SET007 Axioms: SET007+442.ax


%------------------------------------------------------------------------------
% File     : SET007+442 : TPTP v7.5.0. 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 unit)
%            Number of atoms       : 1112 (  48 equality)
%            Maximal formula depth :   23 (  15 average)
%            Number of connectives : 1219 ( 158 ~  ;   0  |; 859  &)
%                                         (   9 <=>; 193 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   42 (   1 propositional; 0-4 arity)
%            Number of functors    :   39 (   6 constant; 0-5 arity)
%            Number of variables   :  188 (   0 singleton; 175 !;  13 ?)
%            Maximal term depth    :    6 (   1 average)
% 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 )).
%------------------------------------------------------------------------------