SET007 Axioms: SET007+770.ax


%------------------------------------------------------------------------------
% File     : SET007+770 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Real Linear Space of Real Sequences
% 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    : rsspace [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   51 (   8 unit)
%            Number of atoms       :  261 (  38 equality)
%            Maximal formula depth :   11 (   6 average)
%            Number of connectives :  228 (  18 ~  ;   4  |; 131  &)
%                                         (   9 <=>;  66 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   1 propositional; 0-3 arity)
%            Number of functors    :   41 (  12 constant; 0-6 arity)
%            Number of variables   :   75 (   0 singleton;  71 !;   4 ?)
%            Maximal term depth    :    4 (   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(fc1_rsspace,axiom,
    ( ~ v3_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace))
    & v2_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace))
    & v3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)) )).

fof(fc2_rsspace,axiom,
    ( ~ v1_xboole_0(k11_rsspace)
    & v1_rlsub_1(k11_rsspace,k7_rsspace) )).

fof(fc3_rsspace,axiom,
    ( ~ v3_struct_0(g1_bhsp_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace),k12_rsspace))
    & v1_bhsp_1(g1_bhsp_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace),k12_rsspace)) )).

fof(fc4_rsspace,axiom,
    ( ~ v3_struct_0(k13_rsspace)
    & v3_rlvect_1(k13_rsspace)
    & v4_rlvect_1(k13_rsspace)
    & v5_rlvect_1(k13_rsspace)
    & v6_rlvect_1(k13_rsspace)
    & v7_rlvect_1(k13_rsspace) )).

fof(d1_rsspace,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( A = k1_rsspace
      <=> ! [B] :
            ( r2_hidden(B,A)
          <=> ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,k1_numbers)
              & m2_relset_1(B,k5_numbers,k1_numbers) ) ) ) ) )).

fof(d2_rsspace,axiom,(
    ! [A] :
      ( r2_hidden(A,k1_rsspace)
     => k2_rsspace(A) = A ) )).

fof(d3_rsspace,axiom,(
    ! [A] :
      ( r2_hidden(A,k1_numbers)
     => k3_rsspace(A) = A ) )).

fof(t1_rsspace,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k1_rsspace,k1_rsspace),k1_rsspace)
      & m2_relset_1(A,k2_zfmisc_1(k1_rsspace,k1_rsspace),k1_rsspace)
      & ! [B] :
          ( m1_subset_1(B,k1_rsspace)
         => ! [C] :
              ( m1_subset_1(C,k1_rsspace)
             => k2_binop_1(k1_rsspace,k1_rsspace,k1_rsspace,A,B,C) = k9_seq_1(k2_rsspace(B),k2_rsspace(C)) ) )
      & v1_binop_1(A,k1_rsspace)
      & v2_binop_1(A,k1_rsspace) ) )).

fof(t2_rsspace,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_rsspace),k1_rsspace)
      & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_rsspace),k1_rsspace)
      & ! [B,C] :
          ( ( r2_hidden(B,k1_numbers)
            & r2_hidden(C,k1_rsspace) )
         => k1_funct_1(A,k4_tarski(B,C)) = k14_seq_1(k2_rsspace(C),k3_rsspace(B)) ) ) )).

fof(d4_rsspace,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_rsspace,k1_rsspace),k1_rsspace)
        & m2_relset_1(A,k2_zfmisc_1(k1_rsspace,k1_rsspace),k1_rsspace) )
     => ( A = k4_rsspace
      <=> ! [B] :
            ( m1_subset_1(B,k1_rsspace)
           => ! [C] :
                ( m1_subset_1(C,k1_rsspace)
               => k2_binop_1(k1_rsspace,k1_rsspace,k1_rsspace,A,B,C) = k9_seq_1(k2_rsspace(B),k2_rsspace(C)) ) ) ) ) )).

fof(d5_rsspace,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_rsspace),k1_rsspace)
        & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_rsspace),k1_rsspace) )
     => ( A = k5_rsspace
      <=> ! [B,C] :
            ( ( r2_hidden(B,k1_numbers)
              & r2_hidden(C,k1_rsspace) )
           => k1_funct_1(A,k4_tarski(B,C)) = k14_seq_1(k2_rsspace(C),k3_rsspace(B)) ) ) ) )).

fof(d6_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_rsspace)
     => ( A = k6_rsspace
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,k2_rsspace(A),B) = np__0 ) ) ) )).

fof(t3_rsspace,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k2_rsspace(A) = A ) )).

fof(t4_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
         => k2_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),A,B) = k9_seq_1(k2_rsspace(A),k2_rsspace(B)) ) ) )).

fof(t5_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
         => k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),B,A) = k14_seq_1(k2_rsspace(B),A) ) ) )).

fof(t6_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
             => k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),A,B),C) = k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),A,k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),B,C)) ) ) ) )).

fof(t7_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
     => k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),A,k1_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace))) = A ) )).

fof(t8_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
     => ? [B] :
          ( m1_subset_1(B,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
          & k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),A,B) = k1_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)) ) ) )).

fof(t9_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
             => k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),B,C),A) = k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),B,A),k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),C,A)) ) ) ) )).

fof(t10_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
             => k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),C,k3_real_1(A,B)) = k4_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),C,A),k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),C,B)) ) ) ) )).

fof(t11_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
             => k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),C,k4_real_1(A,B)) = k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),C,B),A) ) ) ) )).

fof(t12_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,u1_struct_0(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace)))
     => k3_rlvect_1(g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace),A,np__1) = A ) )).

fof(d7_rsspace,axiom,(
    k7_rsspace = g2_rlvect_1(k1_rsspace,k6_rsspace,k4_rsspace,k5_rsspace) )).

fof(d8_rsspace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_rlsub_1(B,A)
           => ( v1_xboole_0(B)
              | k8_rsspace(A,B) = k1_realset1(u1_rlvect_1(A),B) ) ) ) ) )).

fof(d9_rsspace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_rlsub_1(B,A)
           => ( v1_xboole_0(B)
              | k9_rsspace(A,B) = k2_partfun1(k2_zfmisc_1(k1_numbers,u1_struct_0(A)),u1_struct_0(A),u2_rlvect_1(A),k2_zfmisc_1(k1_numbers,B)) ) ) ) ) )).

fof(d10_rsspace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_rlsub_1(B,A)
           => ( v1_xboole_0(B)
              | k10_rsspace(A,B) = k1_rlvect_1(A) ) ) ) ) )).

fof(t13_rsspace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_rlsub_1(B,A)
           => ( v1_xboole_0(B)
              | m1_rlsub_1(g2_rlvect_1(B,k10_rsspace(A,B),k8_rsspace(A,B),k9_rsspace(A,B)),A) ) ) ) ) )).

fof(d11_rsspace,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k7_rsspace)))
     => ( A = k11_rsspace
      <=> ! [B] :
            ( r2_hidden(B,A)
          <=> ( r2_hidden(B,k1_rsspace)
              & v1_series_1(k11_seq_1(k2_rsspace(B),k2_rsspace(B))) ) ) ) ) )).

fof(t14_rsspace,axiom,(
    $true )).

fof(t15_rsspace,axiom,(
    m1_rlsub_1(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace)),k7_rsspace) )).

fof(t16_rsspace,axiom,
    ( ~ v3_struct_0(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace)))
    & v3_rlvect_1(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace)))
    & v4_rlvect_1(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace)))
    & v5_rlvect_1(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace)))
    & v6_rlvect_1(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace)))
    & v7_rlvect_1(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace)))
    & l2_rlvect_1(g2_rlvect_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace))) )).

fof(t17_rsspace,axiom,
    ( u1_struct_0(k7_rsspace) = k1_rsspace
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k7_rsspace))
      <=> ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & m2_relset_1(A,k5_numbers,k1_numbers) ) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k7_rsspace))
       => A = k2_rsspace(A) )
    & ! [A] :
        ( m1_subset_1(A,u1_struct_0(k7_rsspace))
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k7_rsspace))
           => k4_rlvect_1(k7_rsspace,A,B) = k9_seq_1(k2_rsspace(A),k2_rsspace(B)) ) )
    & ! [A] :
        ( m1_subset_1(A,k1_numbers)
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(k7_rsspace))
           => k3_rlvect_1(k7_rsspace,B,A) = k14_seq_1(k2_rsspace(B),A) ) ) )).

fof(t18_rsspace,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers)
      & m2_relset_1(A,k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers)
      & ! [B,C] :
          ( ( r2_hidden(B,k11_rsspace)
            & r2_hidden(C,k11_rsspace) )
         => k2_seq_1(k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers,A,k4_tarski(B,C)) = k2_series_1(k11_seq_1(k2_rsspace(B),k2_rsspace(C))) ) ) )).

fof(d12_rsspace,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers) )
     => ( A = k12_rsspace
      <=> ! [B,C] :
            ( ( r2_hidden(B,k11_rsspace)
              & r2_hidden(C,k11_rsspace) )
           => k2_seq_1(k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers,A,k4_tarski(B,C)) = k2_series_1(k11_seq_1(k2_rsspace(B),k2_rsspace(C))) ) ) ) )).

fof(d13_rsspace,axiom,(
    k13_rsspace = g1_bhsp_1(k11_rsspace,k10_rsspace(k7_rsspace,k11_rsspace),k8_rsspace(k7_rsspace,k11_rsspace),k9_rsspace(k7_rsspace,k11_rsspace),k12_rsspace) )).

fof(t19_rsspace,axiom,(
    ! [A] :
      ( l1_bhsp_1(A)
     => ( ( ~ v3_struct_0(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v3_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v4_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v5_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v6_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & v7_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A)))
          & l2_rlvect_1(g2_rlvect_1(u1_struct_0(A),u2_struct_0(A),u1_rlvect_1(A),u2_rlvect_1(A))) )
       => ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & v4_rlvect_1(A)
          & v5_rlvect_1(A)
          & v6_rlvect_1(A)
          & v7_rlvect_1(A)
          & l2_rlvect_1(A) ) ) ) )).

fof(t20_rsspace,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 )
       => ( v1_series_1(A)
          & k2_series_1(A) = np__0 ) ) ) )).

fof(t21_rsspace,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) )
          & v1_series_1(A)
          & k2_series_1(A) = np__0 )
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = np__0 ) ) ) )).

fof(dt_k1_rsspace,axiom,(
    ~ v1_xboole_0(k1_rsspace) )).

fof(dt_k2_rsspace,axiom,(
    ! [A] :
      ( v1_funct_1(k2_rsspace(A))
      & v1_funct_2(k2_rsspace(A),k5_numbers,k1_numbers)
      & m2_relset_1(k2_rsspace(A),k5_numbers,k1_numbers) ) )).

fof(dt_k3_rsspace,axiom,(
    ! [A] : m1_subset_1(k3_rsspace(A),k1_numbers) )).

fof(dt_k4_rsspace,axiom,
    ( v1_funct_1(k4_rsspace)
    & v1_funct_2(k4_rsspace,k2_zfmisc_1(k1_rsspace,k1_rsspace),k1_rsspace)
    & m2_relset_1(k4_rsspace,k2_zfmisc_1(k1_rsspace,k1_rsspace),k1_rsspace) )).

fof(dt_k5_rsspace,axiom,
    ( v1_funct_1(k5_rsspace)
    & v1_funct_2(k5_rsspace,k2_zfmisc_1(k1_numbers,k1_rsspace),k1_rsspace)
    & m2_relset_1(k5_rsspace,k2_zfmisc_1(k1_numbers,k1_rsspace),k1_rsspace) )).

fof(dt_k6_rsspace,axiom,(
    m1_subset_1(k6_rsspace,k1_rsspace) )).

fof(dt_k7_rsspace,axiom,
    ( ~ v3_struct_0(k7_rsspace)
    & v3_rlvect_1(k7_rsspace)
    & v4_rlvect_1(k7_rsspace)
    & v5_rlvect_1(k7_rsspace)
    & v6_rlvect_1(k7_rsspace)
    & v7_rlvect_1(k7_rsspace)
    & l2_rlvect_1(k7_rsspace) )).

fof(dt_k8_rsspace,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k8_rsspace(A,B))
        & v1_funct_2(k8_rsspace(A,B),k2_zfmisc_1(B,B),B)
        & m2_relset_1(k8_rsspace(A,B),k2_zfmisc_1(B,B),B) ) ) )).

fof(dt_k9_rsspace,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => ( v1_funct_1(k9_rsspace(A,B))
        & v1_funct_2(k9_rsspace(A,B),k2_zfmisc_1(k1_numbers,B),B)
        & m2_relset_1(k9_rsspace(A,B),k2_zfmisc_1(k1_numbers,B),B) ) ) )).

fof(dt_k10_rsspace,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k10_rsspace(A,B),B) ) )).

fof(dt_k11_rsspace,axiom,(
    m1_subset_1(k11_rsspace,k1_zfmisc_1(u1_struct_0(k7_rsspace))) )).

fof(dt_k12_rsspace,axiom,
    ( v1_funct_1(k12_rsspace)
    & v1_funct_2(k12_rsspace,k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers)
    & m2_relset_1(k12_rsspace,k2_zfmisc_1(k11_rsspace,k11_rsspace),k1_numbers) )).

fof(dt_k13_rsspace,axiom,
    ( ~ v3_struct_0(k13_rsspace)
    & l1_bhsp_1(k13_rsspace) )).
%------------------------------------------------------------------------------