SET007 Axioms: SET007+213.ax
%------------------------------------------------------------------------------
% File : SET007+213 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Abelian Groups, Fields and Vector Spaces
% 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 : vectsp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 178 ( 76 unt; 0 def)
% Number of atoms : 945 ( 140 equ)
% Maximal formula atoms : 23 ( 5 avg)
% Number of connectives : 885 ( 118 ~; 5 |; 524 &)
% ( 17 <=>; 221 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 39 ( 37 usr; 1 prp; 0-3 aty)
% Number of functors : 35 ( 35 usr; 7 con; 0-6 aty)
% Number of variables : 268 ( 248 !; 20 ?)
% 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_vectsp_1,axiom,
( ~ v3_struct_0(k1_vectsp_1)
& v1_rlvect_1(k1_vectsp_1) ) ).
fof(fc2_vectsp_1,axiom,
( ~ v3_struct_0(k1_vectsp_1)
& v1_rlvect_1(k1_vectsp_1)
& v3_rlvect_1(k1_vectsp_1)
& v4_rlvect_1(k1_vectsp_1)
& v5_rlvect_1(k1_vectsp_1)
& v6_rlvect_1(k1_vectsp_1) ) ).
fof(rc1_vectsp_1,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v1_rlvect_1(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A) ) ).
fof(rc2_vectsp_1,axiom,
? [A] :
( l1_vectsp_1(A)
& v1_vectsp_1(A) ) ).
fof(rc3_vectsp_1,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v1_vectsp_1(A) ) ).
fof(rc4_vectsp_1,axiom,
? [A] :
( l2_vectsp_1(A)
& v2_vectsp_1(A) ) ).
fof(rc5_vectsp_1,axiom,
? [A] :
( l2_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_vectsp_1(A) ) ).
fof(rc6_vectsp_1,axiom,
? [A] :
( l3_vectsp_1(A)
& v3_vectsp_1(A) ) ).
fof(rc7_vectsp_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_vectsp_1(A) ) ).
fof(fc3_vectsp_1,axiom,
( ~ v3_struct_0(k3_vectsp_1)
& v3_vectsp_1(k3_vectsp_1) ) ).
fof(fc4_vectsp_1,axiom,
( ~ v3_struct_0(k3_vectsp_1)
& v2_group_1(k3_vectsp_1)
& v3_vectsp_1(k3_vectsp_1) ) ).
fof(fc5_vectsp_1,axiom,
( ~ v3_struct_0(k3_vectsp_1)
& v3_rlvect_1(k3_vectsp_1)
& v4_rlvect_1(k3_vectsp_1)
& v5_rlvect_1(k3_vectsp_1)
& v6_rlvect_1(k3_vectsp_1)
& v2_group_1(k3_vectsp_1)
& v4_group_1(k3_vectsp_1)
& v7_group_1(k3_vectsp_1)
& v3_vectsp_1(k3_vectsp_1)
& v6_vectsp_1(k3_vectsp_1)
& v7_vectsp_1(k3_vectsp_1)
& v8_vectsp_1(k3_vectsp_1)
& v9_vectsp_1(k3_vectsp_1)
& ~ v10_vectsp_1(k3_vectsp_1) ) ).
fof(cc1_vectsp_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v7_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A) ) ) ) ).
fof(cc2_vectsp_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v7_vectsp_1(A) ) ) ) ).
fof(cc3_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_group_1(A) )
=> ( ~ v3_struct_0(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A) ) ) ) ).
fof(cc4_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v2_group_1(A) ) ) ) ).
fof(rc8_vectsp_1,axiom,
? [A] :
( l1_group_1(A)
& ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A) ) ).
fof(rc9_vectsp_1,axiom,
? [A] :
( l1_vectsp_1(A)
& ~ v3_struct_0(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v6_vectsp_1(A)
& v8_vectsp_1(A) ) ).
fof(rc10_vectsp_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A) ) ).
fof(rc11_vectsp_1,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l4_vectsp_1(B,A)
& v11_vectsp_1(B,A) ) ) ).
fof(rc12_vectsp_1,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] :
( l4_vectsp_1(B,A)
& ~ v3_struct_0(B)
& v11_vectsp_1(B,A) ) ) ).
fof(fc6_vectsp_1,axiom,
! [A,B,C,D,E] :
( ( l1_struct_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m1_relset_1(C,k2_zfmisc_1(B,B),B)
& m1_subset_1(D,B)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),B),B)
& m1_relset_1(E,k2_zfmisc_1(u1_struct_0(A),B),B) )
=> ( ~ v3_struct_0(g4_vectsp_1(A,B,C,D,E))
& v11_vectsp_1(g4_vectsp_1(A,B,C,D,E),A) ) ) ).
fof(rc13_vectsp_1,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) )
=> ? [B] :
( l4_vectsp_1(B,A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v11_vectsp_1(B,A)
& v12_vectsp_1(B,A) ) ) ).
fof(cc5_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> ( ( ~ v3_struct_0(A)
& v7_group_1(A)
& v8_vectsp_1(A) )
=> ( ~ v3_struct_0(A)
& v6_vectsp_1(A) ) ) ) ).
fof(rc14_vectsp_1,axiom,
? [A] :
( l1_rlvect_1(A)
& ~ v3_struct_0(A)
& v13_vectsp_1(A) ) ).
fof(rc15_vectsp_1,axiom,
? [A] :
( l3_vectsp_1(A)
& ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v2_group_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v3_vectsp_1(A)
& v4_vectsp_1(A)
& v5_vectsp_1(A)
& v6_vectsp_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& v13_vectsp_1(A) ) ).
fof(d1_vectsp_1,axiom,
$true ).
fof(d2_vectsp_1,axiom,
$true ).
fof(d3_vectsp_1,axiom,
$true ).
fof(d4_vectsp_1,axiom,
$true ).
fof(d5_vectsp_1,axiom,
$true ).
fof(d6_vectsp_1,axiom,
k1_vectsp_1 = g1_rlvect_1(k1_numbers,k33_binop_2,np__0) ).
fof(t1_vectsp_1,axiom,
$true ).
fof(t2_vectsp_1,axiom,
$true ).
fof(t3_vectsp_1,axiom,
$true ).
fof(t4_vectsp_1,axiom,
$true ).
fof(t5_vectsp_1,axiom,
$true ).
fof(t6_vectsp_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k1_vectsp_1))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k1_vectsp_1))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k1_vectsp_1))
=> ( k4_rlvect_1(k1_vectsp_1,A,B) = k4_rlvect_1(k1_vectsp_1,B,A)
& k4_rlvect_1(k1_vectsp_1,k4_rlvect_1(k1_vectsp_1,A,B),C) = k4_rlvect_1(k1_vectsp_1,A,k4_rlvect_1(k1_vectsp_1,B,C))
& k4_rlvect_1(k1_vectsp_1,A,k1_rlvect_1(k1_vectsp_1)) = A
& k4_rlvect_1(k1_vectsp_1,A,k5_rlvect_1(k1_vectsp_1,A)) = k1_rlvect_1(k1_vectsp_1) ) ) ) ) ).
fof(t7_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k2_rlvect_1(A,B,C) = k2_rlvect_1(A,C,B)
& k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D))
& k2_rlvect_1(A,B,k1_rlvect_1(A)) = B
& ? [E] :
( m1_subset_1(E,u1_struct_0(A))
& k2_rlvect_1(A,B,E) = k1_rlvect_1(A) ) ) ) ) )
<=> ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) ) ) ) ).
fof(d7_vectsp_1,axiom,
$true ).
fof(d8_vectsp_1,axiom,
$true ).
fof(d9_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> k2_vectsp_1(A) = u1_vectsp_1(A) ) ).
fof(d10_vectsp_1,axiom,
$true ).
fof(d11_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v4_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))
=> k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) ) ).
fof(d12_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v5_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))
=> k1_group_1(A,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ).
fof(d13_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v6_vectsp_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k2_group_1(A)) = B ) ) ) ).
fof(d14_vectsp_1,axiom,
$true ).
fof(d15_vectsp_1,axiom,
k3_vectsp_1 = g3_vectsp_1(k1_numbers,k33_binop_2,k35_binop_2,np__1,np__0) ).
fof(d16_vectsp_1,axiom,
$true ).
fof(d17_vectsp_1,axiom,
$true ).
fof(d18_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ( v7_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))
=> ( k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D))
& k1_group_1(A,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) ) ) ) ).
fof(d19_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_vectsp_1(A) )
=> ( v8_vectsp_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k2_group_1(A),B) = B ) ) ) ).
fof(d20_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( v9_vectsp_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ~ ( B != k1_rlvect_1(A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,C) != k2_group_1(A) ) ) ) ) ) ).
fof(d21_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_vectsp_1(A) )
=> ( v10_vectsp_1(A)
<=> k1_rlvect_1(A) = k2_group_1(A) ) ) ).
fof(t8_vectsp_1,axiom,
$true ).
fof(t9_vectsp_1,axiom,
$true ).
fof(t10_vectsp_1,axiom,
$true ).
fof(t11_vectsp_1,axiom,
$true ).
fof(t12_vectsp_1,axiom,
$true ).
fof(t13_vectsp_1,axiom,
$true ).
fof(t14_vectsp_1,axiom,
$true ).
fof(t15_vectsp_1,axiom,
$true ).
fof(t16_vectsp_1,axiom,
$true ).
fof(t17_vectsp_1,axiom,
$true ).
fof(t18_vectsp_1,axiom,
$true ).
fof(t19_vectsp_1,axiom,
$true ).
fof(t20_vectsp_1,axiom,
k2_group_1(k3_vectsp_1) = np__1 ).
fof(t21_vectsp_1,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k3_vectsp_1))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k3_vectsp_1))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k3_vectsp_1))
=> ( k4_rlvect_1(k3_vectsp_1,A,B) = k4_rlvect_1(k3_vectsp_1,B,A)
& k4_rlvect_1(k3_vectsp_1,k4_rlvect_1(k3_vectsp_1,A,B),C) = k4_rlvect_1(k3_vectsp_1,A,k4_rlvect_1(k3_vectsp_1,B,C))
& k4_rlvect_1(k3_vectsp_1,A,k1_rlvect_1(k3_vectsp_1)) = A
& k4_rlvect_1(k3_vectsp_1,A,k5_rlvect_1(k3_vectsp_1,A)) = k1_rlvect_1(k3_vectsp_1)
& k10_group_1(k3_vectsp_1,A,B) = k10_group_1(k3_vectsp_1,B,A)
& k10_group_1(k3_vectsp_1,k10_group_1(k3_vectsp_1,A,B),C) = k10_group_1(k3_vectsp_1,A,k10_group_1(k3_vectsp_1,B,C))
& k10_group_1(k3_vectsp_1,k2_group_1(k3_vectsp_1),A) = A
& ~ ( A != k1_rlvect_1(k3_vectsp_1)
& ! [D] :
( m1_subset_1(D,u1_struct_0(k3_vectsp_1))
=> k10_group_1(k3_vectsp_1,A,D) != k2_group_1(k3_vectsp_1) ) )
& k10_group_1(k3_vectsp_1,A,k4_rlvect_1(k3_vectsp_1,B,C)) = k4_rlvect_1(k3_vectsp_1,k10_group_1(k3_vectsp_1,A,B),k10_group_1(k3_vectsp_1,A,C))
& k10_group_1(k3_vectsp_1,k4_rlvect_1(k3_vectsp_1,B,C),A) = k4_rlvect_1(k3_vectsp_1,k10_group_1(k3_vectsp_1,B,A),k10_group_1(k3_vectsp_1,C,A)) ) ) ) ) ).
fof(t22_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(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))
=> ( ~ ( B != k1_rlvect_1(A)
& ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> k1_group_1(A,B,E) != k2_group_1(A) ) )
& k1_group_1(A,B,k2_rlvect_1(A,C,D)) = k2_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D))
& k1_group_1(A,k2_rlvect_1(A,C,D),B) = k2_rlvect_1(A,k1_group_1(A,C,B),k1_group_1(A,D,B)) ) ) ) )
<=> ( ~ v3_struct_0(A)
& v7_vectsp_1(A)
& v9_vectsp_1(A)
& l3_vectsp_1(A) ) ) ) ).
fof(t23_vectsp_1,axiom,
$true ).
fof(t24_vectsp_1,axiom,
$true ).
fof(t25_vectsp_1,axiom,
$true ).
fof(t26_vectsp_1,axiom,
$true ).
fof(t27_vectsp_1,axiom,
$true ).
fof(t28_vectsp_1,axiom,
$true ).
fof(t29_vectsp_1,axiom,
$true ).
fof(t30_vectsp_1,axiom,
$true ).
fof(t31_vectsp_1,axiom,
$true ).
fof(t32_vectsp_1,axiom,
$true ).
fof(t33_vectsp_1,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))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k10_group_1(A,B,C) = k10_group_1(A,B,D)
=> ( B = k1_rlvect_1(A)
| C = D ) ) ) ) ) ) ).
fof(d22_vectsp_1,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))
=> ( B != k1_rlvect_1(A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k4_vectsp_1(A,B)
<=> k10_group_1(A,B,C) = k2_group_1(A) ) ) ) ) ) ).
fof(d23_vectsp_1,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))
=> k5_vectsp_1(A,B,C) = k10_group_1(A,B,k4_vectsp_1(A,C)) ) ) ) ).
fof(t34_vectsp_1,axiom,
$true ).
fof(t35_vectsp_1,axiom,
$true ).
fof(t36_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,B,k1_rlvect_1(A)) = k1_rlvect_1(A) ) ) ).
fof(t37_vectsp_1,axiom,
$true ).
fof(t38_vectsp_1,axiom,
$true ).
fof(t39_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_group_1(A,k1_rlvect_1(A),B) = k1_rlvect_1(A) ) ) ).
fof(t40_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,B,k5_rlvect_1(A,C)) = k5_rlvect_1(A,k1_group_1(A,B,C)) ) ) ) ).
fof(t41_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,k5_rlvect_1(A,B),C) = k5_rlvect_1(A,k1_group_1(A,B,C)) ) ) ) ).
fof(t42_vectsp_1,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] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_group_1(A,k5_rlvect_1(A,B),k5_rlvect_1(A,C)) = k1_group_1(A,B,C) ) ) ) ).
fof(t43_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_group_1(A,B,k6_rlvect_1(A,C,D)) = k6_rlvect_1(A,k1_group_1(A,B,C),k1_group_1(A,B,D)) ) ) ) ) ).
fof(t44_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& 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))
=> ( k10_group_1(A,B,C) = k1_rlvect_1(A)
<=> ( B = k1_rlvect_1(A)
| C = k1_rlvect_1(A) ) ) ) ) ) ).
fof(t45_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v5_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( 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))
=> k1_group_1(A,k6_rlvect_1(A,B,C),D) = k6_rlvect_1(A,k1_group_1(A,B,D),k1_group_1(A,C,D)) ) ) ) ) ).
fof(d24_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k6_vectsp_1(A,B,C,D) = k2_binop_1(u1_struct_0(A),u1_struct_0(B),u1_struct_0(B),u2_vectsp_1(A,B),C,D) ) ) ) ) ).
fof(d25_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
=> ( B = k7_vectsp_1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = k5_rlvect_1(A,C) ) ) ) ) ).
fof(d26_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_vectsp_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ( v12_vectsp_1(B,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(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( k6_vectsp_1(A,B,C,k2_rlvect_1(B,E,F)) = k2_rlvect_1(B,k6_vectsp_1(A,B,C,E),k6_vectsp_1(A,B,C,F))
& k6_vectsp_1(A,B,k2_rlvect_1(A,C,D),E) = k2_rlvect_1(B,k6_vectsp_1(A,B,C,E),k6_vectsp_1(A,B,D,E))
& k6_vectsp_1(A,B,k1_group_1(A,C,D),E) = k6_vectsp_1(A,B,C,k6_vectsp_1(A,B,D,E))
& k6_vectsp_1(A,B,k2_group_1(A),E) = E ) ) ) ) ) ) ) ) ).
fof(t46_vectsp_1,axiom,
$true ).
fof(t47_vectsp_1,axiom,
$true ).
fof(t48_vectsp_1,axiom,
$true ).
fof(t49_vectsp_1,axiom,
$true ).
fof(t50_vectsp_1,axiom,
$true ).
fof(t51_vectsp_1,axiom,
$true ).
fof(t52_vectsp_1,axiom,
$true ).
fof(t53_vectsp_1,axiom,
$true ).
fof(t54_vectsp_1,axiom,
$true ).
fof(t55_vectsp_1,axiom,
$true ).
fof(t56_vectsp_1,axiom,
$true ).
fof(t57_vectsp_1,axiom,
$true ).
fof(t58_vectsp_1,axiom,
$true ).
fof(t59_vectsp_1,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) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ( k6_vectsp_1(A,C,k1_rlvect_1(A),D) = k1_rlvect_1(C)
& k6_vectsp_1(A,C,k5_rlvect_1(A,k2_group_1(A)),D) = k5_rlvect_1(C,D)
& k6_vectsp_1(A,C,B,k1_rlvect_1(C)) = k1_rlvect_1(C) ) ) ) ) ) ).
fof(t60_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ( k6_vectsp_1(A,C,B,D) = k1_rlvect_1(C)
<=> ( B = k1_rlvect_1(A)
| D = k1_rlvect_1(C) ) ) ) ) ) ) ).
fof(t61_vectsp_1,axiom,
$true ).
fof(t62_vectsp_1,axiom,
$true ).
fof(t63_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_rlvect_1(A,B,C) = k1_rlvect_1(A)
<=> k5_rlvect_1(A,B) = C ) ) ) ) ).
fof(t64_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_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))
=> ( k5_rlvect_1(A,k2_rlvect_1(A,C,D)) = k6_rlvect_1(A,k5_rlvect_1(A,D),C)
& k5_rlvect_1(A,k2_rlvect_1(A,D,k5_rlvect_1(A,C))) = k6_rlvect_1(A,C,D)
& k5_rlvect_1(A,k6_rlvect_1(A,C,D)) = k2_rlvect_1(A,D,k5_rlvect_1(A,C))
& k5_rlvect_1(A,k6_rlvect_1(A,k5_rlvect_1(A,C),D)) = k2_rlvect_1(A,D,C)
& k6_rlvect_1(A,B,k2_rlvect_1(A,D,C)) = k6_rlvect_1(A,k6_rlvect_1(A,B,C),D) ) ) ) ) ) ).
fof(t65_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k6_rlvect_1(A,k1_rlvect_1(A),B) = k5_rlvect_1(A,B)
& k6_rlvect_1(A,B,k1_rlvect_1(A)) = B ) ) ) ).
fof(t66_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( k2_rlvect_1(A,B,k5_rlvect_1(A,C)) = k1_rlvect_1(A)
=> B = C )
& ( B = C
=> k2_rlvect_1(A,B,k5_rlvect_1(A,C)) = k1_rlvect_1(A) )
& ( k6_rlvect_1(A,B,C) = k1_rlvect_1(A)
=> B = C )
& ( B = C
=> k6_rlvect_1(A,B,C) = k1_rlvect_1(A) ) ) ) ) ) ).
fof(t67_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( ~ v3_struct_0(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v12_vectsp_1(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ( B != k1_rlvect_1(A)
=> k6_vectsp_1(A,C,k4_vectsp_1(A,B),k6_vectsp_1(A,C,B,D)) = D ) ) ) ) ) ).
fof(t68_vectsp_1,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) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(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))
=> ( k5_rlvect_1(B,k6_vectsp_1(A,B,C,D)) = k6_vectsp_1(A,B,k5_rlvect_1(A,C),D)
& k6_rlvect_1(B,E,k6_vectsp_1(A,B,C,D)) = k2_rlvect_1(B,E,k6_vectsp_1(A,B,k5_rlvect_1(A,C),D)) ) ) ) ) ) ) ).
fof(t69_vectsp_1,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)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k6_vectsp_1(A,B,C,k5_rlvect_1(B,D)) = k5_rlvect_1(B,k6_vectsp_1(A,B,C,D)) ) ) ) ) ).
fof(t70_vectsp_1,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)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(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,k6_rlvect_1(B,D,E)) = k6_rlvect_1(B,k6_vectsp_1(A,B,C,D),k6_vectsp_1(A,B,C,E)) ) ) ) ) ) ).
fof(t71_vectsp_1,axiom,
$true ).
fof(t72_vectsp_1,axiom,
$true ).
fof(t73_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> k4_vectsp_1(A,k4_vectsp_1(A,B)) = B ) ) ) ).
fof(t74_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B != k1_rlvect_1(A)
=> ( k4_vectsp_1(A,B) != k1_rlvect_1(A)
& k5_rlvect_1(A,k4_vectsp_1(A,B)) != k1_rlvect_1(A) ) ) ) ) ).
fof(t75_vectsp_1,axiom,
$true ).
fof(t76_vectsp_1,axiom,
$true ).
fof(t77_vectsp_1,axiom,
$true ).
fof(t78_vectsp_1,axiom,
k4_rlvect_1(k3_vectsp_1,k2_group_1(k3_vectsp_1),k2_group_1(k3_vectsp_1)) != k1_rlvect_1(k3_vectsp_1) ).
fof(d27_vectsp_1,axiom,
$true ).
fof(d28_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v13_vectsp_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k2_rlvect_1(A,B,B) = k1_rlvect_1(A)
=> B = k1_rlvect_1(A) ) ) ) ) ).
fof(d29_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_vectsp_1(A)
& l3_vectsp_1(A) )
=> ( v13_vectsp_1(A)
<=> k2_rlvect_1(A,k2_group_1(A),k2_group_1(A)) != k1_rlvect_1(A) ) ) ).
fof(t79_vectsp_1,axiom,
$true ).
fof(t80_vectsp_1,axiom,
$true ).
fof(t81_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_rlvect_1(A,k6_rlvect_1(A,B,C)) = k6_rlvect_1(A,C,B) ) ) ) ).
fof(t82_vectsp_1,axiom,
$true ).
fof(t83_vectsp_1,axiom,
$true ).
fof(t84_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k6_rlvect_1(A,B,C) = k1_rlvect_1(A)
=> B = C ) ) ) ) ).
fof(t85_vectsp_1,axiom,
$true ).
fof(t86_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k5_rlvect_1(A,B) = k1_rlvect_1(A)
=> B = k1_rlvect_1(A) ) ) ) ).
fof(t87_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k6_rlvect_1(A,B,C) = k1_rlvect_1(A)
=> k6_rlvect_1(A,C,B) = k1_rlvect_1(A) ) ) ) ) ).
fof(t88_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& 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))
=> ( ( k6_rlvect_1(A,k10_group_1(A,B,D),C) = k1_rlvect_1(A)
=> ( B = k1_rlvect_1(A)
| D = k10_group_1(A,C,k4_vectsp_1(A,B)) ) )
& ( k6_rlvect_1(A,C,k10_group_1(A,D,B)) = k1_rlvect_1(A)
=> ( B = k1_rlvect_1(A)
| D = k10_group_1(A,C,k4_vectsp_1(A,B)) ) ) ) ) ) ) ) ).
fof(t89_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_rlvect_1(A,B,C) = k5_rlvect_1(A,k2_rlvect_1(A,k5_rlvect_1(A,C),k5_rlvect_1(A,B))) ) ) ) ).
fof(t90_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_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))
=> k6_rlvect_1(A,k2_rlvect_1(A,C,B),k2_rlvect_1(A,D,B)) = k6_rlvect_1(A,C,D) ) ) ) ) ).
fof(t91_vectsp_1,axiom,
$true ).
fof(t92_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_rlvect_1(A,k2_rlvect_1(A,k5_rlvect_1(A,B),C)) = k2_rlvect_1(A,k5_rlvect_1(A,C),B) ) ) ) ).
fof(t93_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_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))
=> k6_rlvect_1(A,k6_rlvect_1(A,B,C),D) = k6_rlvect_1(A,k6_rlvect_1(A,B,D),C) ) ) ) ) ).
fof(t94_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& l1_rlvect_1(A) )
=> ( ~ v3_struct_0(g1_group_1(u1_struct_0(A),u1_rlvect_1(A)))
& v3_group_1(g1_group_1(u1_struct_0(A),u1_rlvect_1(A)))
& v4_group_1(g1_group_1(u1_struct_0(A),u1_rlvect_1(A)))
& v7_group_1(g1_group_1(u1_struct_0(A),u1_rlvect_1(A)))
& l1_group_1(g1_group_1(u1_struct_0(A),u1_rlvect_1(A))) ) ) ).
fof(dt_l1_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> l1_group_1(A) ) ).
fof(existence_l1_vectsp_1,axiom,
? [A] : l1_vectsp_1(A) ).
fof(dt_l2_vectsp_1,axiom,
! [A] :
( l2_vectsp_1(A)
=> ( l1_vectsp_1(A)
& l2_struct_0(A) ) ) ).
fof(existence_l2_vectsp_1,axiom,
? [A] : l2_vectsp_1(A) ).
fof(dt_l3_vectsp_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( l1_rlvect_1(A)
& l2_vectsp_1(A) ) ) ).
fof(existence_l3_vectsp_1,axiom,
? [A] : l3_vectsp_1(A) ).
fof(dt_l4_vectsp_1,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l4_vectsp_1(B,A)
=> l1_rlvect_1(B) ) ) ).
fof(existence_l4_vectsp_1,axiom,
! [A] :
( l1_struct_0(A)
=> ? [B] : l4_vectsp_1(B,A) ) ).
fof(abstractness_v1_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> ( v1_vectsp_1(A)
=> A = g1_vectsp_1(u1_struct_0(A),u1_group_1(A),u1_vectsp_1(A)) ) ) ).
fof(abstractness_v2_vectsp_1,axiom,
! [A] :
( l2_vectsp_1(A)
=> ( v2_vectsp_1(A)
=> A = g2_vectsp_1(u1_struct_0(A),u1_group_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ) ).
fof(abstractness_v3_vectsp_1,axiom,
! [A] :
( l3_vectsp_1(A)
=> ( v3_vectsp_1(A)
=> A = g3_vectsp_1(u1_struct_0(A),u1_rlvect_1(A),u1_group_1(A),u1_vectsp_1(A),u2_struct_0(A)) ) ) ).
fof(abstractness_v11_vectsp_1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l4_vectsp_1(B,A) )
=> ( v11_vectsp_1(B,A)
=> B = g4_vectsp_1(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u2_vectsp_1(A,B)) ) ) ).
fof(dt_k1_vectsp_1,axiom,
( v1_rlvect_1(k1_vectsp_1)
& l1_rlvect_1(k1_vectsp_1) ) ).
fof(dt_k2_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> m1_subset_1(k2_vectsp_1(A),u1_struct_0(A)) ) ).
fof(dt_k3_vectsp_1,axiom,
( v3_vectsp_1(k3_vectsp_1)
& l3_vectsp_1(k3_vectsp_1) ) ).
fof(dt_k4_vectsp_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v4_group_1(A)
& v7_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& l3_vectsp_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k4_vectsp_1(A,B),u1_struct_0(A)) ) ).
fof(dt_k5_vectsp_1,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(k5_vectsp_1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k6_vectsp_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,u1_struct_0(B)) )
=> m1_subset_1(k6_vectsp_1(A,B,C,D),u1_struct_0(B)) ) ).
fof(dt_k7_vectsp_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ( v1_funct_1(k7_vectsp_1(A))
& v1_funct_2(k7_vectsp_1(A),u1_struct_0(A),u1_struct_0(A))
& m2_relset_1(k7_vectsp_1(A),u1_struct_0(A),u1_struct_0(A)) ) ) ).
fof(dt_u1_vectsp_1,axiom,
! [A] :
( l1_vectsp_1(A)
=> m1_subset_1(u1_vectsp_1(A),u1_struct_0(A)) ) ).
fof(dt_u2_vectsp_1,axiom,
! [A,B] :
( ( l1_struct_0(A)
& l4_vectsp_1(B,A) )
=> ( v1_funct_1(u2_vectsp_1(A,B))
& v1_funct_2(u2_vectsp_1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B))
& m2_relset_1(u2_vectsp_1(A,B),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)),u1_struct_0(B)) ) ) ).
fof(dt_g1_vectsp_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,A) )
=> ( v1_vectsp_1(g1_vectsp_1(A,B,C))
& l1_vectsp_1(g1_vectsp_1(A,B,C)) ) ) ).
fof(free_g1_vectsp_1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,A) )
=> ! [D,E,F] :
( g1_vectsp_1(A,B,C) = g1_vectsp_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(dt_g2_vectsp_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,A)
& m1_subset_1(D,A) )
=> ( v2_vectsp_1(g2_vectsp_1(A,B,C,D))
& l2_vectsp_1(g2_vectsp_1(A,B,C,D)) ) ) ).
fof(free_g2_vectsp_1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_subset_1(C,A)
& m1_subset_1(D,A) )
=> ! [E,F,G,H] :
( g2_vectsp_1(A,B,C,D) = g2_vectsp_1(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(dt_g3_vectsp_1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& m1_subset_1(D,A)
& m1_subset_1(E,A) )
=> ( v3_vectsp_1(g3_vectsp_1(A,B,C,D,E))
& l3_vectsp_1(g3_vectsp_1(A,B,C,D,E)) ) ) ).
fof(free_g3_vectsp_1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& m1_subset_1(D,A)
& m1_subset_1(E,A) )
=> ! [F,G,H,I,J] :
( g3_vectsp_1(A,B,C,D,E) = g3_vectsp_1(F,G,H,I,J)
=> ( A = F
& B = G
& C = H
& D = I
& E = J ) ) ) ).
fof(dt_g4_vectsp_1,axiom,
! [A,B,C,D,E] :
( ( 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)
& m1_subset_1(D,B)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),B),B)
& m1_relset_1(E,k2_zfmisc_1(u1_struct_0(A),B),B) )
=> ( v11_vectsp_1(g4_vectsp_1(A,B,C,D,E),A)
& l4_vectsp_1(g4_vectsp_1(A,B,C,D,E),A) ) ) ).
fof(free_g4_vectsp_1,axiom,
! [A,B,C,D,E] :
( ( 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)
& m1_subset_1(D,B)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),B),B)
& m1_relset_1(E,k2_zfmisc_1(u1_struct_0(A),B),B) )
=> ! [F,G,H,I,J] :
( g4_vectsp_1(A,B,C,D,E) = g4_vectsp_1(F,G,H,I,J)
=> ( A = F
& B = G
& C = H
& D = I
& E = J ) ) ) ).
%------------------------------------------------------------------------------