SET007 Axioms: SET007+442.ax
%------------------------------------------------------------------------------
% File : SET007+442 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Steinitz Theorem and the Dimension of a 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_9 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 51 ( 1 unt; 0 def)
% Number of atoms : 1112 ( 48 equ)
% Maximal formula atoms : 31 ( 21 avg)
% Number of connectives : 1219 ( 158 ~; 0 |; 859 &)
% ( 9 <=>; 193 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 15 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 42 ( 40 usr; 1 prp; 0-4 aty)
% Number of functors : 39 ( 39 usr; 6 con; 0-5 aty)
% Number of variables : 188 ( 175 !; 13 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_vectsp_9,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_struct_0(A) )
=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
& ~ v1_xboole_0(B) ) ) ).
fof(rc2_vectsp_9,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)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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) )
=> ? [C] :
( m1_vectsp_4(C,A,B)
& ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& v2_matrlin(C,A) ) ) ).
fof(cc1_vectsp_9,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)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> v2_matrlin(C,A) ) ) ).
fof(rc3_vectsp_9,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)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ? [C] :
( m1_vectsp_4(C,A,B)
& ~ v3_struct_0(C)
& v3_rlvect_1(C)
& v4_rlvect_1(C)
& v5_rlvect_1(C)
& v6_rlvect_1(C)
& v11_vectsp_1(C,A)
& v12_vectsp_1(C,A)
& v2_matrlin(C,A) ) ) ).
fof(t1_vectsp_9,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v1_finset_1(B)
=> ~ ( r1_xreal_0(A,k4_card_1(B))
& ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> k1_card_1(C) != A ) ) ) ) ).
fof(t2_vectsp_9,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( v2_funct_1(B)
& r2_hidden(A,k2_relat_1(B)) )
=> k1_card_1(k10_relat_1(B,k1_tarski(A))) = np__1 ) ) ).
fof(t3_vectsp_9,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ~ r2_hidden(A,k2_relat_1(B))
=> k1_card_1(k10_relat_1(B,k1_tarski(A))) = np__0 ) ) ).
fof(t4_vectsp_9,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k2_relat_1(A) = k2_relat_1(B)
& v2_funct_1(A)
& v2_funct_1(B) )
=> r1_rfinseq(A,B) ) ) ) ).
fof(t5_vectsp_9,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] :
( m2_finseq_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k4_finseq_1(D),k4_finseq_1(D))
& v3_funct_2(F,k4_finseq_1(D),k4_finseq_1(D))
& m2_relset_1(F,k4_finseq_1(D),k4_finseq_1(D)) )
=> ( E = k5_relat_1(F,D)
=> k9_rlvect_1(B,k3_vectsp_6(A,B,D,C)) = k9_rlvect_1(B,k3_vectsp_6(A,B,E,C)) ) ) ) ) ) ) ) ).
fof(t6_vectsp_9,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] :
( m2_finseq_1(D,u1_struct_0(B))
=> ( r1_xboole_0(k1_vectsp_6(A,B,C),k2_relat_1(D))
=> k9_rlvect_1(B,k3_vectsp_6(A,B,D,C)) = k1_rlvect_1(B) ) ) ) ) ) ).
fof(t7_vectsp_9,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] :
( m2_finseq_1(C,u1_struct_0(B))
=> ( v2_funct_1(C)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( r1_tarski(k1_vectsp_6(A,B,D),k2_relat_1(C))
=> k9_rlvect_1(B,k3_vectsp_6(A,B,C,D)) = k4_vectsp_6(A,B,D) ) ) ) ) ) ) ).
fof(t8_vectsp_9,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] :
( m2_finseq_1(D,u1_struct_0(B))
=> ? [E] :
( m1_vectsp_6(E,A,B)
& k1_vectsp_6(A,B,E) = k3_xboole_0(k2_relat_1(D),k1_vectsp_6(A,B,C))
& k3_vectsp_6(A,B,D,C) = k3_vectsp_6(A,B,D,E) ) ) ) ) ) ).
fof(t9_vectsp_9,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,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ~ ( r1_tarski(k2_relat_1(E),u1_struct_0(k1_vectsp_7(A,B,D)))
& ! [F] :
( m2_vectsp_6(F,A,B,D)
=> k9_rlvect_1(B,k3_vectsp_6(A,B,E,C)) != k4_vectsp_6(A,B,F) ) ) ) ) ) ) ) ).
fof(t10_vectsp_9,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,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( r1_tarski(k1_vectsp_6(A,B,C),u1_struct_0(k1_vectsp_7(A,B,D)))
& ! [E] :
( m2_vectsp_6(E,A,B,D)
=> k4_vectsp_6(A,B,C) != k4_vectsp_6(A,B,E) ) ) ) ) ) ) ).
fof(t11_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ( r1_tarski(k1_vectsp_6(A,B,D),u1_struct_0(C))
=> ! [E] :
( m1_vectsp_6(E,A,C)
=> ( E = k7_relat_1(D,u1_struct_0(C))
=> ( k1_vectsp_6(A,B,D) = k1_vectsp_6(A,C,E)
& k4_vectsp_6(A,B,D) = k4_vectsp_6(A,C,E) ) ) ) ) ) ) ) ) ).
fof(t12_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,C)
=> ? [E] :
( m1_vectsp_6(E,A,B)
& k1_vectsp_6(A,C,D) = k1_vectsp_6(A,B,E)
& k4_vectsp_6(A,C,D) = k4_vectsp_6(A,B,E) ) ) ) ) ) ).
fof(t13_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_vectsp_6(D,A,B)
=> ~ ( r1_tarski(k1_vectsp_6(A,B,D),u1_struct_0(C))
& ! [E] :
( m1_vectsp_6(E,A,C)
=> ~ ( k1_vectsp_6(A,C,E) = k1_vectsp_6(A,B,D)
& k4_vectsp_6(A,C,E) = k4_vectsp_6(A,B,D) ) ) ) ) ) ) ) ).
fof(t14_vectsp_9,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_7(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> r1_rlvect_1(k1_vectsp_7(A,B,C),D) ) ) ) ) ).
fof(t15_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
=> ( v1_vectsp_7(D,A,C)
=> ( v1_vectsp_7(D,A,B)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) ) ) ) ) ) ) ).
fof(t16_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( ( v1_vectsp_7(D,A,B)
& r1_tarski(D,u1_struct_0(C)) )
=> ( v1_vectsp_7(D,A,C)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C))) ) ) ) ) ) ) ).
fof(t17_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_vectsp_7(D,A,C)
=> ? [E] :
( m1_vectsp_7(E,A,B)
& r1_tarski(D,E) ) ) ) ) ) ).
fof(t18_vectsp_9,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,k1_zfmisc_1(u1_struct_0(B)))
=> ( v1_vectsp_7(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ( r2_hidden(D,C)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( E = k6_subset_1(u1_struct_0(B),C,k1_struct_0(B,D))
& r1_rlvect_1(k1_vectsp_7(A,B,E),D) ) ) ) ) ) ) ) ) ).
fof(t19_vectsp_9,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_7(C,A,B)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( r1_xboole_0(D,C)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( E = k4_subset_1(u1_struct_0(B),C,D)
& v1_vectsp_7(E,A,B) ) ) ) ) ) ) ) ).
fof(t20_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ( r1_tarski(D,u1_struct_0(C))
=> m1_vectsp_4(k1_vectsp_7(A,B,D),A,C) ) ) ) ) ) ).
fof(t21_vectsp_9,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_4(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(C)))
=> ( D = E
=> k1_vectsp_7(A,B,D) = k1_vectsp_7(A,C,E) ) ) ) ) ) ) ).
fof(t22_vectsp_9,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] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(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))
=> ~ ( r1_rlvect_1(k1_vectsp_7(A,B,k4_subset_1(u1_struct_0(B),C,D)),E)
& ~ r1_rlvect_1(k1_vectsp_7(A,B,D),E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ~ ( r2_hidden(F,C)
& r1_rlvect_1(k1_vectsp_7(A,B,k4_subset_1(u1_struct_0(B),k6_subset_1(u1_struct_0(B),k4_subset_1(u1_struct_0(B),C,D),k1_struct_0(B,F)),k1_struct_0(B,E))),F) ) ) ) ) ) ) ) ) ).
fof(t23_vectsp_9,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] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) )
=> ! [D] :
( ( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
=> ( ( g4_vectsp_1(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u2_vectsp_1(A,B)) = k1_vectsp_7(A,B,C)
& v1_vectsp_7(D,A,B) )
=> ( r1_xreal_0(k4_card_1(D),k4_card_1(C))
& ? [E] :
( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
& r1_tarski(E,C)
& k4_card_1(E) = k6_xcmplx_0(k4_card_1(C),k4_card_1(D))
& g4_vectsp_1(A,u1_struct_0(B),u1_rlvect_1(B),u2_struct_0(B),u2_vectsp_1(A,B)) = k1_vectsp_7(A,B,k4_subset_1(u1_struct_0(B),D,E)) ) ) ) ) ) ) ) ).
fof(d1_vectsp_9,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) )
=> ( v2_matrlin(B,A)
<=> ? [C] :
( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_vectsp_7(C,A,B) ) ) ) ) ).
fof(t24_vectsp_9,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) )
=> ( v2_matrlin(B,A)
=> ! [C] :
( m1_vectsp_7(C,A,B)
=> v1_finset_1(C) ) ) ) ) ).
fof(t25_vectsp_9,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) )
=> ( v2_matrlin(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v1_vectsp_7(C,A,B)
=> v1_finset_1(C) ) ) ) ) ) ).
fof(t26_vectsp_9,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) )
=> ( v2_matrlin(B,A)
=> ! [C] :
( m1_vectsp_7(C,A,B)
=> ! [D] :
( m1_vectsp_7(D,A,B)
=> k1_card_1(C) = k1_card_1(D) ) ) ) ) ) ).
fof(t27_vectsp_9,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) )
=> v2_matrlin(k1_vectsp_4(A,B),A) ) ) ).
fof(t28_vectsp_9,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_4(C,A,B)
=> ( v2_matrlin(B,A)
=> v2_matrlin(C,A) ) ) ) ) ).
fof(d2_vectsp_9,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) )
=> ( v2_matrlin(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k1_vectsp_9(A,B)
<=> ! [D] :
( m1_vectsp_7(D,A,B)
=> C = k1_card_1(D) ) ) ) ) ) ) ).
fof(t29_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> r1_xreal_0(k1_vectsp_9(A,C),k1_vectsp_9(A,B)) ) ) ) ).
fof(t30_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
=> ( v1_vectsp_7(C,A,B)
=> k1_card_1(C) = k1_vectsp_9(A,k1_vectsp_7(A,B,C)) ) ) ) ) ).
fof(t31_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> k1_vectsp_9(A,B) = k1_vectsp_9(A,k2_vectsp_4(A,B)) ) ) ).
fof(t32_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ( k1_vectsp_9(A,B) = k1_vectsp_9(A,C)
<=> k2_vectsp_4(A,B) = k2_vectsp_4(A,C) ) ) ) ) ).
fof(t33_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ( k1_vectsp_9(A,B) = np__0
<=> k2_vectsp_4(A,B) = k1_vectsp_4(A,B) ) ) ) ).
fof(t34_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ( k1_vectsp_9(A,B) = np__1
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& C != k1_rlvect_1(B)
& k2_vectsp_4(A,B) = k1_vectsp_7(A,B,k1_struct_0(B,C)) ) ) ) ) ).
fof(t35_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ( k1_vectsp_9(A,B) = np__2
<=> ? [C] :
( m1_subset_1(C,u1_struct_0(B))
& ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& C != D
& v1_vectsp_7(k2_struct_0(B,C,D),A,B)
& k2_vectsp_4(A,B) = k1_vectsp_7(A,B,k2_struct_0(B,C,D)) ) ) ) ) ) ).
fof(t36_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> k1_nat_1(k1_vectsp_9(A,k1_vectsp_5(A,B,C,D)),k1_vectsp_9(A,k2_vectsp_5(A,B,C,D))) = k1_nat_1(k1_vectsp_9(A,C),k1_vectsp_9(A,D)) ) ) ) ) ).
fof(t37_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> r1_xreal_0(k6_xcmplx_0(k1_nat_1(k1_vectsp_9(A,C),k1_vectsp_9(A,D)),k1_vectsp_9(A,B)),k1_vectsp_9(A,k2_vectsp_5(A,B,C,D))) ) ) ) ) ).
fof(t38_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m1_vectsp_4(C,A,B)
=> ! [D] :
( m1_vectsp_4(D,A,B)
=> ( r1_vectsp_5(A,B,C,D)
=> k1_vectsp_9(A,B) = k1_nat_1(k1_vectsp_9(A,C),k1_vectsp_9(A,D)) ) ) ) ) ) ).
fof(t39_vectsp_9,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] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [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,A)
& v2_matrlin(C,A)
& l4_vectsp_1(C,A) )
=> ( r1_xreal_0(B,k1_vectsp_9(A,C))
<=> ? [D] :
( v11_vectsp_1(D,A)
& m1_vectsp_4(D,A,C)
& k1_vectsp_9(A,D) = B ) ) ) ) ) ).
fof(d3_vectsp_9,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)
& v2_matrlin(B,A)
& l4_vectsp_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( D = k2_vectsp_9(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( v11_vectsp_1(F,A)
& m1_vectsp_4(F,A,B)
& F = E
& k1_vectsp_9(A,F) = C ) ) ) ) ) ) ).
fof(t40_vectsp_9,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] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [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,A)
& v2_matrlin(C,A)
& l4_vectsp_1(C,A) )
=> ~ ( r1_xreal_0(B,k1_vectsp_9(A,C))
& v1_xboole_0(k2_vectsp_9(A,C,B)) ) ) ) ) ).
fof(t41_vectsp_9,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] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [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,A)
& v2_matrlin(C,A)
& l4_vectsp_1(C,A) )
=> ( ~ r1_xreal_0(B,k1_vectsp_9(A,C))
=> k2_vectsp_9(A,C,B) = k1_xboole_0 ) ) ) ) ).
fof(t42_vectsp_9,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] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [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,A)
& v2_matrlin(C,A)
& l4_vectsp_1(C,A) )
=> ! [D] :
( m1_vectsp_4(D,A,C)
=> r1_tarski(k2_vectsp_9(A,D,B),k2_vectsp_9(A,C,B)) ) ) ) ) ).
fof(dt_k1_vectsp_9,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)
& v7_vectsp_1(A)
& v8_vectsp_1(A)
& v9_vectsp_1(A)
& ~ v10_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) )
=> m2_subset_1(k1_vectsp_9(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k2_vectsp_9,axiom,
$true ).
%------------------------------------------------------------------------------