SET007 Axioms: SET007+685.ax
%------------------------------------------------------------------------------
% File : SET007+685 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Algebra 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 : polyalg1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 1 unt; 0 def)
% Number of atoms : 601 ( 57 equ)
% Maximal formula atoms : 41 ( 10 avg)
% Number of connectives : 632 ( 91 ~; 0 |; 378 &)
% ( 8 <=>; 155 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 40 ( 38 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 2 con; 0-7 aty)
% Number of variables : 183 ( 171 !; 12 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l1_polyalg1(B,A)
& v1_polyalg1(B,A) ) ) ).
fof(rc2_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ? [B] :
( l1_polyalg1(B,A)
& ~ v3_struct_0(B)
& v1_polyalg1(B,A) ) ) ).
fof(rc3_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ? [B] :
( l1_polyalg1(B,A)
& ~ v3_struct_0(B)
& v2_group_1(B)
& v4_vectsp_1(B)
& v5_vectsp_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v12_vectsp_1(B,A)
& v2_polyalg1(B,A) ) ) ).
fof(fc1_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v3_rlvect_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc2_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc3_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v5_rlvect_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc4_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v6_rlvect_1(k1_polyalg1(A))
& v3_algstr_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc5_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v7_group_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v3_rlvect_1(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v7_group_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc6_polyalg1,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(k1_polyalg1(A))
& v3_rlvect_1(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v6_rlvect_1(k1_polyalg1(A))
& v4_group_1(k1_polyalg1(A))
& v2_algstr_1(k1_polyalg1(A))
& v3_algstr_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(rc4_polyalg1,axiom,
? [A] :
( l3_vectsp_1(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)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v1_algstr_1(A)
& v3_algstr_1(A) ) ).
fof(fc7_polyalg1,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) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v6_rlvect_1(k1_polyalg1(A))
& v2_group_1(k1_polyalg1(A))
& v6_vectsp_1(k1_polyalg1(A))
& v8_vectsp_1(k1_polyalg1(A))
& v3_algstr_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc8_polyalg1,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(k1_polyalg1(A))
& v3_rlvect_1(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v6_rlvect_1(k1_polyalg1(A))
& v4_vectsp_1(k1_polyalg1(A))
& v2_algstr_1(k1_polyalg1(A))
& v3_algstr_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc9_polyalg1,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(k1_polyalg1(A))
& v3_rlvect_1(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v6_rlvect_1(k1_polyalg1(A))
& v5_vectsp_1(k1_polyalg1(A))
& v2_algstr_1(k1_polyalg1(A))
& v3_algstr_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc10_polyalg1,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_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v3_rlvect_1(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v6_rlvect_1(k1_polyalg1(A))
& v4_vectsp_1(k1_polyalg1(A))
& v5_vectsp_1(k1_polyalg1(A))
& v7_vectsp_1(k1_polyalg1(A))
& v12_vectsp_1(k1_polyalg1(A),A)
& v2_algstr_1(k1_polyalg1(A))
& v3_algstr_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(fc11_polyalg1,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_vectsp_1(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v3_rlvect_1(k1_polyalg1(A))
& v4_rlvect_1(k1_polyalg1(A))
& v5_rlvect_1(k1_polyalg1(A))
& v6_rlvect_1(k1_polyalg1(A))
& v4_vectsp_1(k1_polyalg1(A))
& v5_vectsp_1(k1_polyalg1(A))
& v7_vectsp_1(k1_polyalg1(A))
& v2_algstr_1(k1_polyalg1(A))
& v3_algstr_1(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A)
& v2_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(rc5_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( l1_polyalg1(B,A)
& ~ v3_struct_0(B)
& v1_polyalg1(B,A) ) ) ).
fof(rc6_polyalg1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l1_polyalg1(B,A) )
=> ? [C] :
( m1_polyalg1(C,A,B)
& v1_polyalg1(C,A) ) ) ).
fof(rc7_polyalg1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ? [C] :
( m1_polyalg1(C,A,B)
& ~ v3_struct_0(C)
& v1_polyalg1(C,A) ) ) ).
fof(fc12_polyalg1,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))
& v5_rlvect_1(k16_polynom3(A))
& v6_rlvect_1(k16_polynom3(A))
& v3_vectsp_1(k16_polynom3(A))
& v3_algstr_1(k16_polynom3(A))
& v6_algstr_1(k16_polynom3(A)) ) ) ).
fof(d1_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ( v2_polyalg1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> k6_vectsp_1(A,B,C,k1_group_1(B,D,E)) = k1_group_1(B,k6_vectsp_1(A,B,C,D),E) ) ) ) ) ) ) ).
fof(t1_polyalg1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,B),A)
& m2_relset_1(C,k2_zfmisc_1(A,B),A) )
=> k1_relat_1(C) = k2_zfmisc_1(A,B) ) ).
fof(t2_polyalg1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,B),B)
& m2_relset_1(C,k2_zfmisc_1(A,B),B) )
=> k1_relat_1(C) = k2_zfmisc_1(A,B) ) ).
fof(d2_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v1_polyalg1(B,A)
& l1_polyalg1(B,A) )
=> ( B = k1_polyalg1(A)
<=> ( ! [C] :
( r2_hidden(C,u1_struct_0(B))
<=> ( 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] :
( 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) ) ) ) ) )
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [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)) )
=> ( D = E
=> k6_vectsp_1(A,B,C,D) = k3_polynom5(A,E,C) ) ) ) )
& k1_rlvect_1(B) = k12_polynom3(A)
& k2_vectsp_1(B) = k13_polynom3(A) ) ) ) ) ).
fof(t3_polyalg1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,A) )
=> k1_rfinseq(A,B,np__1) = k1_polynom3(A,np__1,B) ) ) ).
fof(t4_polyalg1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m2_finseq_1(B,A) )
=> B = k7_finseq_1(k9_finseq_1(k1_funct_1(B,np__1)),k1_polynom3(A,np__1,B)) ) ) ).
fof(t5_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_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))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k14_polynom3(A,k13_polynom3(A),B) = B ) ) ).
fof(t6_polyalg1,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) )
=> ! [B] :
( m1_subset_1(B,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)) )
=> k3_polynom5(A,k9_polynom3(A,C,D),B) = k9_polynom3(A,k3_polynom5(A,C,B),k3_polynom5(A,D,B)) ) ) ) ) ).
fof(t7_polyalg1,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) )
=> ! [B] :
( m1_subset_1(B,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)) )
=> k3_polynom5(A,D,k4_rlvect_1(A,B,C)) = k9_polynom3(A,k3_polynom5(A,D,B),k3_polynom5(A,D,C)) ) ) ) ) ).
fof(t8_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_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] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,u1_struct_0(A))
& m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
=> k3_polynom5(A,D,k1_group_1(A,B,C)) = k3_polynom5(A,k3_polynom5(A,D,C),B) ) ) ) ) ).
fof(t9_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_group_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))
& m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
=> k3_polynom5(A,B,k2_group_1(A)) = B ) ) ).
fof(t10_polyalg1,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_vectsp_1(A)
& v1_algstr_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,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)) )
=> k3_polynom5(A,k14_polynom3(A,C,D),B) = k14_polynom3(A,k3_polynom5(A,C,B),D) ) ) ) ) ).
fof(d3_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_polyalg1(B,A)
=> ! [C] :
( l1_polyalg1(C,A)
=> ( m1_polyalg1(C,A,B)
<=> ( r1_tarski(u1_struct_0(C),u1_struct_0(B))
& k2_vectsp_1(C) = k2_vectsp_1(B)
& k1_rlvect_1(C) = k1_rlvect_1(B)
& u1_rlvect_1(C) = k1_realset1(u1_rlvect_1(B),u1_struct_0(C))
& u1_group_1(C) = k1_realset1(u1_group_1(B),u1_struct_0(C))
& u2_vectsp_1(A,C) = k7_relat_1(u2_vectsp_1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(C))) ) ) ) ) ) ).
fof(t11_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_polyalg1(B,A)
=> m1_polyalg1(B,A,B) ) ) ).
fof(t12_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_polyalg1(B,A)
=> ! [C] :
( l1_polyalg1(C,A)
=> ! [D] :
( l1_polyalg1(D,A)
=> ( ( m1_polyalg1(B,A,C)
& m1_polyalg1(C,A,D) )
=> m1_polyalg1(B,A,D) ) ) ) ) ) ).
fof(t13_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_polyalg1(B,A)
=> ! [C] :
( l1_polyalg1(C,A)
=> ( ( m1_polyalg1(B,A,C)
& m1_polyalg1(C,A,B) )
=> g1_polyalg1(A,u1_struct_0(B),u1_rlvect_1(B),u1_group_1(B),u2_struct_0(B),u1_vectsp_1(B),u2_vectsp_1(A,B)) = g1_polyalg1(A,u1_struct_0(C),u1_rlvect_1(C),u1_group_1(C),u2_struct_0(C),u1_vectsp_1(C),u2_vectsp_1(A,C)) ) ) ) ) ).
fof(t14_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_polyalg1(B,A)
=> ! [C] :
( l1_polyalg1(C,A)
=> ( g1_polyalg1(A,u1_struct_0(B),u1_rlvect_1(B),u1_group_1(B),u2_struct_0(B),u1_vectsp_1(B),u2_vectsp_1(A,B)) = g1_polyalg1(A,u1_struct_0(C),u1_rlvect_1(C),u1_group_1(C),u2_struct_0(C),u1_vectsp_1(C),u2_vectsp_1(A,C))
=> ( m1_polyalg1(B,A,C)
& m1_polyalg1(C,A,B) ) ) ) ) ) ).
fof(d4_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v3_polyalg1(C,A,B)
<=> ( v1_vectsp_4(C,A,B)
& ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( ( r2_hidden(D,C)
& r2_hidden(E,C) )
=> r2_hidden(k1_group_1(B,D,E),C) ) ) )
& r2_hidden(k2_vectsp_1(B),C)
& r2_hidden(k1_rlvect_1(B),C) ) ) ) ) ) ).
fof(t15_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_polyalg1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(C))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(C))
=> ( ( D = F
& E = G )
=> k2_rlvect_1(B,D,E) = k2_rlvect_1(C,F,G) ) ) ) ) ) ) ) ) ).
fof(t16_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_polyalg1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(C))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(C))
=> ( ( D = F
& E = G )
=> k1_group_1(B,D,E) = k1_group_1(C,F,G) ) ) ) ) ) ) ) ) ).
fof(t17_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_polyalg1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(C))
=> ( E = F
=> k6_vectsp_1(A,B,D,E) = k6_vectsp_1(A,C,D,F) ) ) ) ) ) ) ) ).
fof(t18_polyalg1,axiom,
$true ).
fof(t19_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& m1_polyalg1(C,A,B) )
=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& u1_struct_0(C) = D
& v3_polyalg1(D,A,B) ) ) ) ) ).
fof(t20_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( v3_polyalg1(C,A,B)
& ! [D] :
( ( v1_polyalg1(D,A)
& m1_polyalg1(D,A,B) )
=> u1_struct_0(D) != C ) ) ) ) ) ).
fof(t21_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B))))
=> ( ! [E] :
( r2_hidden(E,D)
<=> ( r1_tarski(E,u1_struct_0(B))
& ? [F] :
( m1_polyalg1(F,A,B)
& E = u1_struct_0(F)
& r1_tarski(C,E) ) ) )
=> v3_polyalg1(k6_setfam_1(u1_struct_0(B),D),A,B) ) ) ) ) ) ).
fof(d5_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [D] :
( ( ~ v3_struct_0(D)
& v1_polyalg1(D,A)
& m1_polyalg1(D,A,B) )
=> ( D = k2_polyalg1(A,B,C)
<=> ( r1_tarski(C,u1_struct_0(D))
& ! [E] :
( m1_polyalg1(E,A,B)
=> ( r1_tarski(C,u1_struct_0(E))
=> r1_tarski(u1_struct_0(D),u1_struct_0(E)) ) ) ) ) ) ) ) ) ).
fof(t22_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_group_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_polyalg1(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( v3_polyalg1(C,A,B)
=> u1_struct_0(k2_polyalg1(A,B,C)) = C ) ) ) ) ).
fof(d6_polyalg1,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)
& v1_polyalg1(B,A)
& l1_polyalg1(B,A) )
=> ( B = k3_polyalg1(A)
<=> ? [C] :
( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k1_polyalg1(A))))
& C = u1_struct_0(k16_polynom3(A))
& B = k2_polyalg1(A,k1_polyalg1(A),C) ) ) ) ) ).
fof(t23_polyalg1,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_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k1_polyalg1(A)))) )
=> ( B = u1_struct_0(k16_polynom3(A))
=> v3_polyalg1(B,A,k1_polyalg1(A)) ) ) ) ).
fof(t24_polyalg1,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) )
=> g3_vectsp_1(u1_struct_0(k3_polyalg1(A)),u1_rlvect_1(k3_polyalg1(A)),u1_group_1(k3_polyalg1(A)),u1_vectsp_1(k3_polyalg1(A)),u2_struct_0(k3_polyalg1(A))) = k16_polynom3(A) ) ).
fof(t25_polyalg1,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) )
=> k2_group_1(k1_polyalg1(A)) = k13_polynom3(A) ) ).
fof(dt_m1_polyalg1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l1_polyalg1(B,A) )
=> ! [C] :
( m1_polyalg1(C,A,B)
=> l1_polyalg1(C,A) ) ) ).
fof(existence_m1_polyalg1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l1_polyalg1(B,A) )
=> ? [C] : m1_polyalg1(C,A,B) ) ).
fof(dt_l1_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_polyalg1(B,A)
=> ( l3_vectsp_1(B)
& l4_vectsp_1(B,A) ) ) ) ).
fof(existence_l1_polyalg1,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] : l1_polyalg1(B,A) ) ).
fof(abstractness_v1_polyalg1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l1_polyalg1(B,A) )
=> ( v1_polyalg1(B,A)
=> B = g1_polyalg1(A,u1_struct_0(B),u1_rlvect_1(B),u1_group_1(B),u2_struct_0(B),u1_vectsp_1(B),u2_vectsp_1(A,B)) ) ) ).
fof(dt_k1_polyalg1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k1_polyalg1(A))
& v1_polyalg1(k1_polyalg1(A),A)
& l1_polyalg1(k1_polyalg1(A),A) ) ) ).
fof(dt_k2_polyalg1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_group_1(A)
& ~ v3_struct_0(B)
& l1_polyalg1(B,A)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ( ~ v3_struct_0(k2_polyalg1(A,B,C))
& v1_polyalg1(k2_polyalg1(A,B,C),A)
& m1_polyalg1(k2_polyalg1(A,B,C),A,B) ) ) ).
fof(dt_k3_polyalg1,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(k3_polyalg1(A))
& v1_polyalg1(k3_polyalg1(A),A)
& l1_polyalg1(k3_polyalg1(A),A) ) ) ).
fof(dt_g1_polyalg1,axiom,
! [A,B,C,D,E,F,G] :
( ( l1_struct_0(A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m1_relset_1(C,k2_zfmisc_1(B,B),B)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m1_relset_1(D,k2_zfmisc_1(B,B),B)
& m1_subset_1(E,B)
& m1_subset_1(F,B)
& v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(u1_struct_0(A),B),B)
& m1_relset_1(G,k2_zfmisc_1(u1_struct_0(A),B),B) )
=> ( v1_polyalg1(g1_polyalg1(A,B,C,D,E,F,G),A)
& l1_polyalg1(g1_polyalg1(A,B,C,D,E,F,G),A) ) ) ).
fof(free_g1_polyalg1,axiom,
! [A,B,C,D,E,F,G] :
( ( l1_struct_0(A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m1_relset_1(C,k2_zfmisc_1(B,B),B)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m1_relset_1(D,k2_zfmisc_1(B,B),B)
& m1_subset_1(E,B)
& m1_subset_1(F,B)
& v1_funct_1(G)
& v1_funct_2(G,k2_zfmisc_1(u1_struct_0(A),B),B)
& m1_relset_1(G,k2_zfmisc_1(u1_struct_0(A),B),B) )
=> ! [H,I,J,K,L,M,N] :
( g1_polyalg1(A,B,C,D,E,F,G) = g1_polyalg1(H,I,J,K,L,M,N)
=> ( A = H
& B = I
& C = J
& D = K
& E = L
& F = M
& G = N ) ) ) ).
%------------------------------------------------------------------------------