SET007 Axioms: SET007+767.ax


%------------------------------------------------------------------------------
% File     : SET007+767 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Some Properties of Real Hilbert Space. Part I
% 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    : bhsp_6 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   20 (   0 unit)
%            Number of atoms       :  396 (  13 equality)
%            Maximal formula depth :   30 (  17 average)
%            Number of connectives :  435 (  59 ~  ;   0  |; 269  &)
%                                         (   8 <=>;  99 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   0 propositional; 1-3 arity)
%            Number of functors    :   27 (   6 constant; 0-5 arity)
%            Number of variables   :   86 (   0 singleton;  79 !;   7 ?)
%            Maximal term depth    :    5 (   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(d1_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k1_bhsp_6(A,B)
                <=> ? [D] :
                      ( m2_finseq_1(D,u1_struct_0(A))
                      & v2_funct_1(D)
                      & k5_relset_1(k5_numbers,u1_struct_0(A),D) = B
                      & C = k2_finsop_1(u1_struct_0(A),D,u1_rlvect_1(A)) ) ) ) ) ) ) )).

fof(t1_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                  & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
               => ( ( r1_tarski(B,k4_relset_1(u1_struct_0(A),u1_struct_0(A),C))
                    & ! [D] :
                        ( r2_hidden(D,k4_relset_1(u1_struct_0(A),u1_struct_0(A),C))
                       => k1_funct_1(C,D) = D ) )
                 => k1_bhsp_6(A,B) = k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C) ) ) ) ) ) )).

fof(t2_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] :
                ( ( v1_finset_1(C)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ( r1_xboole_0(B,C)
                 => ! [D] :
                      ( ( v1_finset_1(D)
                        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                     => ( D = k4_subset_1(u1_struct_0(A),B,C)
                       => k1_bhsp_6(A,D) = k4_rlvect_1(A,k1_bhsp_6(A,B),k1_bhsp_6(A,C)) ) ) ) ) ) ) ) )).

fof(d2_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_bhsp_6(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( ( v1_finset_1(E)
                              & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                           => ~ ( ~ v1_xboole_0(E)
                                & r1_tarski(E,B)
                                & ! [F] :
                                    ( ( v1_finset_1(F)
                                      & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                                   => ~ ( r1_tarski(E,F)
                                        & r1_tarski(F,B)
                                        & r1_xreal_0(D,k3_bhsp_1(A,k6_rlvect_1(A,C,k1_bhsp_6(A,F)))) ) ) ) ) ) ) ) ) ) ) )).

fof(d3_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_bhsp_6(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k2_bhsp_6(A,B)
                <=> ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( ( v1_finset_1(E)
                                & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                             => ~ ( ~ v1_xboole_0(E)
                                  & r1_tarski(E,B)
                                  & ! [F] :
                                      ( ( v1_finset_1(F)
                                        & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                                     => ~ ( r1_tarski(E,F)
                                          & r1_tarski(F,B)
                                          & r1_xreal_0(D,k3_bhsp_1(A,k6_rlvect_1(A,C,k1_bhsp_6(A,F)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
            & v2_hahnban(B,A)
            & v3_hahnban(B,A)
            & m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
         => ( v2_bhsp_6(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ~ r1_xreal_0(C,np__0)
                & ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => r1_xreal_0(k18_complex1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,D)),k11_binop_2(C,k3_bhsp_1(A,D))) ) ) ) ) ) )).

fof(d5_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_bhsp_6(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),k6_supinf_1)
                      & v2_hahnban(D,A)
                      & v3_hahnban(D,A)
                      & m2_relset_1(D,u1_struct_0(A),k6_supinf_1) )
                   => ( v2_bhsp_6(D,A)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & ! [F] :
                                  ( ( v1_finset_1(F)
                                    & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                                 => ~ ( ~ v1_xboole_0(F)
                                      & r1_tarski(F,B)
                                      & ! [G] :
                                          ( ( v1_finset_1(G)
                                            & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                                         => ~ ( r1_tarski(F,G)
                                              & r1_tarski(G,B)
                                              & r1_xreal_0(E,k18_complex1(k8_funct_2(u1_struct_0(A),k6_supinf_1,D,k6_rlvect_1(A,C,k1_bhsp_6(A,G))))) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
             => ( r1_bhsp_6(A,B,C)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ~ ( ~ r1_xreal_0(E,np__0)
                            & ! [F] :
                                ( ( v1_finset_1(F)
                                  & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                               => ~ ( ~ v1_xboole_0(F)
                                    & r1_tarski(F,B)
                                    & ! [G] :
                                        ( ( v1_finset_1(G)
                                          & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                                       => ~ ( r1_tarski(F,G)
                                            & r1_tarski(G,B)
                                            & r1_xreal_0(E,k18_complex1(k10_binop_2(D,k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,C)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d7_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
             => ( r1_bhsp_6(A,B,C)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ( D = k3_bhsp_6(A,B,C)
                    <=> ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & ! [F] :
                                  ( ( v1_finset_1(F)
                                    & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                                 => ~ ( ~ v1_xboole_0(F)
                                      & r1_tarski(F,B)
                                      & ! [G] :
                                          ( ( v1_finset_1(G)
                                            & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                                         => ~ ( r1_tarski(F,G)
                                              & r1_tarski(G,B)
                                              & r1_xreal_0(E,k18_complex1(k10_binop_2(D,k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,C)))) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t3_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_bhsp_6(B,A)
           => v3_bhsp_6(B,A) ) ) ) )).

fof(t4_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
            & v2_hahnban(B,A)
            & v3_hahnban(B,A)
            & m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
         => ! [C] :
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( r1_xreal_0(np__1,k3_finseq_1(C))
               => ! [D] :
                    ( m2_finseq_1(D,k1_numbers)
                   => ( ( k5_finsop_1(C) = k5_finsop_1(D)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( r2_hidden(E,k5_finsop_1(D))
                             => k1_funct_1(D,E) = k1_funct_1(B,k1_funct_1(C,E)) ) ) )
                     => k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k2_finsop_1(u1_struct_0(A),C,u1_rlvect_1(A))) = k2_finsop_1(k1_numbers,D,k33_binop_2) ) ) ) ) ) ) )).

fof(t5_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ( ~ v1_xboole_0(B)
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & v2_hahnban(C,A)
                    & v3_hahnban(C,A)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => k8_funct_2(u1_struct_0(A),k6_supinf_1,C,k1_bhsp_6(A,B)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,B,C) ) ) ) ) ) )).

fof(t6_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ~ ( v3_bhsp_6(B,A)
                & ! [C] :
                    ( m1_subset_1(C,u1_struct_0(A))
                   => ? [D] :
                        ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),k6_supinf_1)
                        & v2_hahnban(D,A)
                        & v3_hahnban(D,A)
                        & m2_relset_1(D,u1_struct_0(A),k6_supinf_1)
                        & v2_bhsp_6(D,A)
                        & ? [E] :
                            ( m1_subset_1(E,k1_numbers)
                            & ~ r1_xreal_0(E,np__0)
                            & ! [F] :
                                ( ( v1_finset_1(F)
                                  & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                               => ~ ( ~ v1_xboole_0(F)
                                    & r1_tarski(F,B)
                                    & ! [G] :
                                        ( ( v1_finset_1(G)
                                          & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                                       => ~ ( r1_tarski(F,G)
                                            & r1_tarski(G,B)
                                            & r1_xreal_0(E,k18_complex1(k10_binop_2(k8_funct_2(u1_struct_0(A),k6_supinf_1,D,C),k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,D)))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v3_bhsp_6(B,A)
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & v2_hahnban(C,A)
                    & v3_hahnban(C,A)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => ( v2_bhsp_6(C,A)
                   => r1_bhsp_6(A,B,C) ) ) ) ) ) ) )).

fof(t8_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v1_bhsp_6(B,A)
             => ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & v2_hahnban(C,A)
                    & v3_hahnban(C,A)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => ( v2_bhsp_6(C,A)
                   => r1_bhsp_6(A,B,C) ) ) ) ) ) ) )).

fof(t9_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( ~ v1_xboole_0(B)
           => v1_bhsp_6(B,A) ) ) ) )).

fof(t10_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A))
          & v4_bhsp_3(A) )
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v1_bhsp_6(B,A)
            <=> ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( ( v1_finset_1(D)
                            & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                         => ~ ( ~ v1_xboole_0(D)
                              & r1_tarski(D,B)
                              & ! [E] :
                                  ( ( v1_finset_1(E)
                                    & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                                 => ~ ( ~ v1_xboole_0(E)
                                      & r1_tarski(E,B)
                                      & r1_xboole_0(D,E)
                                      & r1_xreal_0(C,k3_bhsp_1(A,k1_bhsp_6(A,E))) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_k1_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A)
        & v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k1_bhsp_6(A,B),u1_struct_0(A)) ) )).

fof(dt_k2_bhsp_6,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)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k2_bhsp_6(A,B),u1_struct_0(A)) ) )).

fof(dt_k3_bhsp_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)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & v1_funct_1(C)
        & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
        & m1_relset_1(C,u1_struct_0(A),k6_supinf_1) )
     => m1_subset_1(k3_bhsp_6(A,B,C),k1_numbers) ) )).
%------------------------------------------------------------------------------