SET007 Axioms: SET007+249.ax
%------------------------------------------------------------------------------
% File : SET007+249 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Linear Combinations in Vector Space
% 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_6 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 113 ( 37 unt; 0 def)
% Number of atoms : 1305 ( 58 equ)
% Maximal formula atoms : 25 ( 11 avg)
% Number of connectives : 1354 ( 162 ~; 1 |; 906 &)
% ( 13 <=>; 272 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-4 aty)
% Number of functors : 38 ( 38 usr; 3 con; 0-4 aty)
% Number of variables : 301 ( 295 !; 6 ?)
% 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(d1_vectsp_6,axiom,
$true ).
fof(d2_vectsp_6,axiom,
$true ).
fof(d3_vectsp_6,axiom,
$true ).
fof(d4_vectsp_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_fraenkel(C,u1_struct_0(B),u1_struct_0(A),k1_fraenkel(u1_struct_0(B),u1_struct_0(A)))
=> ( m1_vectsp_6(C,A,B)
<=> ? [D] :
( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
& ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ( ~ r2_hidden(E,D)
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E) = k1_rlvect_1(A) ) ) ) ) ) ) ) ).
fof(t1_vectsp_6,axiom,
$true ).
fof(t2_vectsp_6,axiom,
$true ).
fof(t3_vectsp_6,axiom,
$true ).
fof(t4_vectsp_6,axiom,
$true ).
fof(t5_vectsp_6,axiom,
$true ).
fof(t6_vectsp_6,axiom,
$true ).
fof(t7_vectsp_6,axiom,
$true ).
fof(t8_vectsp_6,axiom,
$true ).
fof(t9_vectsp_6,axiom,
$true ).
fof(t10_vectsp_6,axiom,
$true ).
fof(t11_vectsp_6,axiom,
$true ).
fof(t12_vectsp_6,axiom,
$true ).
fof(t13_vectsp_6,axiom,
$true ).
fof(t14_vectsp_6,axiom,
$true ).
fof(t15_vectsp_6,axiom,
$true ).
fof(t16_vectsp_6,axiom,
$true ).
fof(t17_vectsp_6,axiom,
$true ).
fof(t18_vectsp_6,axiom,
$true ).
fof(t19_vectsp_6,axiom,
! [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_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)
& v12_vectsp_1(C,B)
& l4_vectsp_1(C,B) )
=> ! [D] :
( m1_vectsp_6(D,B,C)
=> ( r2_hidden(A,k1_vectsp_6(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(C))
& A = E
& k8_funct_2(u1_struct_0(C),u1_struct_0(B),D,E) != k1_rlvect_1(B) ) ) ) ) ) ).
fof(t20_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C) = k1_rlvect_1(A)
<=> ~ r2_hidden(C,k1_vectsp_6(A,B,D)) ) ) ) ) ) ).
fof(d6_vectsp_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_6(C,A,B)
=> ( C = k2_vectsp_6(A,B)
<=> k1_vectsp_6(A,B,C) = k1_xboole_0 ) ) ) ) ).
fof(t21_vectsp_6,axiom,
$true ).
fof(t22_vectsp_6,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)
& v3_rlvect_1(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(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),k2_vectsp_6(A,B),C) = k1_rlvect_1(A) ) ) ) ).
fof(d7_vectsp_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( m2_vectsp_6(D,A,B,C)
<=> r1_tarski(k1_vectsp_6(A,B,D),C) ) ) ) ) ) ).
fof(t23_vectsp_6,axiom,
$true ).
fof(t24_vectsp_6,axiom,
$true ).
fof(t25_vectsp_6,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)
& v3_rlvect_1(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,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( m2_vectsp_6(E,A,B,C)
=> ( r1_tarski(C,D)
=> m2_vectsp_6(E,A,B,D) ) ) ) ) ) ) ).
fof(t26_vectsp_6,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)
& v3_rlvect_1(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,k1_zfmisc_1(u1_struct_0(B)))
=> m2_vectsp_6(k2_vectsp_6(A,B),A,B,C) ) ) ) ).
fof(t27_vectsp_6,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)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_vectsp_6(C,A,B,k1_subset_1(u1_struct_0(B)))
=> C = k2_vectsp_6(A,B) ) ) ) ).
fof(t28_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> m2_vectsp_6(C,A,B,k1_vectsp_6(A,B,C)) ) ) ) ).
fof(d8_vectsp_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(B))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ( E = k3_vectsp_6(A,B,C,D)
<=> ( k3_finseq_1(E) = k3_finseq_1(C)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k1_relat_1(E))
=> k1_funct_1(E,F) = k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,k4_finseq_4(k5_numbers,u1_struct_0(B),C,F)),k4_finseq_4(k5_numbers,u1_struct_0(B),C,F)) ) ) ) ) ) ) ) ) ) ).
fof(t29_vectsp_6,axiom,
$true ).
fof(t30_vectsp_6,axiom,
$true ).
fof(t31_vectsp_6,axiom,
$true ).
fof(t32_vectsp_6,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [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_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)
& v12_vectsp_1(C,B)
& l4_vectsp_1(C,B) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(C))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(C))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(C),u1_struct_0(B))
& m2_relset_1(F,u1_struct_0(C),u1_struct_0(B)) )
=> ( ( r2_hidden(A,k1_relat_1(E))
& D = k1_funct_1(E,A) )
=> k1_funct_1(k3_vectsp_6(B,C,E,F),A) = k6_vectsp_1(B,C,k8_funct_2(u1_struct_0(C),u1_struct_0(B),F,D),D) ) ) ) ) ) ) ) ).
fof(t33_vectsp_6,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)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(C,u1_struct_0(B),u1_struct_0(A)) )
=> k3_vectsp_6(A,B,k6_finseq_1(u1_struct_0(B)),C) = k6_finseq_1(u1_struct_0(B)) ) ) ) ).
fof(t34_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> k3_vectsp_6(A,B,k12_finseq_1(u1_struct_0(B),C),D) = k12_finseq_1(u1_struct_0(B),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),C)) ) ) ) ) ).
fof(t35_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(A)) )
=> k3_vectsp_6(A,B,k2_finseq_4(u1_struct_0(B),C,D),E) = k2_finseq_4(u1_struct_0(B),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C),C),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,D),D)) ) ) ) ) ) ).
fof(t36_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(F,u1_struct_0(B),u1_struct_0(A)) )
=> k3_vectsp_6(A,B,k3_finseq_4(u1_struct_0(B),C,D,E),F) = k3_finseq_4(u1_struct_0(B),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),F,C),C),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),F,D),D),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),F,E),E)) ) ) ) ) ) ) ).
fof(t37_vectsp_6,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)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_finseq_1(C,u1_struct_0(B))
=> ! [D] :
( m2_finseq_1(D,u1_struct_0(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(B),u1_struct_0(A))
& m2_relset_1(E,u1_struct_0(B),u1_struct_0(A)) )
=> k3_vectsp_6(A,B,k8_finseq_1(u1_struct_0(B),C,D),E) = k8_finseq_1(u1_struct_0(B),k3_vectsp_6(A,B,C,E),k3_vectsp_6(A,B,D,E)) ) ) ) ) ) ).
fof(d9_vectsp_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_6(C,A,B)
=> ( ( v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B) )
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( D = k4_vectsp_6(A,B,C)
<=> ? [E] :
( m2_finseq_1(E,u1_struct_0(B))
& v2_funct_1(E)
& k2_relat_1(E) = k1_vectsp_6(A,B,C)
& D = k9_rlvect_1(B,k3_vectsp_6(A,B,E,C)) ) ) ) ) ) ) ) ).
fof(t38_vectsp_6,axiom,
$true ).
fof(t39_vectsp_6,axiom,
$true ).
fof(t40_vectsp_6,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)
& v3_rlvect_1(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,k1_zfmisc_1(u1_struct_0(B)))
=> ( k1_rlvect_1(A) != k2_group_1(A)
=> ( ( C != k1_xboole_0
& v1_vectsp_4(C,A,B) )
<=> ! [D] :
( m2_vectsp_6(D,A,B,C)
=> r2_hidden(k4_vectsp_6(A,B,D),C) ) ) ) ) ) ) ).
fof(t41_vectsp_6,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)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> k4_vectsp_6(A,B,k2_vectsp_6(A,B)) = k1_rlvect_1(B) ) ) ).
fof(t42_vectsp_6,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)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_vectsp_6(C,A,B,k1_subset_1(u1_struct_0(B)))
=> k4_vectsp_6(A,B,C) = k1_rlvect_1(B) ) ) ) ).
fof(t43_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m2_vectsp_6(D,A,B,k1_struct_0(B,C))
=> k4_vectsp_6(A,B,D) = k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),C) ) ) ) ) ).
fof(t44_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( C != D
=> ! [E] :
( m2_vectsp_6(E,A,B,k2_struct_0(B,C,D))
=> k4_vectsp_6(A,B,E) = k4_rlvect_1(B,k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C),C),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,D),D)) ) ) ) ) ) ) ).
fof(t45_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ( k1_vectsp_6(A,B,C) = k1_xboole_0
=> k4_vectsp_6(A,B,C) = k1_rlvect_1(B) ) ) ) ) ).
fof(t46_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( k1_vectsp_6(A,B,D) = k1_struct_0(B,C)
=> k4_vectsp_6(A,B,D) = k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),C) ) ) ) ) ) ).
fof(t47_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> ( k1_vectsp_6(A,B,E) = k2_struct_0(B,C,D)
=> ( C = D
| k4_vectsp_6(A,B,E) = k4_rlvect_1(B,k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C),C),k6_vectsp_1(A,B,k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,D),D)) ) ) ) ) ) ) ) ).
fof(d10_vectsp_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( r1_vectsp_6(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,E) = k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,E) ) ) ) ) ) ) ).
fof(d11_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> ( E = k5_vectsp_6(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,F) = k4_rlvect_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),C,F),k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,F)) ) ) ) ) ) ) ) ).
fof(t48_vectsp_6,axiom,
$true ).
fof(t49_vectsp_6,axiom,
$true ).
fof(t50_vectsp_6,axiom,
$true ).
fof(t51_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> r1_tarski(k1_vectsp_6(A,B,k5_vectsp_6(A,B,C,D)),k4_subset_1(u1_struct_0(B),k1_vectsp_6(A,B,C),k1_vectsp_6(A,B,D))) ) ) ) ) ).
fof(t52_vectsp_6,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)
& v3_rlvect_1(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,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> ( ( m2_vectsp_6(D,A,B,C)
& m2_vectsp_6(E,A,B,C) )
=> m2_vectsp_6(k5_vectsp_6(A,B,D,E),A,B,C) ) ) ) ) ) ) ).
fof(t53_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> r1_vectsp_6(A,B,k5_vectsp_6(A,B,C,D),k5_vectsp_6(A,B,D,C)) ) ) ) ) ).
fof(t54_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> r1_vectsp_6(A,B,k5_vectsp_6(A,B,C,k5_vectsp_6(A,B,D,E)),k5_vectsp_6(A,B,k5_vectsp_6(A,B,C,D),E)) ) ) ) ) ) ).
fof(t55_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ( r1_vectsp_6(A,B,k5_vectsp_6(A,B,C,k2_vectsp_6(A,B)),C)
& r1_vectsp_6(A,B,k5_vectsp_6(A,B,k2_vectsp_6(A,B),C),C) ) ) ) ) ).
fof(d12_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(D,A,B)
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> ( E = k6_vectsp_6(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,F) = k1_group_1(A,C,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,F)) ) ) ) ) ) ) ) ).
fof(t56_vectsp_6,axiom,
$true ).
fof(t57_vectsp_6,axiom,
$true ).
fof(t58_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(D,A,B)
=> r1_tarski(k1_vectsp_6(A,B,k6_vectsp_6(A,B,C,D)),k1_vectsp_6(A,B,D)) ) ) ) ) ).
fof(t59_vectsp_6,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] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(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_vectsp_6(D,A,B)
=> ( C != k1_rlvect_1(A)
=> k1_vectsp_6(A,B,k6_vectsp_6(A,B,C,D)) = k1_vectsp_6(A,B,D) ) ) ) ) ) ).
fof(t60_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> r1_vectsp_6(A,B,k6_vectsp_6(A,B,k1_rlvect_1(A),C),k2_vectsp_6(A,B)) ) ) ) ).
fof(t61_vectsp_6,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)
& v3_rlvect_1(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,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> ( m2_vectsp_6(E,A,B,D)
=> m2_vectsp_6(k6_vectsp_6(A,B,C,E),A,B,D) ) ) ) ) ) ) ).
fof(t62_vectsp_6,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)
& v3_rlvect_1(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(A))
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> r1_vectsp_6(A,B,k6_vectsp_6(A,B,k4_rlvect_1(A,C,D),E),k5_vectsp_6(A,B,k6_vectsp_6(A,B,C,E),k6_vectsp_6(A,B,D,E))) ) ) ) ) ) ).
fof(t63_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(D,A,B)
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> r1_vectsp_6(A,B,k6_vectsp_6(A,B,C,k5_vectsp_6(A,B,D,E)),k5_vectsp_6(A,B,k6_vectsp_6(A,B,C,D),k6_vectsp_6(A,B,C,E))) ) ) ) ) ) ).
fof(t64_vectsp_6,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)
& v3_rlvect_1(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(A))
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> r1_vectsp_6(A,B,k6_vectsp_6(A,B,C,k6_vectsp_6(A,B,D,E)),k6_vectsp_6(A,B,k1_group_1(A,C,D),E)) ) ) ) ) ) ).
fof(t65_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> r1_vectsp_6(A,B,k6_vectsp_6(A,B,k2_group_1(A),C),C) ) ) ) ).
fof(d13_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> k7_vectsp_6(A,B,C) = k6_vectsp_6(A,B,k5_rlvect_1(A,k2_group_1(A)),C) ) ) ) ).
fof(t66_vectsp_6,axiom,
$true ).
fof(t67_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),k7_vectsp_6(A,B,D),C) = k5_rlvect_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C)) ) ) ) ) ).
fof(t68_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( r1_vectsp_6(A,B,k5_vectsp_6(A,B,C,D),k2_vectsp_6(A,B))
=> r1_vectsp_6(A,B,D,k7_vectsp_6(A,B,C)) ) ) ) ) ) ).
fof(t69_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> k1_vectsp_6(A,B,k7_vectsp_6(A,B,C)) = k1_vectsp_6(A,B,C) ) ) ) ).
fof(t70_vectsp_6,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)
& v3_rlvect_1(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,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( m2_vectsp_6(D,A,B,C)
=> m2_vectsp_6(k7_vectsp_6(A,B,D),A,B,C) ) ) ) ) ) ).
fof(d14_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> k8_vectsp_6(A,B,C,D) = k5_vectsp_6(A,B,C,k7_vectsp_6(A,B,D)) ) ) ) ) ).
fof(t71_vectsp_6,axiom,
$true ).
fof(t72_vectsp_6,axiom,
$true ).
fof(t73_vectsp_6,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)
& v3_rlvect_1(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(B))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> k8_funct_2(u1_struct_0(B),u1_struct_0(A),k8_vectsp_6(A,B,D,E),C) = k6_rlvect_1(A,k8_funct_2(u1_struct_0(B),u1_struct_0(A),D,C),k8_funct_2(u1_struct_0(B),u1_struct_0(A),E,C)) ) ) ) ) ) ).
fof(t74_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> r1_tarski(k1_vectsp_6(A,B,k8_vectsp_6(A,B,C,D)),k4_subset_1(u1_struct_0(B),k1_vectsp_6(A,B,C),k1_vectsp_6(A,B,D))) ) ) ) ) ).
fof(t75_vectsp_6,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)
& v3_rlvect_1(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,k1_zfmisc_1(u1_struct_0(B)))
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ! [E] :
( m1_vectsp_6(E,A,B)
=> ( ( m2_vectsp_6(D,A,B,C)
& m2_vectsp_6(E,A,B,C) )
=> m2_vectsp_6(k8_vectsp_6(A,B,D,E),A,B,C) ) ) ) ) ) ) ).
fof(t76_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> r1_vectsp_6(A,B,k8_vectsp_6(A,B,C,C),k2_vectsp_6(A,B)) ) ) ) ).
fof(t77_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> k4_vectsp_6(A,B,k5_vectsp_6(A,B,C,D)) = k4_rlvect_1(B,k4_vectsp_6(A,B,C),k4_vectsp_6(A,B,D)) ) ) ) ) ).
fof(t78_vectsp_6,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] :
( ( ~ v3_struct_0(B)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k4_vectsp_6(A,B,k6_vectsp_6(A,B,D,C)) = k6_vectsp_1(A,B,D,k4_vectsp_6(A,B,C)) ) ) ) ) ).
fof(t79_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> k4_vectsp_6(A,B,k7_vectsp_6(A,B,C)) = k5_rlvect_1(B,k4_vectsp_6(A,B,C)) ) ) ) ).
fof(t80_vectsp_6,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)
& v3_rlvect_1(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_vectsp_6(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> k4_vectsp_6(A,B,k8_vectsp_6(A,B,C,D)) = k6_rlvect_1(B,k4_vectsp_6(A,B,C),k4_vectsp_6(A,B,D)) ) ) ) ) ).
fof(t81_vectsp_6,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))
=> k1_group_1(A,k5_rlvect_1(A,k2_group_1(A)),B) = k5_rlvect_1(A,B) ) ) ).
fof(t82_vectsp_6,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) )
=> k5_rlvect_1(A,k2_group_1(A)) != k1_rlvect_1(A) ) ).
fof(dt_m1_vectsp_6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_6(C,A,B)
=> m2_fraenkel(C,u1_struct_0(B),u1_struct_0(A),k1_fraenkel(u1_struct_0(B),u1_struct_0(A))) ) ) ).
fof(existence_m1_vectsp_6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ? [C] : m1_vectsp_6(C,A,B) ) ).
fof(dt_m2_vectsp_6,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [D] :
( m2_vectsp_6(D,A,B,C)
=> m1_vectsp_6(D,A,B) ) ) ).
fof(existence_m2_vectsp_6,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ? [D] : m2_vectsp_6(D,A,B,C) ) ).
fof(symmetry_r1_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B)
& m1_vectsp_6(D,A,B) )
=> ( r1_vectsp_6(A,B,C,D)
=> r1_vectsp_6(A,B,D,C) ) ) ).
fof(reflexivity_r1_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B)
& m1_vectsp_6(D,A,B) )
=> r1_vectsp_6(A,B,C,C) ) ).
fof(redefinition_r1_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B)
& m1_vectsp_6(D,A,B) )
=> ( r1_vectsp_6(A,B,C,D)
<=> C = D ) ) ).
fof(dt_k1_vectsp_6,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B) )
=> ( v1_finset_1(k1_vectsp_6(A,B,C))
& m1_subset_1(k1_vectsp_6(A,B,C),k1_zfmisc_1(u1_struct_0(B))) ) ) ).
fof(dt_k2_vectsp_6,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> m1_vectsp_6(k2_vectsp_6(A,B),A,B) ) ).
fof(dt_k3_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_finseq_1(C,u1_struct_0(B))
& v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
& m1_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
=> m2_finseq_1(k3_vectsp_6(A,B,C,D),u1_struct_0(B)) ) ).
fof(dt_k4_vectsp_6,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_rlvect_1(A)
& ~ v3_struct_0(B)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B) )
=> m1_subset_1(k4_vectsp_6(A,B,C),u1_struct_0(B)) ) ).
fof(dt_k5_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B)
& m1_vectsp_6(D,A,B) )
=> m1_vectsp_6(k5_vectsp_6(A,B,C,D),A,B) ) ).
fof(dt_k6_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_subset_1(C,u1_struct_0(A))
& m1_vectsp_6(D,A,B) )
=> m1_vectsp_6(k6_vectsp_6(A,B,C,D),A,B) ) ).
fof(dt_k7_vectsp_6,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_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B) )
=> m1_vectsp_6(k7_vectsp_6(A,B,C),A,B) ) ).
fof(involutiveness_k7_vectsp_6,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_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B) )
=> k7_vectsp_6(A,B,k7_vectsp_6(A,B,C)) = C ) ).
fof(dt_k8_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v3_rlvect_1(A)
& v4_rlvect_1(A)
& v5_rlvect_1(A)
& v6_rlvect_1(A)
& v4_group_1(A)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& l3_vectsp_1(A)
& ~ v3_struct_0(B)
& v3_rlvect_1(B)
& v4_rlvect_1(B)
& v5_rlvect_1(B)
& v6_rlvect_1(B)
& v12_vectsp_1(B,A)
& l4_vectsp_1(B,A)
& m1_vectsp_6(C,A,B)
& m1_vectsp_6(D,A,B) )
=> m1_vectsp_6(k8_vectsp_6(A,B,C,D),A,B) ) ).
fof(d5_vectsp_6,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_struct_0(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_6(C,A,B)
=> k1_vectsp_6(A,B,C) = a_3_0_vectsp_6(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_vectsp_6,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l2_struct_0(B)
& ~ v3_struct_0(C)
& l4_vectsp_1(C,B)
& m1_vectsp_6(D,B,C) )
=> ( r2_hidden(A,a_3_0_vectsp_6(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(C))
& A = E
& k8_funct_2(u1_struct_0(C),u1_struct_0(B),D,E) != k1_rlvect_1(B) ) ) ) ).
%------------------------------------------------------------------------------