SET007 Axioms: SET007+657.ax
%------------------------------------------------------------------------------
% File : SET007+657 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Evaluation of 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 : polynom4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 40 ( 3 unt; 0 def)
% Number of atoms : 473 ( 38 equ)
% Maximal formula atoms : 25 ( 11 avg)
% Number of connectives : 489 ( 56 ~; 0 |; 334 &)
% ( 3 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 46 ( 44 usr; 1 prp; 0-3 aty)
% Number of functors : 41 ( 41 usr; 4 con; 0-6 aty)
% Number of variables : 96 ( 94 !; 2 ?)
% 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(cc1_polynom4,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& v1_algstr_1(A)
& v3_algstr_1(A) ) ) ) ).
fof(rc1_polynom4,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)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& ~ v3_realset2(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) ) ).
fof(fc1_polynom4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_relat_1(k1_polynom4(A,B))
& v1_funct_1(k1_polynom4(A,B))
& v1_funct_2(k1_polynom4(A,B),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k1_polynom4(A,B),A) ) ) ).
fof(fc2_polynom4,axiom,
! [A,B] :
( ( ~ 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)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k3_polynom4(A,B))
& v1_funct_1(k3_polynom4(A,B))
& v1_funct_2(k3_polynom4(A,B),u1_struct_0(k16_polynom3(A)),u1_struct_0(A))
& v1_endalg(k3_polynom4(A,B),k16_polynom3(A),A) ) ) ).
fof(fc3_polynom4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k3_polynom4(A,B))
& v1_funct_1(k3_polynom4(A,B))
& v1_funct_2(k3_polynom4(A,B),u1_struct_0(k16_polynom3(A)),u1_struct_0(A))
& v1_grcat_1(k3_polynom4(A,B),k16_polynom3(A),A) ) ) ).
fof(fc4_polynom4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k3_polynom4(A,B))
& v1_funct_1(k3_polynom4(A,B))
& v1_funct_2(k3_polynom4(A,B),u1_struct_0(k16_polynom3(A)),u1_struct_0(A))
& v1_grcat_1(k3_polynom4(A,B),k16_polynom3(A),A)
& v1_group_6(k3_polynom4(A,B),k16_polynom3(A),A) ) ) ).
fof(fc5_polynom4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_relat_1(k3_polynom4(A,B))
& v1_funct_1(k3_polynom4(A,B))
& v1_funct_2(k3_polynom4(A,B),u1_struct_0(k16_polynom3(A)),u1_struct_0(A))
& v1_quofield(k3_polynom4(A,B),k16_polynom3(A),A)
& v1_grcat_1(k3_polynom4(A,B),k16_polynom3(A),A)
& v1_endalg(k3_polynom4(A,B),k16_polynom3(A),A)
& v1_group_6(k3_polynom4(A,B),k16_polynom3(A),A) ) ) ).
fof(t1_polynom4,axiom,
$true ).
fof(t2_polynom4,axiom,
$true ).
fof(t3_polynom4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B)) )
=> B = k7_finseq_1(k7_finseq_1(k16_finseq_1(A,B,k5_binarith(C,np__1)),k9_finseq_1(k1_funct_1(B,C))),k1_rfinseq(A,B,C)) ) ) ) ) ).
fof(t4_polynom4,axiom,
$true ).
fof(t5_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k14_polynom3(A,k12_polynom3(A),B) = k12_polynom3(A) ) ) ).
fof(t6_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> k3_algseq_1(A,k12_polynom3(A)) = np__0 ) ).
fof(t7_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> k3_algseq_1(A,k13_polynom3(A)) = np__1 ) ).
fof(t8_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( k3_algseq_1(A,B) = np__0
=> B = k12_polynom3(A) ) ) ) ).
fof(t9_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(k3_algseq_1(A,B),D)
& r1_xreal_0(k3_algseq_1(A,C),D) )
=> r1_xreal_0(k3_algseq_1(A,k8_polynom3(A,B,C)),D) ) ) ) ) ) ).
fof(t10_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( k3_algseq_1(A,B) != k3_algseq_1(A,C)
=> k3_algseq_1(A,k8_polynom3(A,B,C)) = k4_square_1(k3_algseq_1(A,B),k3_algseq_1(A,C)) ) ) ) ) ).
fof(t11_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k3_algseq_1(A,k10_polynom3(A,B)) = k3_algseq_1(A,B) ) ) ).
fof(t12_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(k3_algseq_1(A,B),D)
& r1_xreal_0(k3_algseq_1(A,C),D) )
=> r1_xreal_0(k3_algseq_1(A,k11_polynom3(A,B,C)),D) ) ) ) ) ) ).
fof(t13_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( k10_group_1(A,k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1)),k2_normsp_1(A,C,k5_binarith(k3_algseq_1(A,C),np__1))) != k1_rlvect_1(A)
=> k3_algseq_1(A,k14_polynom3(A,B,C)) = k6_xcmplx_0(k1_nat_1(k3_algseq_1(A,B),k3_algseq_1(A,C)),np__1) ) ) ) ) ).
fof(d1_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( C = k1_polynom4(A,B)
<=> ( k2_normsp_1(A,C,k5_binarith(k3_algseq_1(A,B),np__1)) = k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( D != k5_binarith(k3_algseq_1(A,B),np__1)
=> k2_normsp_1(A,C,D) = k1_rlvect_1(A) ) ) ) ) ) ) ) ).
fof(t14_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k1_polynom4(A,B) = k1_polynom1(k5_numbers,u1_struct_0(A),k12_polynom3(A),k5_binarith(k3_algseq_1(A,B),np__1),k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1))) ) ) ).
fof(t15_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( k3_algseq_1(A,B) = np__0
=> k1_polynom4(A,B) = k12_polynom3(A) ) ) ) ).
fof(t16_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> k1_polynom4(A,k12_polynom3(A)) = k12_polynom3(A) ) ).
fof(t17_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> k1_polynom4(A,k13_polynom3(A)) = k13_polynom3(A) ) ).
fof(t18_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k3_algseq_1(A,k1_polynom4(A,B)) = k3_algseq_1(A,B) ) ) ).
fof(t19_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ~ ( k3_algseq_1(A,B) != np__0
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ~ ( ~ r1_xreal_0(k3_algseq_1(A,B),k3_algseq_1(A,C))
& B = k8_polynom3(A,C,k1_polynom4(A,B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k6_xcmplx_0(k3_algseq_1(A,B),np__1),D)
=> k2_normsp_1(A,C,D) = k2_normsp_1(A,B,D) ) ) ) ) ) ) ) ).
fof(d2_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( D = k2_polynom4(A,B,C)
<=> ? [E] :
( m2_finseq_1(E,u1_struct_0(A))
& D = k9_rlvect_1(A,E)
& k3_finseq_1(E) = k3_algseq_1(A,B)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k5_finsop_1(E))
=> k1_funct_1(E,F) = k1_group_1(A,k2_normsp_1(A,B,k5_binarith(F,np__1)),k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),C,k5_binarith(F,np__1))) ) ) ) ) ) ) ) ) ).
fof(t20_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_polynom4(A,k12_polynom3(A),B) = k1_rlvect_1(A) ) ) ).
fof(t21_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_polynom4(A,k13_polynom3(A),B) = k2_group_1(A) ) ) ).
fof(t22_polynom4,axiom,
! [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)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k9_polynom3(A,B,C),D) = k4_rlvect_1(A,k2_polynom4(A,B,D),k2_polynom4(A,C,D)) ) ) ) ) ).
fof(t23_polynom4,axiom,
! [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)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_polynom4(A,k10_polynom3(A,B),C) = k5_rlvect_1(A,k2_polynom4(A,B,C)) ) ) ) ).
fof(t24_polynom4,axiom,
! [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)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k11_polynom3(A,B,C),D) = k6_rlvect_1(A,k2_polynom4(A,B,D),k2_polynom4(A,C,D)) ) ) ) ) ).
fof(t25_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_polynom4(A,k1_polynom4(A,B),C) = k1_group_1(A,k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1)),k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),C,k5_binarith(k3_algseq_1(A,B),np__1))) ) ) ) ).
fof(t26_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k15_polynom3(A,k1_polynom4(A,B),C),D) = k10_group_1(A,k2_polynom4(A,k1_polynom4(A,B),D),k2_polynom4(A,C,D)) ) ) ) ) ).
fof(t27_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v3_realset2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k15_polynom3(A,B,C),D) = k10_group_1(A,k2_polynom4(A,B,D),k2_polynom4(A,C,D)) ) ) ) ) ).
fof(d3_polynom4,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k16_polynom3(A)),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(k16_polynom3(A)),u1_struct_0(A)) )
=> ( C = k3_polynom4(A,B)
<=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& v1_algseq_1(D,A)
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> k1_funct_1(C,D) = k2_polynom4(A,D,B) ) ) ) ) ) ).
fof(dt_k1_polynom4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k1_polynom4(A,B))
& v1_funct_2(k1_polynom4(A,B),k5_numbers,u1_struct_0(A))
& m2_relset_1(k1_polynom4(A,B),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k2_polynom4,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k2_polynom4(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k3_polynom4,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v1_funct_1(k3_polynom4(A,B))
& v1_funct_2(k3_polynom4(A,B),u1_struct_0(k16_polynom3(A)),u1_struct_0(A))
& m2_relset_1(k3_polynom4(A,B),u1_struct_0(k16_polynom3(A)),u1_struct_0(A)) ) ) ).
%------------------------------------------------------------------------------