SET007 Axioms: SET007+650.ax
%------------------------------------------------------------------------------
% File : SET007+650 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Ring 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 : polynom3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 94 ( 0 unt; 0 def)
% Number of atoms : 909 ( 86 equ)
% Maximal formula atoms : 39 ( 9 avg)
% Number of connectives : 916 ( 101 ~; 6 |; 562 &)
% ( 13 <=>; 234 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 10 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 51 ( 50 usr; 0 prp; 1-3 aty)
% Number of functors : 72 ( 72 usr; 10 con; 0-6 aty)
% Number of variables : 278 ( 273 !; 5 ?)
% 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_polynom3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers) )
=> ! [C] :
( m1_finseq_1(C,k4_finseq_2(B,A))
=> v1_matrlin(C) ) ) ).
fof(fc1_polynom3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_relat_1(k5_polynom3(A))
& v1_relat_2(k5_polynom3(A))
& v4_relat_2(k5_polynom3(A))
& v8_relat_2(k5_polynom3(A))
& v1_partfun1(k5_polynom3(A),k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers))
& v3_orders_1(k5_polynom3(A)) ) ) ).
fof(fc2_polynom3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( ~ v1_xboole_0(k6_polynom3(A,B))
& v1_relat_1(k6_polynom3(A,B))
& v1_funct_1(k6_polynom3(A,B))
& v2_funct_1(k6_polynom3(A,B))
& v1_finset_1(k6_polynom3(A,B))
& v1_finseq_1(k6_polynom3(A,B))
& v1_funcop_1(k6_polynom3(A,B))
& v1_matrlin(k6_polynom3(A,B))
& v1_polynom1(k6_polynom3(A,B)) ) ) ).
fof(fc3_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l1_rlvect_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_xboole_0(k8_polynom3(A,B,C))
& v1_relat_1(k8_polynom3(A,B,C))
& v1_funct_1(k8_polynom3(A,B,C))
& v1_funct_2(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& v1_partfun1(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k8_polynom3(A,B,C),A) ) ) ).
fof(fc4_polynom3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_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(k10_polynom3(A,B))
& v1_relat_1(k10_polynom3(A,B))
& v1_funct_1(k10_polynom3(A,B))
& v1_funct_2(k10_polynom3(A,B),k5_numbers,u1_struct_0(A))
& v1_partfun1(k10_polynom3(A,B),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k10_polynom3(A,B),A) ) ) ).
fof(fc5_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_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_xboole_0(k11_polynom3(A,B,C))
& v1_relat_1(k11_polynom3(A,B,C))
& v1_funct_1(k11_polynom3(A,B,C))
& v1_funct_2(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& v1_partfun1(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k11_polynom3(A,B,C),A) ) ) ).
fof(fc6_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(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) ) ) ).
fof(fc7_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( ~ v1_xboole_0(k13_polynom3(A))
& v1_relat_1(k13_polynom3(A))
& v1_funct_1(k13_polynom3(A))
& v1_funct_2(k13_polynom3(A),k5_numbers,u1_struct_0(A))
& v1_partfun1(k13_polynom3(A),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k13_polynom3(A),A) ) ) ).
fof(fc8_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_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_xboole_0(k14_polynom3(A,B,C))
& v1_relat_1(k14_polynom3(A,B,C))
& v1_funct_1(k14_polynom3(A,B,C))
& v1_funct_2(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& v1_partfun1(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& v1_algseq_1(k14_polynom3(A,B,C),A) ) ) ).
fof(fc9_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v3_rlvect_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A)) ) ) ).
fof(fc10_polynom3,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) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v4_rlvect_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A)) ) ) ).
fof(fc11_polynom3,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) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v5_rlvect_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A)) ) ) ).
fof(fc12_polynom3,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) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v6_rlvect_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A)) ) ) ).
fof(fc13_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v3_rlvect_1(k16_polynom3(A))
& v4_rlvect_1(k16_polynom3(A))
& v5_rlvect_1(k16_polynom3(A))
& v6_rlvect_1(k16_polynom3(A))
& v7_group_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A)) ) ) ).
fof(fc14_polynom3,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)
& v4_group_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v3_rlvect_1(k16_polynom3(A))
& v4_rlvect_1(k16_polynom3(A))
& v5_rlvect_1(k16_polynom3(A))
& v6_rlvect_1(k16_polynom3(A))
& v4_group_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A)) ) ) ).
fof(fc15_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v3_rlvect_1(k16_polynom3(A))
& v4_rlvect_1(k16_polynom3(A))
& v5_rlvect_1(k16_polynom3(A))
& v6_rlvect_1(k16_polynom3(A))
& v2_group_1(k16_polynom3(A))
& v7_group_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A))
& v6_vectsp_1(k16_polynom3(A))
& v8_vectsp_1(k16_polynom3(A)) ) ) ).
fof(fc16_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v3_rlvect_1(k16_polynom3(A))
& v4_rlvect_1(k16_polynom3(A))
& v5_rlvect_1(k16_polynom3(A))
& v6_rlvect_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A))
& v4_vectsp_1(k16_polynom3(A))
& v5_vectsp_1(k16_polynom3(A))
& v7_vectsp_1(k16_polynom3(A)) ) ) ).
fof(t1_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> ( ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k5_finsop_1(B))
=> k1_funct_1(B,C) = k1_rlvect_1(A) ) )
=> k9_rlvect_1(A,B) = k1_rlvect_1(A) ) ) ) ).
fof(t2_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m2_finseq_1(B,u1_struct_0(A))
=> k9_rlvect_1(A,B) = k9_rlvect_1(A,k4_finseq_5(u1_struct_0(A),B)) ) ) ).
fof(t3_polynom3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> k15_rvsum_1(A) = k15_rvsum_1(k4_finseq_5(k1_numbers,A)) ) ).
fof(t4_polynom3,axiom,
! [A] :
( m1_trees_4(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k5_finsop_1(A))
=> r1_xreal_0(k3_wsierp_1(A,B),k9_wsierp_1(A)) ) ) ) ).
fof(d1_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( r1_polynom3(A,B,C)
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r2_hidden(D,k2_finseq_1(A))
& ~ r1_xreal_0(k3_wsierp_1(C,D),k3_wsierp_1(B,D))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(D,E)
| k3_wsierp_1(B,E) = k3_wsierp_1(C,E) ) ) ) ) ) ) ) ) ).
fof(d2_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( r2_polynom3(A,B,C)
<=> ( r1_polynom3(A,B,C)
| B = C ) ) ) ) ) ).
fof(t5_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [D] :
( m2_finseq_2(D,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( ( ( r1_polynom3(A,B,C)
& r1_polynom3(A,C,D) )
=> r1_polynom3(A,B,D) )
& ( ~ ( ~ ( r1_polynom3(A,B,C)
& r2_polynom3(A,C,D) )
& ~ ( r2_polynom3(A,B,C)
& r1_polynom3(A,C,D) )
& ~ ( r2_polynom3(A,B,C)
& r2_polynom3(A,C,D) ) )
=> r2_polynom3(A,B,D) ) ) ) ) ) ) ).
fof(t6_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ~ ( B != C
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k2_finseq_1(A))
& k3_wsierp_1(B,D) != k3_wsierp_1(C,D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(D,E)
| k3_wsierp_1(B,E) = k3_wsierp_1(C,E) ) ) ) ) ) ) ) ) ) ).
fof(t7_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( r2_polynom3(A,B,C)
| r1_polynom3(A,C,B) ) ) ) ) ).
fof(d3_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_2(B)
& v4_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers))
& m2_relset_1(B,k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers)) )
=> ( B = k5_polynom3(A)
<=> ! [C] :
( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ! [D] :
( m2_finseq_2(D,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( r2_hidden(k4_tarski(C,D),B)
<=> r2_polynom3(A,C,D) ) ) ) ) ) ) ).
fof(d4_polynom3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_finseq_1(C,k4_finseq_2(A,k5_numbers))
=> ( C = k6_polynom3(A,B)
<=> ? [D] :
( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k4_finseq_2(A,k5_numbers)))
& C = k2_triang_1(k4_finseq_2(A,k5_numbers),D,k5_polynom3(A))
& ! [E] :
( m2_finseq_2(E,k5_numbers,k4_finseq_2(A,k5_numbers))
=> ( r2_hidden(E,D)
<=> k9_wsierp_1(E) = B ) ) ) ) ) ) ) ).
fof(t8_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_finseq_1(k6_polynom3(np__1,A)) = np__1 ) ).
fof(t9_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_finseq_1(k6_polynom3(np__2,A)) = k1_nat_1(A,np__1) ) ).
fof(t10_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k6_polynom3(np__1,A) = k13_binarith(k4_finseq_2(np__1,k5_numbers),k13_binarith(k5_numbers,A)) ) ).
fof(t11_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( k1_funct_1(k6_polynom3(np__2,C),A) = k2_polynom3(k5_numbers,D,k5_binarith(C,D))
& k1_funct_1(k6_polynom3(np__2,C),B) = k2_polynom3(k5_numbers,E,k5_binarith(C,E)) )
=> ( ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(E,D) )
& ~ ( ~ r1_xreal_0(E,D)
& r1_xreal_0(B,A) ) ) ) ) ) ) ) ) ).
fof(t12_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( k1_funct_1(k6_polynom3(np__2,B),A) = k2_polynom3(k5_numbers,C,k5_binarith(B,C))
& k1_funct_1(k6_polynom3(np__2,B),k1_nat_1(A,np__1)) = k2_polynom3(k5_numbers,D,k5_binarith(B,D)) )
=> D = k1_nat_1(C,np__1) ) ) ) ) ) ).
fof(t13_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_funct_1(k6_polynom3(np__2,A),np__1) = k2_polynom3(k5_numbers,np__0,A) ) ).
fof(t14_polynom3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k2_finseq_1(k1_nat_1(A,np__1)))
=> k1_funct_1(k6_polynom3(np__2,A),B) = k2_polynom3(k5_numbers,k5_binarith(B,np__1),k5_binarith(k1_nat_1(A,np__1),B)) ) ) ) ).
fof(d5_polynom3,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(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))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ! [E] :
( m2_finseq_1(E,k4_finseq_2(np__3,k5_numbers))
=> ! [F] :
( m2_finseq_2(F,u1_struct_0(A),k3_finseq_2(u1_struct_0(A)))
=> ( F = k7_polynom3(A,B,C,D,E)
<=> ( k3_finseq_1(F) = k3_finseq_1(E)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k5_finsop_1(E))
=> k1_funct_1(F,G) = k1_group_1(A,k1_group_1(A,k2_normsp_1(A,B,k4_finseq_4(k5_numbers,k5_numbers,k3_polynom1(k5_numbers,k5_numbers,k4_finseq_2(np__3,k5_numbers),E,G),np__1)),k2_normsp_1(A,C,k4_finseq_4(k5_numbers,k5_numbers,k3_polynom1(k5_numbers,k5_numbers,k4_finseq_2(np__3,k5_numbers),E,G),np__2))),k2_normsp_1(A,D,k4_finseq_4(k5_numbers,k5_numbers,k3_polynom1(k5_numbers,k5_numbers,k4_finseq_2(np__3,k5_numbers),E,G),np__3))) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_polynom3,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(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))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ! [E] :
( m2_finseq_1(E,k4_finseq_2(np__3,k5_numbers))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_finsop_1(E),k5_finsop_1(E))
& v3_funct_2(F,k5_finsop_1(E),k5_finsop_1(E))
& m2_relset_1(F,k5_finsop_1(E),k5_finsop_1(E)) )
=> ! [G] :
( m2_finseq_1(G,k4_finseq_2(np__3,k5_numbers))
=> ( G = k1_partfun1(k5_finsop_1(E),k5_finsop_1(E),k5_numbers,k4_finseq_2(np__3,k5_numbers),F,E)
=> k7_polynom3(A,B,C,D,G) = k1_partfun1(k5_finsop_1(E),k5_finsop_1(E),k5_numbers,u1_struct_0(A),F,k7_polynom3(A,B,C,D,E)) ) ) ) ) ) ) ) ) ).
fof(t16_polynom3,axiom,
! [A,B] :
( m2_finseq_1(B,k3_finseq_2(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k5_polynom1(A,k16_finseq_1(k3_finseq_2(A),B,C)) = k16_finseq_1(k5_numbers,k5_polynom1(A,B),C) ) ) ).
fof(t17_polynom3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m1_trees_4(B,k1_numbers,k5_numbers)
=> ( A = B
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k16_finseq_1(k1_numbers,A,C) = k16_finseq_1(k5_numbers,B,C) ) ) ) ) ).
fof(t18_polynom3,axiom,
! [A] :
( m1_trees_4(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,C)
=> r1_xreal_0(k9_wsierp_1(k16_finseq_1(k5_numbers,A,B)),k9_wsierp_1(k16_finseq_1(k5_numbers,A,C))) ) ) ) ) ).
fof(t19_polynom3,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(B),C)
=> k16_finseq_1(A,B,k1_nat_1(C,np__1)) = k7_finseq_1(k16_finseq_1(A,B,C),k9_finseq_1(k1_funct_1(B,k1_nat_1(C,np__1)))) ) ) ) ).
fof(t20_polynom3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k3_finseq_1(A),B)
=> k15_rvsum_1(k16_finseq_1(k1_numbers,A,k1_nat_1(B,np__1))) = k2_xcmplx_0(k15_rvsum_1(k16_finseq_1(k1_numbers,A,B)),k1_wsierp_1(A,k1_nat_1(B,np__1))) ) ) ) ).
fof(t21_polynom3,axiom,
! [A] :
( m1_trees_4(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(np__1,E)
& r1_xreal_0(D,k3_wsierp_1(A,k1_nat_1(B,np__1)))
& r1_xreal_0(E,k3_wsierp_1(A,k1_nat_1(C,np__1)))
& k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,A,B)),D) = k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,A,C)),E) )
=> ( r1_xreal_0(k3_finseq_1(A),B)
| r1_xreal_0(k3_finseq_1(A),C)
| ( B = C
& D = E ) ) ) ) ) ) ) ) ).
fof(t22_polynom3,axiom,
! [A,B,C] :
( m2_finseq_1(C,k3_finseq_2(A))
=> ! [D] :
( m2_finseq_1(D,k3_finseq_2(B))
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( ( r2_hidden(E,k5_finsop_1(C))
& r2_hidden(F,k5_finsop_1(D))
& r2_hidden(G,k5_finsop_1(k1_funct_1(C,E)))
& r2_hidden(H,k5_finsop_1(k1_funct_1(D,F)))
& k5_polynom1(A,C) = k5_polynom1(B,D)
& k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,k5_polynom1(A,C),k5_binarith(E,np__1))),G) = k1_nat_1(k9_wsierp_1(k16_finseq_1(k5_numbers,k5_polynom1(B,D),k5_binarith(F,np__1))),H) )
=> ( E = F
& G = H ) ) ) ) ) ) ) ) ).
fof(t23_polynom3,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] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k3_algseq_1(A,B),C)
<=> r1_algseq_1(A,B,C) ) ) ) ) ).
fof(d6_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(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))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> ( D = k8_polynom3(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_normsp_1(A,D,E) = k2_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ).
fof(t24_polynom3,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_algseq_1(A,B,D)
& r1_algseq_1(A,C,D) )
=> r1_algseq_1(A,k8_polynom3(A,B,C),D) ) ) ) ) ) ).
fof(t25_polynom3,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)) )
=> r1_tarski(k4_algseq_1(A,k8_polynom3(A,B,C)),k4_subset_1(k5_numbers,k4_algseq_1(A,B),k4_algseq_1(A,C))) ) ) ) ).
fof(t26_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& l1_rlvect_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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(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))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> k8_polynom3(A,k8_polynom3(A,B,C),D) = k8_polynom3(A,B,k8_polynom3(A,C,D)) ) ) ) ) ).
fof(d7_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_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] :
( ( 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 = k10_polynom3(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,C,D) = k5_rlvect_1(A,k2_normsp_1(A,B,D)) ) ) ) ) ) ).
fof(d8_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k11_polynom3(A,B,C) = k8_polynom3(A,B,k10_polynom3(A,C)) ) ) ) ).
fof(t27_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_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] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_normsp_1(A,k11_polynom3(A,B,C),D) = k6_rlvect_1(A,k2_normsp_1(A,B,D),k2_normsp_1(A,C,D)) ) ) ) ) ).
fof(d9_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> k12_polynom3(A) = k3_finsop_1(u1_struct_0(A),k5_numbers,k1_rlvect_1(A)) ) ).
fof(t28_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_normsp_1(A,k12_polynom3(A),B) = k1_rlvect_1(A) ) ) ).
fof(t29_polynom3,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))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k8_polynom3(A,B,k12_polynom3(A)) = B ) ) ).
fof(t30_polynom3,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))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k11_polynom3(A,B,B) = k12_polynom3(A) ) ) ).
fof(d10_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> k13_polynom3(A) = k1_polynom1(k5_numbers,u1_struct_0(A),k12_polynom3(A),np__0,k2_group_1(A)) ) ).
fof(t31_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( k2_normsp_1(A,k13_polynom3(A),np__0) = k2_group_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B != np__0
=> k2_normsp_1(A,k13_polynom3(A),B) = k1_rlvect_1(A) ) ) ) ) ).
fof(d11_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(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)) )
=> ! [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)) )
=> ! [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 = k14_polynom3(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_finseq_1(F,u1_struct_0(A))
& k3_finseq_1(F) = k1_nat_1(E,np__1)
& k2_normsp_1(A,D,E) = k9_rlvect_1(A,F)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r2_hidden(G,k5_finsop_1(F))
=> k1_funct_1(F,G) = k1_group_1(A,k2_normsp_1(A,B,k5_binarith(G,np__1)),k2_normsp_1(A,C,k5_binarith(k1_nat_1(E,np__1),G))) ) ) ) ) ) ) ) ) ) ).
fof(t32_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_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))
& 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)) )
=> ! [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)) )
=> k14_polynom3(A,B,k9_polynom3(A,C,D)) = k9_polynom3(A,k14_polynom3(A,B,C),k14_polynom3(A,B,D)) ) ) ) ) ).
fof(t33_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(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)) )
=> ! [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)) )
=> ! [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)) )
=> k14_polynom3(A,k9_polynom3(A,B,C),D) = k9_polynom3(A,k14_polynom3(A,B,D),k14_polynom3(A,C,D)) ) ) ) ) ).
fof(t34_polynom3,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)
& v4_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))
& 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)) )
=> ! [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)) )
=> k14_polynom3(A,k14_polynom3(A,B,C),D) = k14_polynom3(A,B,k14_polynom3(A,C,D)) ) ) ) ) ).
fof(t35_polynom3,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] :
( ( 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,B,k12_polynom3(A)) = k12_polynom3(A) ) ) ).
fof(t36_polynom3,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] :
( ( 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,B,k13_polynom3(A)) = B ) ) ).
fof(d12_polynom3,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] :
( ( ~ v3_struct_0(B)
& v3_vectsp_1(B)
& l3_vectsp_1(B) )
=> ( B = k16_polynom3(A)
<=> ( ! [C] :
( r2_hidden(C,u1_struct_0(B))
<=> ( 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)) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,u1_struct_0(A))
& m2_relset_1(F,k5_numbers,u1_struct_0(A)) )
=> ( ( C = E
& D = F )
=> k2_rlvect_1(B,C,D) = k8_polynom3(A,E,F) ) ) ) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,u1_struct_0(A))
& m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_numbers,u1_struct_0(A))
& m2_relset_1(F,k5_numbers,u1_struct_0(A)) )
=> ( ( C = E
& D = F )
=> k1_group_1(B,C,D) = k14_polynom3(A,E,F) ) ) ) ) )
& k1_rlvect_1(B) = k12_polynom3(A)
& k2_vectsp_1(B) = k13_polynom3(A) ) ) ) ) ).
fof(t37_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& l3_vectsp_1(A) )
=> k2_group_1(k16_polynom3(A)) = k13_polynom3(A) ) ).
fof(s1_polynom3,axiom,
? [A] :
( m2_finseq_1(A,k3_finseq_2(f1_s1_polynom3))
& k3_finseq_1(A) = f2_s1_polynom3
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k2_finseq_1(f2_s1_polynom3))
=> ( k3_finseq_1(k3_polynom1(f1_s1_polynom3,k5_numbers,k3_finseq_2(f1_s1_polynom3),A,B)) = f3_s1_polynom3(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k5_finsop_1(k3_polynom1(f1_s1_polynom3,k5_numbers,k3_finseq_2(f1_s1_polynom3),A,B)))
=> k1_funct_1(k3_polynom1(f1_s1_polynom3,k5_numbers,k3_finseq_2(f1_s1_polynom3),A,B),C) = f4_s1_polynom3(B,C) ) ) ) ) ) ) ).
fof(s2_polynom3,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,u1_struct_0(f1_s2_polynom3))
& v1_algseq_1(A,f1_s2_polynom3)
& m2_relset_1(A,k5_numbers,u1_struct_0(f1_s2_polynom3))
& r1_xreal_0(k3_algseq_1(f1_s2_polynom3,A),f2_s2_polynom3)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(f2_s2_polynom3,B)
=> k2_normsp_1(f1_s2_polynom3,A,B) = f3_s2_polynom3(B) ) ) ) ).
fof(antisymmetry_r1_polynom3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k5_numbers))
& m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
=> ( r1_polynom3(A,B,C)
=> ~ r1_polynom3(A,C,B) ) ) ).
fof(reflexivity_r2_polynom3,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k5_numbers))
& m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
=> r2_polynom3(A,B,B) ) ).
fof(dt_k1_polynom3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_finseq_1(C,A) )
=> m2_finseq_1(k1_polynom3(A,B,C),A) ) ).
fof(redefinition_k1_polynom3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_finseq_1(C,A) )
=> k1_polynom3(A,B,C) = k2_finseq_3(B,C) ) ).
fof(dt_k2_polynom3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m2_finseq_2(k2_polynom3(A,B,C),A,k4_finseq_2(np__2,A)) ) ).
fof(redefinition_k2_polynom3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k2_polynom3(A,B,C) = k10_finseq_1(B,C) ) ).
fof(dt_k3_polynom3,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k4_finseq_2(B,A))
& m1_subset_1(E,k4_finseq_2(C,A)) )
=> m2_finseq_2(k3_polynom3(A,B,C,D,E),A,k4_finseq_2(k1_nat_1(B,C),A)) ) ).
fof(redefinition_k3_polynom3,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k4_finseq_2(B,A))
& m1_subset_1(E,k4_finseq_2(C,A)) )
=> k3_polynom3(A,B,C,D,E) = k7_finseq_1(D,E) ) ).
fof(dt_k4_polynom3,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_finseq_1(D,k4_finseq_2(B,A))
& m1_finseq_1(E,k4_finseq_2(C,A)) )
=> m2_finseq_2(k4_polynom3(A,B,C,D,E),k4_finseq_2(k1_nat_1(B,C),A),k3_finseq_2(k4_finseq_2(k1_nat_1(B,C),A))) ) ).
fof(redefinition_k4_polynom3,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,k5_numbers)
& m1_subset_1(C,k5_numbers)
& m1_finseq_1(D,k4_finseq_2(B,A))
& m1_finseq_1(E,k4_finseq_2(C,A)) )
=> k4_polynom3(A,B,C,D,E) = k4_matrlin(D,E) ) ).
fof(dt_k5_polynom3,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_relat_2(k5_polynom3(A))
& v4_relat_2(k5_polynom3(A))
& v8_relat_2(k5_polynom3(A))
& v1_partfun1(k5_polynom3(A),k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers))
& m2_relset_1(k5_polynom3(A),k4_finseq_2(A,k5_numbers),k4_finseq_2(A,k5_numbers)) ) ) ).
fof(dt_k6_polynom3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_finseq_1(k6_polynom3(A,B),k4_finseq_2(A,k5_numbers)) ) ).
fof(dt_k7_polynom3,axiom,
! [A,B,C,D,E] :
( ( ~ 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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A))
& v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m1_relset_1(D,k5_numbers,u1_struct_0(A))
& m1_finseq_1(E,k4_finseq_2(np__3,k5_numbers)) )
=> m2_finseq_2(k7_polynom3(A,B,C,D,E),u1_struct_0(A),k3_finseq_2(u1_struct_0(A))) ) ).
fof(dt_k8_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k8_polynom3(A,B,C))
& v1_funct_2(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k8_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k9_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k9_polynom3(A,B,C))
& v1_funct_2(k9_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k9_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(commutativity_k9_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k9_polynom3(A,B,C) = k9_polynom3(A,C,B) ) ).
fof(redefinition_k9_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l1_rlvect_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k9_polynom3(A,B,C) = k8_polynom3(A,B,C) ) ).
fof(dt_k10_polynom3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_rlvect_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)) )
=> ( v1_funct_1(k10_polynom3(A,B))
& v1_funct_2(k10_polynom3(A,B),k5_numbers,u1_struct_0(A))
& m2_relset_1(k10_polynom3(A,B),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k11_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k11_polynom3(A,B,C))
& v1_funct_2(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k11_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k12_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ( v1_funct_1(k12_polynom3(A))
& v1_funct_2(k12_polynom3(A),k5_numbers,u1_struct_0(A))
& m2_relset_1(k12_polynom3(A),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k13_polynom3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( v1_funct_1(k13_polynom3(A))
& v1_funct_2(k13_polynom3(A),k5_numbers,u1_struct_0(A))
& m2_relset_1(k13_polynom3(A),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k14_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l3_vectsp_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k14_polynom3(A,B,C))
& v1_funct_2(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k14_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(dt_k15_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_group_1(A)
& l3_vectsp_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> ( v1_funct_1(k15_polynom3(A,B,C))
& v1_funct_2(k15_polynom3(A,B,C),k5_numbers,u1_struct_0(A))
& m2_relset_1(k15_polynom3(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ).
fof(commutativity_k15_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_group_1(A)
& l3_vectsp_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k15_polynom3(A,B,C) = k15_polynom3(A,C,B) ) ).
fof(redefinition_k15_polynom3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_group_1(A)
& l3_vectsp_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))
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,u1_struct_0(A))
& m1_relset_1(C,k5_numbers,u1_struct_0(A)) )
=> k15_polynom3(A,B,C) = k14_polynom3(A,B,C) ) ).
fof(dt_k16_polynom3,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) )
=> ( ~ v3_struct_0(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A))
& l3_vectsp_1(k16_polynom3(A)) ) ) ).
%------------------------------------------------------------------------------