(B)
& ~ v3_realset2(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,u1_struct_0(B))
& m2_relset_1(D,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k4_polynom7(A,B,C),D) = C ) ) ) ) ).
fof(d10_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(B)) )
=> ( E = k5_polynom7(A,B,C,D)
<=> ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> k15_polynom1(A,B,E,F) = k1_group_1(B,D,k15_polynom1(A,B,C,F)) ) ) ) ) ) ) ).
fof(d11_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l2_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(E,k14_polynom1(A),u1_struct_0(B)) )
=> ( E = k6_polynom7(A,B,C,D)
<=> ! [F] :
( ( v7_seqm_3(F)
& v1_polynom1(F)
& m1_pboole(F,A) )
=> k15_polynom1(A,B,E,F) = k1_group_1(B,k15_polynom1(A,B,C,F),D) ) ) ) ) ) ) ).
fof(t26_polynom7,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& v7_group_1(B)
& l2_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k5_polynom7(A,B,C,D) = k6_polynom7(A,B,C,D) ) ) ) ).
fof(t27_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v5_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k5_polynom7(A,B,C,D) = k28_polynom1(A,B,k4_polynom7(A,B,D),C) ) ) ) ) ).
fof(t28_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k6_polynom7(A,B,C,D) = k28_polynom1(A,B,C,k4_polynom7(A,B,D)) ) ) ) ) ).
fof(t29_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& ~ v3_realset2(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)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& m2_relset_1(E,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k5_polynom7(A,B,C,D),E) = k10_group_1(B,D,k5_polynom2(A,B,C,E)) ) ) ) ) ) ).
fof(t30_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& v2_algstr_1(B)
& v2_vectsp_2(B)
& ~ v3_realset2(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)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& m2_relset_1(E,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k5_polynom7(A,B,C,D),E) = k1_group_1(B,D,k5_polynom2(A,B,C,E)) ) ) ) ) ) ).
fof(t31_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& ~ v3_realset2(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)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& m2_relset_1(E,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k6_polynom7(A,B,C,D),E) = k10_group_1(B,k5_polynom2(A,B,C,E),D) ) ) ) ) ) ).
fof(t32_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& v2_algstr_1(B)
& v2_vectsp_2(B)
& ~ v3_realset2(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)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& m2_relset_1(E,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k6_polynom7(A,B,C,D),E) = k10_group_1(B,k5_polynom2(A,B,C,E),D) ) ) ) ) ) ).
fof(t33_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& ~ v3_realset2(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)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& m2_relset_1(E,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k29_polynom1(A,B,k4_polynom7(A,B,D),C),E) = k10_group_1(B,D,k5_polynom2(A,B,C,E)) ) ) ) ) ) ).
fof(t34_polynom7,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v2_group_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v1_algstr_1(B)
& ~ v3_realset2(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)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,u1_struct_0(B))
& m2_relset_1(E,A,u1_struct_0(B)) )
=> k5_polynom2(A,B,k29_polynom1(A,B,C,k4_polynom7(A,B,D)),E) = k10_group_1(B,k5_polynom2(A,B,C,E),D) ) ) ) ) ) ).
fof(dt_k1_polynom7,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B)
& m1_subset_1(C,u1_struct_0(B))
& v7_seqm_3(D)
& v1_polynom1(D)
& m1_pboole(D,A) )
=> ( v1_funct_1(k1_polynom7(A,B,C,D))
& v1_funct_2(k1_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(k1_polynom7(A,B,C,D),A,B)
& m2_relset_1(k1_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) ).
fof(dt_k2_polynom7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> ( v7_seqm_3(k2_polynom7(A,B,C))
& v1_polynom1(k2_polynom7(A,B,C))
& m1_pboole(k2_polynom7(A,B,C),A) ) ) ).
fof(dt_k3_polynom7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& v3_polynom7(C,A,B)
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B)) )
=> m1_subset_1(k3_polynom7(A,B,C),u1_struct_0(B)) ) ).
fof(dt_k4_polynom7,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( v1_funct_1(k4_polynom7(A,B,C))
& v1_funct_2(k4_polynom7(A,B,C),k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(k4_polynom7(A,B,C),k14_polynom1(A),u1_struct_0(B)) ) ) ).
fof(dt_k5_polynom7,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l2_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_funct_1(k5_polynom7(A,B,C,D))
& v1_funct_2(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(k5_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) ).
fof(dt_k6_polynom7,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l2_vectsp_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k14_polynom1(A),u1_struct_0(B))
& m1_relset_1(C,k14_polynom1(A),u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( v1_funct_1(k6_polynom7(A,B,C,D))
& v1_funct_2(k6_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B))
& m2_relset_1(k6_polynom7(A,B,C,D),k14_polynom1(A),u1_struct_0(B)) ) ) ).
%------------------------------------------------------------------------------