SET007 Axioms: SET007+547.ax
%------------------------------------------------------------------------------
% File : SET007+547 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Field of Quotients Over an Integral Domain
% 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 : quofield [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 142 ( 6 unt; 0 def)
% Number of atoms : 2095 ( 124 equ)
% Maximal formula atoms : 45 ( 14 avg)
% Number of connectives : 2271 ( 318 ~; 0 |;1588 &)
% ( 31 <=>; 334 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 13 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 50 ( 48 usr; 1 prp; 0-3 aty)
% Number of functors : 51 ( 51 usr; 0 con; 1-6 aty)
% Number of variables : 363 ( 348 !; 15 ?)
% 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_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ( ~ v1_xboole_0(k1_quofield(A))
& v1_relat_1(k1_quofield(A)) ) ) ).
fof(fc2_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> ~ v1_xboole_0(k8_quofield(A,B)) ) ).
fof(fc3_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ~ v1_xboole_0(k9_quofield(A)) ) ).
fof(fc4_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k21_quofield(A))
& v3_vectsp_1(k21_quofield(A)) ) ) ).
fof(fc5_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k21_quofield(A))
& v4_rlvect_1(k21_quofield(A))
& v5_rlvect_1(k21_quofield(A))
& v6_rlvect_1(k21_quofield(A))
& v3_vectsp_1(k21_quofield(A))
& v1_algstr_1(k21_quofield(A))
& v2_algstr_1(k21_quofield(A))
& v3_algstr_1(k21_quofield(A))
& v4_algstr_1(k21_quofield(A))
& v5_algstr_1(k21_quofield(A))
& v6_algstr_1(k21_quofield(A)) ) ) ).
fof(fc6_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k21_quofield(A))
& v4_rlvect_1(k21_quofield(A))
& v5_rlvect_1(k21_quofield(A))
& v6_rlvect_1(k21_quofield(A))
& v7_group_1(k21_quofield(A))
& v3_vectsp_1(k21_quofield(A))
& v1_algstr_1(k21_quofield(A))
& v2_algstr_1(k21_quofield(A))
& v3_algstr_1(k21_quofield(A))
& v4_algstr_1(k21_quofield(A))
& v5_algstr_1(k21_quofield(A))
& v6_algstr_1(k21_quofield(A)) ) ) ).
fof(fc7_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k21_quofield(A))
& v4_rlvect_1(k21_quofield(A))
& v5_rlvect_1(k21_quofield(A))
& v6_rlvect_1(k21_quofield(A))
& v2_group_1(k21_quofield(A))
& v7_group_1(k21_quofield(A))
& v3_vectsp_1(k21_quofield(A))
& v6_vectsp_1(k21_quofield(A))
& v8_vectsp_1(k21_quofield(A))
& v1_algstr_1(k21_quofield(A))
& v2_algstr_1(k21_quofield(A))
& v3_algstr_1(k21_quofield(A))
& v4_algstr_1(k21_quofield(A))
& v5_algstr_1(k21_quofield(A))
& v6_algstr_1(k21_quofield(A)) ) ) ).
fof(fc8_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k21_quofield(A))
& v3_rlvect_1(k21_quofield(A))
& v4_rlvect_1(k21_quofield(A))
& v5_rlvect_1(k21_quofield(A))
& v6_rlvect_1(k21_quofield(A))
& v2_group_1(k21_quofield(A))
& v4_group_1(k21_quofield(A))
& v7_group_1(k21_quofield(A))
& v3_vectsp_1(k21_quofield(A))
& v4_vectsp_1(k21_quofield(A))
& v5_vectsp_1(k21_quofield(A))
& v6_vectsp_1(k21_quofield(A))
& v7_vectsp_1(k21_quofield(A))
& v8_vectsp_1(k21_quofield(A))
& v9_vectsp_1(k21_quofield(A))
& ~ v10_vectsp_1(k21_quofield(A))
& v3_gcd_1(k21_quofield(A))
& v1_algstr_1(k21_quofield(A))
& v2_algstr_1(k21_quofield(A))
& v3_algstr_1(k21_quofield(A))
& v4_algstr_1(k21_quofield(A))
& v5_algstr_1(k21_quofield(A))
& v6_algstr_1(k21_quofield(A)) ) ) ).
fof(cc1_quofield,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A)
& v6_algstr_1(A) ) ) ) ).
fof(rc1_quofield,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& v3_gcd_1(A)
& v1_algstr_1(A)
& v2_algstr_1(A)
& v3_algstr_1(A)
& v4_algstr_1(A)
& v5_algstr_1(A)
& v6_algstr_1(A) ) ).
fof(cc2_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_quofield(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_grcat_1(C,A,B)
& v1_endalg(C,A,B)
& v1_group_6(C,A,B) ) ) ) ) ).
fof(cc3_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_grcat_1(C,A,B)
& v1_endalg(C,A,B)
& v1_group_6(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v1_quofield(C,A,B) ) ) ) ) ).
fof(cc4_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v4_quofield(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v2_quofield(C,A,B)
& v3_quofield(C,A,B) ) ) ) ) ).
fof(cc5_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B))
=> ( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v2_quofield(C,A,B)
& v3_quofield(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v4_quofield(C,A,B) ) ) ) ) ).
fof(fc9_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v1_relat_1(k7_grcat_1(A))
& v1_funct_1(k7_grcat_1(A))
& v1_funct_2(k7_grcat_1(A),u1_struct_0(A),u1_struct_0(A))
& v1_grcat_1(k7_grcat_1(A),A,A)
& v1_endalg(k7_grcat_1(A),A,A)
& v1_group_6(k7_grcat_1(A),A,A)
& v1_quofield(k7_grcat_1(A),A,A) ) ) ).
fof(fc10_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k21_quofield(A))
& v3_rlvect_1(k21_quofield(A))
& v4_rlvect_1(k21_quofield(A))
& v5_rlvect_1(k21_quofield(A))
& v6_rlvect_1(k21_quofield(A))
& v2_group_1(k21_quofield(A))
& v4_group_1(k21_quofield(A))
& v7_group_1(k21_quofield(A))
& v3_vectsp_1(k21_quofield(A))
& v4_vectsp_1(k21_quofield(A))
& v5_vectsp_1(k21_quofield(A))
& v6_vectsp_1(k21_quofield(A))
& v7_vectsp_1(k21_quofield(A))
& v8_vectsp_1(k21_quofield(A))
& v9_vectsp_1(k21_quofield(A))
& ~ v10_vectsp_1(k21_quofield(A))
& v2_vectsp_2(k21_quofield(A))
& v3_gcd_1(k21_quofield(A))
& v1_algstr_1(k21_quofield(A))
& v2_algstr_1(k21_quofield(A))
& v3_algstr_1(k21_quofield(A))
& v4_algstr_1(k21_quofield(A))
& v5_algstr_1(k21_quofield(A))
& v6_algstr_1(k21_quofield(A)) ) ) ).
fof(d1_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))))
=> ( B = k1_quofield(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(A))
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& C = k1_domain_1(u1_struct_0(A),u1_struct_0(A),D,E)
& E != k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(t1_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ~ v1_xboole_0(k1_quofield(A)) ) ).
fof(t2_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> k3_domain_1(u1_struct_0(A),u1_struct_0(A),B) != k1_rlvect_1(A) ) ) ).
fof(d2_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> k4_quofield(A,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),k2_rlvect_1(A,k1_group_1(A,k2_quofield(A,B),k3_quofield(A,C)),k1_group_1(A,k2_quofield(A,C),k3_quofield(A,B))),k1_group_1(A,k3_quofield(A,B),k3_quofield(A,C))) ) ) ) ).
fof(d3_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> k5_quofield(A,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),k1_group_1(A,k2_quofield(A,B),k2_quofield(A,C)),k1_group_1(A,k3_quofield(A,B),k3_quofield(A,C))) ) ) ) ).
fof(t3_quofield,axiom,
$true ).
fof(t4_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( k4_quofield(A,B,k4_quofield(A,C,D)) = k4_quofield(A,k4_quofield(A,B,C),D)
& k4_quofield(A,B,C) = k4_quofield(A,C,B) ) ) ) ) ) ).
fof(t5_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( k5_quofield(A,B,k5_quofield(A,C,D)) = k5_quofield(A,k5_quofield(A,B,C),D)
& k5_quofield(A,B,C) = k5_quofield(A,C,B) ) ) ) ) ) ).
fof(d4_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_quofield(A)))
=> ( C = k8_quofield(A,B)
<=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( r2_hidden(D,C)
<=> k1_group_1(A,k2_quofield(A,D),k3_quofield(A,B)) = k1_group_1(A,k3_quofield(A,D),k2_quofield(A,B)) ) ) ) ) ) ) ).
fof(t6_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v7_group_1(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> r2_hidden(B,k8_quofield(A,B)) ) ) ).
fof(d5_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_quofield(A))))
=> ( B = k9_quofield(A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_quofield(A)))
=> ( r2_hidden(C,B)
<=> ? [D] :
( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
& C = k8_quofield(A,D) ) ) ) ) ) ) ).
fof(t7_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> ~ v1_xboole_0(k9_quofield(A)) ) ).
fof(t8_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( ? [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
& r2_hidden(B,D)
& r2_hidden(C,D) )
=> k10_group_1(A,k2_quofield(A,B),k3_quofield(A,C)) = k10_group_1(A,k2_quofield(A,C),k3_quofield(A,B)) ) ) ) ) ).
fof(t9_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( ~ r1_xboole_0(B,C)
=> B = C ) ) ) ) ).
fof(d6_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( D = k10_quofield(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( r2_hidden(E,D)
<=> ? [F] :
( m2_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
& ? [G] :
( m2_subset_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
& r2_hidden(F,B)
& r2_hidden(G,C)
& k10_group_1(A,k2_quofield(A,E),k10_group_1(A,k3_quofield(A,F),k3_quofield(A,G))) = k10_group_1(A,k3_quofield(A,E),k4_rlvect_1(A,k10_group_1(A,k2_quofield(A,F),k3_quofield(A,G)),k10_group_1(A,k2_quofield(A,G),k3_quofield(A,F)))) ) ) ) ) ) ) ) ) ) ).
fof(d7_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( D = k11_quofield(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( r2_hidden(E,D)
<=> ? [F] :
( m2_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
& ? [G] :
( m2_subset_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
& r2_hidden(F,B)
& r2_hidden(G,C)
& k10_group_1(A,k2_quofield(A,E),k10_group_1(A,k3_quofield(A,F),k3_quofield(A,G))) = k10_group_1(A,k3_quofield(A,E),k10_group_1(A,k2_quofield(A,F),k2_quofield(A,G))) ) ) ) ) ) ) ) ) ) ).
fof(t10_quofield,axiom,
$true ).
fof(t11_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> k10_quofield(A,k12_quofield(A,B),k12_quofield(A,C)) = k12_quofield(A,k6_quofield(A,B,C)) ) ) ) ).
fof(t12_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> k11_quofield(A,k12_quofield(A,B),k12_quofield(A,C)) = k12_quofield(A,k7_quofield(A,B,C)) ) ) ) ).
fof(d8_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( B = k13_quofield(A)
<=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( r2_hidden(C,B)
<=> k2_quofield(A,C) = k1_rlvect_1(A) ) ) ) ) ) ).
fof(d9_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( B = k14_quofield(A)
<=> ! [C] :
( m2_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( r2_hidden(C,B)
<=> k2_quofield(A,C) = k3_quofield(A,C) ) ) ) ) ) ).
fof(d10_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( C = k15_quofield(A,B)
<=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( r2_hidden(D,C)
<=> ? [E] :
( m2_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
& r2_hidden(E,B)
& k10_group_1(A,k2_quofield(A,D),k3_quofield(A,E)) = k10_group_1(A,k3_quofield(A,D),k5_rlvect_1(A,k2_quofield(A,E))) ) ) ) ) ) ) ) ).
fof(d11_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( B != k13_quofield(A)
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( C = k16_quofield(A,B)
<=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( r2_hidden(D,C)
<=> ? [E] :
( m2_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
& r2_hidden(E,B)
& k10_group_1(A,k2_quofield(A,D),k2_quofield(A,E)) = k10_group_1(A,k3_quofield(A,D),k3_quofield(A,E)) ) ) ) ) ) ) ) ) ).
fof(t13_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k10_quofield(A,B,k10_quofield(A,C,D)) = k10_quofield(A,k10_quofield(A,B,C),D)
& k10_quofield(A,B,C) = k10_quofield(A,C,B) ) ) ) ) ) ).
fof(t14_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k10_quofield(A,B,k13_quofield(A)) = B
& k10_quofield(A,k13_quofield(A),B) = B ) ) ) ).
fof(t15_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k11_quofield(A,B,k11_quofield(A,C,D)) = k11_quofield(A,k11_quofield(A,B,C),D)
& k11_quofield(A,B,C) = k11_quofield(A,C,B) ) ) ) ) ) ).
fof(t16_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k11_quofield(A,B,k14_quofield(A)) = B
& k11_quofield(A,k14_quofield(A),B) = B ) ) ) ).
fof(t17_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k11_quofield(A,k10_quofield(A,B,C),D) = k10_quofield(A,k11_quofield(A,B,D),k11_quofield(A,C,D)) ) ) ) ) ).
fof(t18_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k11_quofield(A,B,k10_quofield(A,C,D)) = k10_quofield(A,k11_quofield(A,B,C),k11_quofield(A,B,D)) ) ) ) ) ).
fof(t19_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k10_quofield(A,B,k15_quofield(A,B)) = k13_quofield(A)
& k10_quofield(A,k15_quofield(A,B),B) = k13_quofield(A) ) ) ) ).
fof(t20_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( B != k13_quofield(A)
=> ( k11_quofield(A,B,k16_quofield(A,B)) = k14_quofield(A)
& k11_quofield(A,k16_quofield(A,B),B) = k14_quofield(A) ) ) ) ) ).
fof(t21_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> k14_quofield(A) != k13_quofield(A) ) ).
fof(d12_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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,k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A))
& m2_relset_1(B,k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) )
=> ( B = k17_quofield(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),B,C,D) = k10_quofield(A,C,D) ) ) ) ) ) ).
fof(d13_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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,k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A))
& m2_relset_1(B,k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) )
=> ( B = k18_quofield(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),B,C,D) = k11_quofield(A,C,D) ) ) ) ) ) ).
fof(d14_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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,k9_quofield(A),k9_quofield(A))
& m2_relset_1(B,k9_quofield(A),k9_quofield(A)) )
=> ( B = k19_quofield(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k8_funct_2(k9_quofield(A),k9_quofield(A),B,C) = k15_quofield(A,C) ) ) ) ) ).
fof(d15_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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,k9_quofield(A),k9_quofield(A))
& m2_relset_1(B,k9_quofield(A),k9_quofield(A)) )
=> ( B = k20_quofield(A)
<=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k8_funct_2(k9_quofield(A),k9_quofield(A),B,C) = k16_quofield(A,C) ) ) ) ) ).
fof(t22_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),B,C),D) = k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),B,k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),C,D)) ) ) ) ) ).
fof(t23_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),B,C) = k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),C,B) ) ) ) ).
fof(t24_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),B,k13_quofield(A)) = B
& k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),k13_quofield(A),B) = B ) ) ) ).
fof(t25_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,C),D) = k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),C,D)) ) ) ) ) ).
fof(t26_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,C) = k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),C,B) ) ) ) ).
fof(t27_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,k14_quofield(A)) = B
& k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),k14_quofield(A),B) = B ) ) ) ).
fof(t28_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),B,C),D) = k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,D),k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),C,D)) ) ) ) ) ).
fof(t29_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),C,D)) = k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,C),k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,D)) ) ) ) ) ).
fof(t30_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),B,k8_funct_2(k9_quofield(A),k9_quofield(A),k19_quofield(A),B)) = k13_quofield(A)
& k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k17_quofield(A),k8_funct_2(k9_quofield(A),k9_quofield(A),k19_quofield(A),B),B) = k13_quofield(A) ) ) ) ).
fof(t31_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m2_subset_1(B,k1_zfmisc_1(k1_quofield(A)),k9_quofield(A))
=> ( B != k13_quofield(A)
=> ( k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),B,k8_funct_2(k9_quofield(A),k9_quofield(A),k20_quofield(A),B)) = k14_quofield(A)
& k2_binop_1(k9_quofield(A),k9_quofield(A),k9_quofield(A),k18_quofield(A),k8_funct_2(k9_quofield(A),k9_quofield(A),k20_quofield(A),B),B) = k14_quofield(A) ) ) ) ) ).
fof(d16_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> k21_quofield(A) = g3_vectsp_1(k9_quofield(A),k17_quofield(A),k18_quofield(A),k14_quofield(A),k13_quofield(A)) ) ).
fof(t32_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( u1_struct_0(k21_quofield(A)) = k9_quofield(A)
& u1_rlvect_1(k21_quofield(A)) = k17_quofield(A)
& u1_group_1(k21_quofield(A)) = k18_quofield(A)
& u2_struct_0(k21_quofield(A)) = k13_quofield(A)
& u1_vectsp_1(k21_quofield(A)) = k14_quofield(A) ) ) ).
fof(t33_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> m1_subset_1(k1_binop_1(k17_quofield(A),B,C),u1_struct_0(k21_quofield(A))) ) ) ) ).
fof(t34_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> m1_subset_1(k1_funct_1(k19_quofield(A),B),u1_struct_0(k21_quofield(A))) ) ) ).
fof(t35_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> m1_subset_1(k1_binop_1(k18_quofield(A),B,C),u1_struct_0(k21_quofield(A))) ) ) ) ).
fof(t36_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> m1_subset_1(k1_funct_1(k20_quofield(A),B),u1_struct_0(k21_quofield(A))) ) ) ).
fof(t37_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> k2_rlvect_1(k21_quofield(A),B,C) = k1_binop_1(k17_quofield(A),B,C) ) ) ) ).
fof(t38_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> k5_rlvect_1(k21_quofield(A),B) = k1_funct_1(k19_quofield(A),B) ) ) ).
fof(t39_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> k1_group_1(k21_quofield(A),B,C) = k1_binop_1(k18_quofield(A),B,C) ) ) ) ).
fof(t40_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( k2_group_1(k21_quofield(A)) = k14_quofield(A)
& k1_rlvect_1(k21_quofield(A)) = k13_quofield(A) ) ) ).
fof(t41_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k21_quofield(A)))
=> k2_rlvect_1(k21_quofield(A),k2_rlvect_1(k21_quofield(A),B,C),D) = k2_rlvect_1(k21_quofield(A),B,k2_rlvect_1(k21_quofield(A),C,D)) ) ) ) ) ).
fof(t42_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> k2_rlvect_1(k21_quofield(A),B,C) = k2_rlvect_1(k21_quofield(A),C,B) ) ) ) ).
fof(t43_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> k2_rlvect_1(k21_quofield(A),B,k1_rlvect_1(k21_quofield(A))) = B ) ) ).
fof(t44_quofield,axiom,
$true ).
fof(t45_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> k10_group_1(k21_quofield(A),k2_group_1(k21_quofield(A)),B) = B ) ) ).
fof(t46_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> k10_group_1(k21_quofield(A),B,C) = k10_group_1(k21_quofield(A),C,B) ) ) ) ).
fof(t47_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k21_quofield(A)))
=> k10_group_1(k21_quofield(A),k10_group_1(k21_quofield(A),B,C),D) = k10_group_1(k21_quofield(A),B,k10_group_1(k21_quofield(A),C,D)) ) ) ) ) ).
fof(t48_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ~ ( B != k1_rlvect_1(k21_quofield(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(k21_quofield(A)))
=> k10_group_1(k21_quofield(A),B,C) != k2_group_1(k21_quofield(A)) ) ) ) ) ).
fof(t49_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( ~ v3_struct_0(k21_quofield(A))
& v3_rlvect_1(k21_quofield(A))
& v4_rlvect_1(k21_quofield(A))
& v5_rlvect_1(k21_quofield(A))
& v6_rlvect_1(k21_quofield(A))
& v2_group_1(k21_quofield(A))
& v4_group_1(k21_quofield(A))
& v7_vectsp_1(k21_quofield(A))
& v9_vectsp_1(k21_quofield(A))
& ~ v10_vectsp_1(k21_quofield(A))
& l3_vectsp_1(k21_quofield(A)) ) ) ).
fof(t50_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( m1_subset_1(B,u1_struct_0(k21_quofield(A)))
=> ( B != k1_rlvect_1(k21_quofield(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C != k1_rlvect_1(A)
=> ! [D] :
( m2_subset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( ( B = k12_quofield(A,D)
& D = k1_domain_1(u1_struct_0(A),u1_struct_0(A),C,k2_group_1(A)) )
=> ! [E] :
( m2_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A))
=> ( E = k1_domain_1(u1_struct_0(A),u1_struct_0(A),k2_group_1(A),C)
=> k4_vectsp_1(k21_quofield(A),B) = k12_quofield(A,E) ) ) ) ) ) ) ) ) ) ).
fof(d17_quofield,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] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k22_quofield(A,B,C) = k10_group_1(A,B,k4_vectsp_1(A,C)) ) ) ) ).
fof(t51_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( C != k1_rlvect_1(A)
& E != k1_rlvect_1(A)
& k10_group_1(A,k22_quofield(A,B,C),k22_quofield(A,D,E)) != k22_quofield(A,k10_group_1(A,B,D),k10_group_1(A,C,E)) ) ) ) ) ) ) ).
fof(t52_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ~ ( C != k1_rlvect_1(A)
& E != k1_rlvect_1(A)
& k4_rlvect_1(A,k22_quofield(A,B,C),k22_quofield(A,D,E)) != k22_quofield(A,k4_rlvect_1(A,k10_group_1(A,B,E),k10_group_1(A,D,C)),k10_group_1(A,C,E)) ) ) ) ) ) ) ).
fof(d18_quofield,axiom,
$true ).
fof(d19_quofield,axiom,
$true ).
fof(d20_quofield,axiom,
$true ).
fof(d21_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_quofield(C,A,B)
<=> ( v1_grcat_1(C,A,B)
& v1_group_6(C,A,B)
& v1_endalg(C,A,B) ) ) ) ) ) ).
fof(d22_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v2_quofield(C,A,B)
<=> ( v1_quofield(C,A,B)
& k2_relat_1(C) = u1_struct_0(B) ) ) ) ) ) ).
fof(d23_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_quofield(C,A,B)
<=> ( v1_quofield(C,A,B)
& v2_funct_1(C) ) ) ) ) ) ).
fof(d24_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v4_quofield(C,A,B)
<=> ( v3_quofield(C,A,B)
& v2_quofield(C,A,B) ) ) ) ) ) ).
fof(t53_quofield,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)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_quofield(C,A,B)
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_rlvect_1(A)) = k1_rlvect_1(B) ) ) ) ) ).
fof(t54_quofield,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)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v3_quofield(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D) = k1_rlvect_1(B)
<=> D = k1_rlvect_1(A) ) ) ) ) ) ) ).
fof(t55_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_quofield(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( D != k1_rlvect_1(A)
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k4_vectsp_1(A,D)) = k4_vectsp_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D)) ) ) ) ) ) ) ).
fof(t56_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_quofield(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( E != k1_rlvect_1(A)
=> k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,k10_group_1(A,D,k4_vectsp_1(A,E))) = k10_group_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),k4_vectsp_1(B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E))) ) ) ) ) ) ) ) ).
fof(t57_quofield,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)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v6_vectsp_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v6_vectsp_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ( v1_quofield(D,A,B)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) )
=> ( v1_quofield(E,B,C)
=> v1_quofield(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ) ) ).
fof(t58_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> v1_quofield(k7_grcat_1(A),A,A) ) ).
fof(d25_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ( r1_quofield(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& v3_quofield(C,A,B) ) ) ) ) ).
fof(d26_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ( r2_quofield(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& v4_quofield(C,A,B) ) ) ) ) ).
fof(d27_quofield,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)
=> k23_quofield(A,B,C) = k1_domain_1(u1_struct_0(A),u1_struct_0(A),B,C) ) ) ) ) ).
fof(d28_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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,u1_struct_0(A),u1_struct_0(k21_quofield(A)))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(k21_quofield(A))) )
=> ( B = k24_quofield(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(k21_quofield(A)),B,C) = k12_quofield(A,k23_quofield(A,C,k2_group_1(A))) ) ) ) ) ).
fof(t59_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> v1_quofield(k24_quofield(A),A,k21_quofield(A)) ) ).
fof(t60_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> v3_quofield(k24_quofield(A),A,k21_quofield(A)) ) ).
fof(t61_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> r1_quofield(A,k21_quofield(A)) ) ).
fof(t62_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> r2_quofield(A,k21_quofield(A)) ) ).
fof(t63_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> r2_quofield(k21_quofield(k21_quofield(A)),k21_quofield(A)) ) ).
fof(d29_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ( r3_quofield(A,B,C)
<=> ( v3_quofield(C,A,B)
& ! [D] :
( ( ~ v3_struct_0(D)
& v3_rlvect_1(D)
& v4_rlvect_1(D)
& v5_rlvect_1(D)
& v6_rlvect_1(D)
& v4_group_1(D)
& v7_group_1(D)
& v7_vectsp_1(D)
& v8_vectsp_1(D)
& v9_vectsp_1(D)
& ~ v10_vectsp_1(D)
& l3_vectsp_1(D) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(D))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(D)) )
=> ~ ( v3_quofield(E,A,D)
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(B),u1_struct_0(D))
& m2_relset_1(F,u1_struct_0(B),u1_struct_0(D)) )
=> ~ ( v1_quofield(F,B,D)
& k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(D),C,F) = E
& ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(B),u1_struct_0(D))
& m2_relset_1(G,u1_struct_0(B),u1_struct_0(D)) )
=> ( ( v1_quofield(G,B,D)
& k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(D),C,G) = E )
=> G = F ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t64_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& 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] :
( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B)
& ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& r3_quofield(A,B,C) ) ) ) ).
fof(t65_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v4_group_1(B)
& v7_group_1(B)
& v7_vectsp_1(B)
& v8_vectsp_1(B)
& v9_vectsp_1(B)
& ~ v10_vectsp_1(B)
& l3_vectsp_1(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v4_group_1(C)
& v7_group_1(C)
& v7_vectsp_1(C)
& v8_vectsp_1(C)
& v9_vectsp_1(C)
& ~ v10_vectsp_1(C)
& l3_vectsp_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(C))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) )
=> ( ( r3_quofield(A,B,D)
& r3_quofield(A,C,E) )
=> r2_quofield(B,C) ) ) ) ) ) ) ).
fof(symmetry_r2_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& l3_vectsp_1(B) )
=> ( r2_quofield(A,B)
=> r2_quofield(B,A) ) ) ).
fof(dt_k1_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> m1_subset_1(k1_quofield(A),k1_zfmisc_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)))) ) ).
fof(dt_k2_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> m1_subset_1(k2_quofield(A,B),u1_struct_0(A)) ) ).
fof(redefinition_k2_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> k2_quofield(A,B) = k1_mcart_1(B) ) ).
fof(dt_k3_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> m1_subset_1(k3_quofield(A,B),u1_struct_0(A)) ) ).
fof(redefinition_k3_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> k3_quofield(A,B) = k2_mcart_1(B) ) ).
fof(dt_k4_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> m2_subset_1(k4_quofield(A,B,C),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A)) ) ).
fof(dt_k5_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> m2_subset_1(k5_quofield(A,B,C),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A)) ) ).
fof(dt_k6_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> m2_subset_1(k6_quofield(A,B,C),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A)) ) ).
fof(commutativity_k6_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> k6_quofield(A,B,C) = k6_quofield(A,C,B) ) ).
fof(redefinition_k6_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> k6_quofield(A,B,C) = k4_quofield(A,B,C) ) ).
fof(dt_k7_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> m2_subset_1(k7_quofield(A,B,C),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_quofield(A)) ) ).
fof(commutativity_k7_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> k7_quofield(A,B,C) = k7_quofield(A,C,B) ) ).
fof(redefinition_k7_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A))
& m1_subset_1(C,k1_quofield(A)) )
=> k7_quofield(A,B,C) = k5_quofield(A,B,C) ) ).
fof(dt_k8_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> m1_subset_1(k8_quofield(A,B),k1_zfmisc_1(k1_quofield(A))) ) ).
fof(dt_k9_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A) )
=> m1_subset_1(k9_quofield(A),k1_zfmisc_1(k1_zfmisc_1(k1_quofield(A)))) ) ).
fof(dt_k10_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k9_quofield(A))
& m1_subset_1(C,k9_quofield(A)) )
=> m2_subset_1(k10_quofield(A,B,C),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ).
fof(dt_k11_quofield,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k9_quofield(A))
& m1_subset_1(C,k9_quofield(A)) )
=> m2_subset_1(k11_quofield(A,B,C),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ).
fof(dt_k12_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> m2_subset_1(k12_quofield(A,B),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ).
fof(redefinition_k12_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v10_vectsp_1(A)
& l2_vectsp_1(A)
& m1_subset_1(B,k1_quofield(A)) )
=> k12_quofield(A,B) = k8_quofield(A,B) ) ).
fof(dt_k13_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> m2_subset_1(k13_quofield(A),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ).
fof(dt_k14_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> m2_subset_1(k14_quofield(A),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ).
fof(dt_k15_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k9_quofield(A)) )
=> m2_subset_1(k15_quofield(A,B),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ).
fof(dt_k16_quofield,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A)
& m1_subset_1(B,k9_quofield(A)) )
=> m2_subset_1(k16_quofield(A,B),k1_zfmisc_1(k1_quofield(A)),k9_quofield(A)) ) ).
fof(dt_k17_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k17_quofield(A))
& v1_funct_2(k17_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A))
& m2_relset_1(k17_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) ) ) ).
fof(dt_k18_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k18_quofield(A))
& v1_funct_2(k18_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A))
& m2_relset_1(k18_quofield(A),k2_zfmisc_1(k9_quofield(A),k9_quofield(A)),k9_quofield(A)) ) ) ).
fof(dt_k19_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k19_quofield(A))
& v1_funct_2(k19_quofield(A),k9_quofield(A),k9_quofield(A))
& m2_relset_1(k19_quofield(A),k9_quofield(A),k9_quofield(A)) ) ) ).
fof(dt_k20_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k20_quofield(A))
& v1_funct_2(k20_quofield(A),k9_quofield(A),k9_quofield(A))
& m2_relset_1(k20_quofield(A),k9_quofield(A),k9_quofield(A)) ) ) ).
fof(dt_k21_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( v3_vectsp_1(k21_quofield(A))
& l3_vectsp_1(k21_quofield(A)) ) ) ).
fof(dt_k22_quofield,axiom,
! [A,B,C] :
( ( ~ 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)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k22_quofield(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k23_quofield,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)) )
=> m1_subset_1(k23_quofield(A,B,C),k1_quofield(A)) ) ).
fof(dt_k24_quofield,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v2_vectsp_2(A)
& l3_vectsp_1(A) )
=> ( v1_funct_1(k24_quofield(A))
& v1_funct_2(k24_quofield(A),u1_struct_0(A),u1_struct_0(k21_quofield(A)))
& m2_relset_1(k24_quofield(A),u1_struct_0(A),u1_struct_0(k21_quofield(A))) ) ) ).
%------------------------------------------------------------------------------