SET007 Axioms: SET007+667.ax
%------------------------------------------------------------------------------
% File : SET007+667 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Fundamental Theorem of Algebra
% 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 : polynom5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 104 ( 4 unt; 0 def)
% Number of atoms : 1078 ( 102 equ)
% Maximal formula atoms : 28 ( 10 avg)
% Number of connectives : 1101 ( 127 ~; 1 |; 706 &)
% ( 13 <=>; 254 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 10 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 49 ( 47 usr; 1 prp; 0-3 aty)
% Number of functors : 70 ( 70 usr; 8 con; 0-6 aty)
% Number of variables : 257 ( 248 !; 9 ?)
% 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_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> ( ~ v1_xboole_0(k2_polynom5(A,B,C))
& v1_relat_1(k2_polynom5(A,B,C))
& v1_funct_1(k2_polynom5(A,B,C))
& v1_funct_2(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& v1_partfun1(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k2_polynom5(A,B,C),A) ) ) ).
fof(fc2_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_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)) )
=> ( ~ v1_xboole_0(k3_polynom5(A,B,C))
& v1_relat_1(k3_polynom5(A,B,C))
& v1_funct_1(k3_polynom5(A,B,C))
& v1_funct_2(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& v1_partfun1(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k3_polynom5(A,B,C),A) ) ) ).
fof(fc3_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( ~ v1_xboole_0(k4_polynom5(A,B,C))
& v1_relat_1(k4_polynom5(A,B,C))
& v1_funct_1(k4_polynom5(A,B,C))
& v1_funct_2(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& v1_partfun1(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k4_polynom5(A,B,C),A) ) ) ).
fof(fc4_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ( ~ v1_xboole_0(k12_polynom3(A))
& v1_relat_1(k12_polynom3(A))
& v1_funct_1(k12_polynom3(A))
& v1_funct_2(k12_polynom3(A),k5_numbers,u1_struct_0(A))
& v1_partfun1(k12_polynom3(A),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k12_polynom3(A),A)
& v1_polynom5(k12_polynom3(A),A) ) ) ).
fof(rc1_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ? [B] :
( m1_relset_1(B,k5_numbers,u1_struct_0(A))
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_partfun1(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& v1_polynom5(B,A) ) ) ).
fof(fc5_polynom5,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_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)) )
=> ( ~ v1_xboole_0(k7_polynom5(A,B))
& v1_relat_1(k7_polynom5(A,B))
& v1_funct_1(k7_polynom5(A,B))
& v1_funct_2(k7_polynom5(A,B),k5_numbers,u1_struct_0(A))
& v1_partfun1(k7_polynom5(A,B),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k7_polynom5(A,B),A) ) ) ).
fof(fc6_polynom5,axiom,
( ~ v3_struct_0(k1_complfld)
& v2_group_1(k1_complfld)
& v4_group_1(k1_complfld)
& v7_group_1(k1_complfld)
& v3_rlvect_1(k1_complfld)
& v4_rlvect_1(k1_complfld)
& v5_rlvect_1(k1_complfld)
& v6_rlvect_1(k1_complfld)
& v3_vectsp_1(k1_complfld)
& v4_vectsp_1(k1_complfld)
& v5_vectsp_1(k1_complfld)
& v6_vectsp_1(k1_complfld)
& v7_vectsp_1(k1_complfld)
& v8_vectsp_1(k1_complfld)
& v9_vectsp_1(k1_complfld)
& ~ v10_vectsp_1(k1_complfld)
& v2_vectsp_2(k1_complfld)
& v1_algstr_1(k1_complfld)
& v2_algstr_1(k1_complfld)
& v3_algstr_1(k1_complfld)
& v4_algstr_1(k1_complfld)
& v5_algstr_1(k1_complfld)
& v6_algstr_1(k1_complfld)
& ~ v3_realset2(k1_complfld)
& v2_polynom5(k1_complfld) ) ).
fof(rc2_polynom5,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A)
& v6_algstr_1(A)
& ~ v3_realset2(A)
& v2_polynom5(A) ) ).
fof(t1_polynom5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& B != np__0
& ~ r1_xreal_0(np__0,k3_real_1(k5_real_1(k5_real_1(k2_nat_1(A,B),A),B),np__1)) ) ) ) ).
fof(t2_polynom5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,np__0)
=> r1_xreal_0(k7_xcmplx_0(k1_square_1(A,B),k2_square_1(A,B)),np__1) ) ) ) ).
fof(t3_polynom5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(np__1,C)
& ~ r1_xreal_0(B,k3_xcmplx_0(C,A)) ) )
=> r1_xreal_0(B,np__0) ) ) ) ).
fof(t4_polynom5,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k5_finsop_1(A))
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k5_finsop_1(A))
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k15_rvsum_1(A)) ) ) ) ) ).
fof(t5_polynom5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> k5_rlvect_1(k1_complfld,k1_hahnban1(A,B)) = k1_hahnban1(k1_real_1(A),k1_real_1(B)) ) ) ).
fof(t6_polynom5,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k6_rlvect_1(k1_complfld,k1_hahnban1(A,B),k1_hahnban1(C,D)) = k1_hahnban1(k5_real_1(A,C),k5_real_1(B,D)) ) ) ) ) ).
fof(t7_polynom5,axiom,
$true ).
fof(t8_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ( A != k1_rlvect_1(k1_complfld)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k3_complfld(k2_binop_1(u1_struct_0(k1_complfld),k5_numbers,u1_struct_0(k1_complfld),k5_group_1(k1_complfld),A,B)) = k4_power(k3_complfld(A),B) ) ) ) ).
fof(d1_polynom5,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( B = k1_polynom5(A)
<=> ( k3_finseq_1(B) = k3_finseq_1(A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k5_finsop_1(A))
=> k4_finseq_4(k5_numbers,k1_numbers,B,C) = k3_complfld(k4_finseq_4(k5_numbers,u1_struct_0(k1_complfld),A,C)) ) ) ) ) ) ) ).
fof(t9_polynom5,axiom,
k1_polynom5(k6_finseq_1(u1_struct_0(k1_complfld))) = k6_finseq_1(k1_numbers) ).
fof(t10_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> k1_polynom5(k13_binarith(u1_struct_0(k1_complfld),A)) = k13_binarith(k1_numbers,k3_complfld(A)) ) ).
fof(t11_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> k1_polynom5(k2_polynom3(u1_struct_0(k1_complfld),A,B)) = k2_polynom3(k1_numbers,k3_complfld(A),k3_complfld(B)) ) ) ).
fof(t12_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> k1_polynom5(k3_finseq_4(u1_struct_0(k1_complfld),A,B,C)) = k3_finseq_4(k1_numbers,k3_complfld(A),k3_complfld(B),k3_complfld(C)) ) ) ) ).
fof(t13_polynom5,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(k1_complfld))
=> k1_polynom5(k8_finseq_1(u1_struct_0(k1_complfld),A,B)) = k8_finseq_1(k1_numbers,k1_polynom5(A),k1_polynom5(B)) ) ) ).
fof(t14_polynom5,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ( k1_polynom5(k8_finseq_1(u1_struct_0(k1_complfld),A,k13_binarith(u1_struct_0(k1_complfld),B))) = k8_finseq_1(k1_numbers,k1_polynom5(A),k13_binarith(k1_numbers,k3_complfld(B)))
& k1_polynom5(k8_finseq_1(u1_struct_0(k1_complfld),k13_binarith(u1_struct_0(k1_complfld),B),A)) = k8_finseq_1(k1_numbers,k13_binarith(k1_numbers,k3_complfld(B)),k1_polynom5(A)) ) ) ) ).
fof(t15_polynom5,axiom,
! [A] :
( m2_finseq_1(A,u1_struct_0(k1_complfld))
=> r1_xreal_0(k3_complfld(k9_rlvect_1(k1_complfld,A)),k15_rvsum_1(k1_polynom5(A))) ) ).
fof(d2_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_polynom5(A,B,C) = k1_binop_1(k5_group_1(k16_polynom3(A)),B,C) ) ) ) ).
fof(t16_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [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)) )
=> k2_polynom5(A,B,np__0) = k13_polynom3(A) ) ) ).
fof(t17_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [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)) )
=> k2_polynom5(A,B,np__1) = B ) ) ).
fof(t18_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [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)) )
=> k2_polynom5(A,B,np__2) = k15_polynom3(A,B,B) ) ) ).
fof(t19_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [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)) )
=> k2_polynom5(A,B,np__3) = k15_polynom3(A,k15_polynom3(A,B,B),B) ) ) ).
fof(t20_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_polynom5(A,B,k1_nat_1(C,np__1)) = k15_polynom3(A,k2_polynom5(A,B,C),B) ) ) ) ).
fof(t21_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_polynom5(A,k12_polynom3(A),k1_nat_1(B,np__1)) = k12_polynom3(A) ) ) ).
fof(t22_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_polynom5(A,k13_polynom3(A),B) = k13_polynom3(A) ) ) ).
fof(t23_polynom5,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] :
( ( 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] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_polynom4(A,k2_polynom5(A,B,D),C) = k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),k2_polynom4(A,B,C),D) ) ) ) ) ).
fof(t24_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( k3_algseq_1(A,B) != np__0
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k3_algseq_1(A,k2_polynom5(A,B,C)) = k3_real_1(k5_real_1(k2_nat_1(C,k3_algseq_1(A,B)),C),np__1) ) ) ) ) ).
fof(d3_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_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)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(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)) )
=> ( D = k3_polynom5(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k1_group_1(A,C,k2_normsp_1(A,B,E)) ) ) ) ) ) ) ).
fof(t25_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k3_algseq_1(A,k3_polynom5(A,B,k1_rlvect_1(A))) = np__0 ) ) ).
fof(t26_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_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))
=> ( C != k1_rlvect_1(A)
=> k3_algseq_1(A,k3_polynom5(A,B,C)) = k3_algseq_1(A,B) ) ) ) ) ).
fof(t27_polynom5,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)) )
=> k3_polynom5(A,B,k1_rlvect_1(A)) = k12_polynom3(A) ) ) ).
fof(t28_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v8_vectsp_1(A)
& l1_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)) )
=> k3_polynom5(A,B,k2_group_1(A)) = B ) ) ).
fof(t29_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_polynom5(A,k12_polynom3(A),B) = k12_polynom3(A) ) ) ).
fof(t30_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& v6_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_polynom5(A,k13_polynom3(A),B) = k5_algseq_1(A,B) ) ) ).
fof(t31_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_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))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k3_polynom5(A,B,C),D) = k10_group_1(A,C,k2_polynom4(A,B,D)) ) ) ) ) ).
fof(t32_polynom5,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)
& 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)) )
=> k2_polynom4(A,B,k1_rlvect_1(A)) = k2_normsp_1(A,B,np__0) ) ) ).
fof(d4_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k4_polynom5(A,B,C) = k1_polynom1(k5_numbers,u1_struct_0(A),k1_polynom1(k5_numbers,u1_struct_0(A),k12_polynom3(A),np__0,B),np__1,C) ) ) ) ).
fof(t33_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k2_normsp_1(A,k5_algseq_1(A,B),np__0) = B
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,C)
=> k2_normsp_1(A,k5_algseq_1(A,B),C) = k1_rlvect_1(A) ) ) ) ) ) ).
fof(t34_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> k3_algseq_1(A,k5_algseq_1(A,B)) = np__1 ) ) ) ).
fof(t35_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> k5_algseq_1(A,k1_rlvect_1(A)) = k12_polynom3(A) ) ).
fof(t36_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k14_polynom3(A,k5_algseq_1(A,B),k5_algseq_1(A,C)) = k5_algseq_1(A,k10_group_1(A,B,C)) ) ) ) ).
fof(t37_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v9_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k2_polynom5(A,k5_algseq_1(A,B),C) = k5_algseq_1(A,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),B,C)) ) ) ) ).
fof(t38_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_polynom4(A,k5_algseq_1(A,B),C) = B ) ) ) ).
fof(t39_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_normsp_1(A,k4_polynom5(A,B,C),np__0) = B
& k2_normsp_1(A,k4_polynom5(A,B,C),np__1) = C
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__2,D)
=> k2_normsp_1(A,k4_polynom5(A,B,C),D) = k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(t40_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_xreal_0(k3_algseq_1(A,k4_polynom5(A,B,C)),np__2) ) ) ) ).
fof(t41_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C != k1_rlvect_1(A)
=> k3_algseq_1(A,k4_polynom5(A,B,C)) = np__2 ) ) ) ) ).
fof(t42_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> k3_algseq_1(A,k4_polynom5(A,B,k1_rlvect_1(A))) = np__1 ) ) ) ).
fof(t43_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> k4_polynom5(A,k1_rlvect_1(A),k1_rlvect_1(A)) = k12_polynom3(A) ) ).
fof(t44_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_polynom5(A,B,k1_rlvect_1(A)) = k5_algseq_1(A,B) ) ) ).
fof(t45_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k4_polynom5(A,B,C),D) = k2_rlvect_1(A,B,k1_group_1(A,C,D)) ) ) ) ) ).
fof(t46_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k4_polynom5(A,B,k1_rlvect_1(A)),D) = B ) ) ) ) ).
fof(t47_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k4_polynom5(A,k1_rlvect_1(A),C),D) = k1_group_1(A,C,D) ) ) ) ) ).
fof(t48_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k4_polynom5(A,B,k2_group_1(A)),D) = k2_rlvect_1(A,B,D) ) ) ) ) ).
fof(t49_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k2_polynom4(A,k4_polynom5(A,k1_rlvect_1(A),k2_group_1(A)),D) = D ) ) ) ) ).
fof(d5_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& v1_algseq_1(D,A)
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( D = k5_polynom5(A,B,C)
<=> ? [E] :
( m2_finseq_1(E,u1_struct_0(k16_polynom3(A)))
& D = k9_rlvect_1(k16_polynom3(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) = k3_polynom5(A,k2_polynom5(A,C,k5_binarith(F,np__1)),k2_normsp_1(A,B,k5_binarith(F,np__1))) ) ) ) ) ) ) ) ) ).
fof(t50_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k5_polynom5(A,k12_polynom3(A),B) = k12_polynom3(A) ) ) ).
fof(t51_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k5_polynom5(A,B,k12_polynom3(A)) = k5_algseq_1(A,k2_normsp_1(A,B,np__0)) ) ) ).
fof(t52_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v9_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))
=> r1_xreal_0(k3_algseq_1(A,k5_polynom5(A,B,k5_algseq_1(A,C))),np__1) ) ) ) ).
fof(t53_polynom5,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] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ~ ( k3_algseq_1(A,B) != np__0
& ~ r1_xreal_0(k3_algseq_1(A,C),np__1)
& k3_algseq_1(A,k5_polynom5(A,B,C)) != k3_real_1(k5_real_1(k5_real_1(k2_nat_1(k3_algseq_1(A,B),k3_algseq_1(A,C)),k3_algseq_1(A,B)),k3_algseq_1(A,C)),np__2) ) ) ) ) ).
fof(t54_polynom5,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] :
( ( 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,k5_polynom5(A,B,C),D) = k2_polynom4(A,B,k2_polynom4(A,C,D)) ) ) ) ) ).
fof(d6_polynom5,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))
=> ( r1_polynom5(A,B,C)
<=> k2_polynom4(A,B,C) = k1_rlvect_1(A) ) ) ) ) ).
fof(d7_polynom5,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)) )
=> ( v1_polynom5(B,A)
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r1_polynom5(A,B,C) ) ) ) ) ).
fof(t55_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> v1_polynom5(k12_polynom3(A),A) ) ).
fof(t56_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r1_polynom5(A,k12_polynom3(A),B) ) ) ).
fof(d8_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l3_vectsp_1(A) )
=> ( v2_polynom5(A)
<=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( ~ r1_xreal_0(k3_algseq_1(A,B),np__1)
=> v1_polynom5(B,A) ) ) ) ) ).
fof(d9_polynom5,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,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = k6_polynom5(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( r2_hidden(D,C)
<=> r1_polynom5(A,B,D) ) ) ) ) ) ) ).
fof(d10_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_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))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( C = k7_polynom5(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,C,D) = k5_vectsp_1(A,k2_normsp_1(A,B,D),k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1))) ) ) ) ) ) ).
fof(t57_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> ( k3_algseq_1(A,B) != np__0
=> k2_normsp_1(A,k7_polynom5(A,B),k5_binarith(k3_algseq_1(A,B),np__1)) = k2_group_1(A) ) ) ) ).
fof(t58_polynom5,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] :
( ( 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
=> k3_algseq_1(A,k7_polynom5(A,B)) = k3_algseq_1(A,B) ) ) ) ).
fof(t59_polynom5,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] :
( ( 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] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_polynom4(A,k7_polynom5(A,B),C) = k5_vectsp_1(A,k2_polynom4(A,B,C),k2_normsp_1(A,B,k5_binarith(k3_algseq_1(A,B),np__1))) ) ) ) ) ).
fof(t60_polynom5,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] :
( ( 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] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_polynom5(A,B,C)
<=> r1_polynom5(A,k7_polynom5(A,B),C) ) ) ) ) ) ).
fof(t61_polynom5,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] :
( ( 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
=> ( v1_polynom5(B,A)
<=> v1_polynom5(k7_polynom5(A,B),A) ) ) ) ) ).
fof(t62_polynom5,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] :
( ( 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
=> k6_polynom5(A,B) = k6_polynom5(A,k7_polynom5(A,B)) ) ) ) ).
fof(t63_polynom5,axiom,
r2_cfcont_1(k6_partfun1(k2_numbers),k2_numbers) ).
fof(t64_polynom5,axiom,
! [A] :
( m1_subset_1(A,k2_numbers)
=> r2_cfcont_1(k3_finsop_1(k2_numbers,k2_numbers,A),k2_numbers) ) ).
fof(d11_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
=> ( D = k8_polynom5(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),D,E) = k1_group_1(A,B,k2_binop_1(u1_struct_0(A),k5_numbers,u1_struct_0(A),k5_group_1(A),E,C)) ) ) ) ) ) ) ).
fof(t65_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> k8_polynom5(A,k2_group_1(A),np__1) = k6_partfun1(u1_struct_0(A)) ) ).
fof(t66_polynom5,axiom,
k8_polynom5(k1_complfld,k2_group_1(k1_complfld),np__2) = k3_comseq_1(k2_numbers,k6_partfun1(k2_numbers),k6_partfun1(k2_numbers)) ).
fof(t67_polynom5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k8_polynom5(A,B,np__0) = k3_finsop_1(u1_struct_0(A),u1_struct_0(A),B) ) ) ).
fof(t68_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ? [B] :
( m1_subset_1(B,k2_numbers)
& A = B
& k8_polynom5(k1_complfld,A,np__1) = k4_comseq_1(k2_numbers,k6_partfun1(k2_numbers),B) ) ) ).
fof(t69_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ? [B] :
( m1_subset_1(B,k2_numbers)
& A = B
& k8_polynom5(k1_complfld,A,np__2) = k4_comseq_1(k2_numbers,k3_comseq_1(k2_numbers,k6_partfun1(k2_numbers),k6_partfun1(k2_numbers)),B) ) ) ).
fof(t70_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k2_numbers,k2_numbers)
& m2_relset_1(C,k2_numbers,k2_numbers)
& C = k8_polynom5(k1_complfld,A,B)
& k8_polynom5(k1_complfld,A,k1_nat_1(B,np__1)) = k3_comseq_1(k2_numbers,C,k6_partfun1(k2_numbers)) ) ) ) ).
fof(t71_polynom5,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_complfld))
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k2_numbers,k2_numbers)
& m2_relset_1(C,k2_numbers,k2_numbers)
& C = k8_polynom5(k1_complfld,A,B)
& r2_cfcont_1(C,k2_numbers) ) ) ) ).
fof(d12_polynom5,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
=> ( C = k9_polynom5(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),C,D) = k2_polynom4(A,B,D) ) ) ) ) ) ).
fof(t72_polynom5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
& v1_algseq_1(A,k1_complfld)
& m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k2_numbers,k2_numbers)
& m2_relset_1(B,k2_numbers,k2_numbers)
& B = k9_polynom5(k1_complfld,A)
& r2_cfcont_1(B,k2_numbers) ) ) ).
fof(t73_polynom5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
& v1_algseq_1(A,k1_complfld)
& m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
=> ( k3_complfld(k2_normsp_1(k1_complfld,A,k5_binarith(k3_algseq_1(k1_complfld,A),np__1))) = np__1
=> ( r1_xreal_0(k3_algseq_1(k1_complfld,A),np__2)
| ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( k3_finseq_1(B) = k3_algseq_1(k1_complfld,A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k5_finsop_1(B))
=> k2_seq_1(k5_numbers,k1_numbers,B,C) = k3_complfld(k2_normsp_1(k1_complfld,A,k5_binarith(C,np__1))) ) ) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> ~ ( ~ r1_xreal_0(k3_complfld(C),k15_rvsum_1(B))
& r1_xreal_0(k3_complfld(k2_polynom4(k1_complfld,A,C)),k3_real_1(k3_complfld(k2_normsp_1(k1_complfld,A,np__0)),np__1)) ) ) ) ) ) ) ) ).
fof(t74_polynom5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
& v1_algseq_1(A,k1_complfld)
& m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
=> ~ ( ~ r1_xreal_0(k3_algseq_1(k1_complfld,A),np__2)
& ! [B] :
( m1_subset_1(B,u1_struct_0(k1_complfld))
=> ~ ! [C] :
( m1_subset_1(C,u1_struct_0(k1_complfld))
=> r1_xreal_0(k3_complfld(k2_polynom4(k1_complfld,A,B)),k3_complfld(k2_polynom4(k1_complfld,A,C))) ) ) ) ) ).
fof(t75_polynom5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(k1_complfld))
& v1_algseq_1(A,k1_complfld)
& m2_relset_1(A,k5_numbers,u1_struct_0(k1_complfld)) )
=> ( ~ r1_xreal_0(k3_algseq_1(k1_complfld,A),np__1)
=> v1_polynom5(A,k1_complfld) ) ) ).
fof(dt_k1_polynom5,axiom,
! [A] :
( m1_finseq_1(A,u1_struct_0(k1_complfld))
=> m2_finseq_1(k1_polynom5(A),k1_numbers) ) ).
fof(dt_k2_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& v1_algseq_1(B,A)
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> ( v1_funct_1(k2_polynom5(A,B,C))
& v1_funct_2(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k2_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k3_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,u1_struct_0(A))
& m1_relset_1(B,k5_numbers,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( v1_funct_1(k3_polynom5(A,B,C))
& v1_funct_2(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k3_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k4_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( v1_funct_1(k4_polynom5(A,B,C))
& v1_funct_2(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k4_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k5_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A)
& 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(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& v1_algseq_1(C,A)
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k5_polynom5(A,B,C))
& v1_funct_2(k5_polynom5(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k5_polynom5(A,B,C),A)
& m2_relset_1(k5_polynom5(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k6_polynom5,axiom,
! [A,B] :
( ( ~ 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(k6_polynom5(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ).
fof(dt_k7_polynom5,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_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)) )
=> ( v1_funct_1(k7_polynom5(A,B))
& v1_funct_2(k7_polynom5(A,B),k5_numbers,u1_struct_0(A))
& m2_relset_1(k7_polynom5(A,B),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k8_polynom5,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_group_1(A)
& l1_group_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,k5_numbers) )
=> ( v1_funct_1(k8_polynom5(A,B,C))
& v1_funct_2(k8_polynom5(A,B,C),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k8_polynom5(A,B,C),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_k9_polynom5,axiom,
! [A,B] :
( ( ~ 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)) )
=> ( v1_funct_1(k9_polynom5(A,B))
& v1_funct_2(k9_polynom5(A,B),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k9_polynom5(A,B),u1_struct_0(A),u1_struct_0(A)) ) ) ).
%------------------------------------------------------------------------------