SET007 Axioms: SET007+332.ax


%------------------------------------------------------------------------------
% File     : SET007+332 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Series in Banach and Hilbert Spaces
% 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_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   74 (   2 unit)
%            Number of atoms       : 1070 (  50 equality)
%            Maximal formula depth :   19 (  11 average)
%            Number of connectives : 1081 (  85 ~  ;   2  |; 745  &)
%                                         (   8 <=>; 241 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-3 arity)
%            Number of functors    :   42 (   6 constant; 0-4 arity)
%            Number of variables   :  241 (   0 singleton; 228 !;  13 ?)
%            Maximal term depth    :    7 (   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_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( C = k1_bhsp_4(A,B)
              <=> ( k2_normsp_1(A,C,np__0) = k2_normsp_1(A,B,np__0)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_normsp_1(A,C,k1_nat_1(D,np__1)) = k4_rlvect_1(A,k2_normsp_1(A,C,D),k2_normsp_1(A,B,k1_nat_1(D,np__1))) ) ) ) ) ) ) )).

fof(t1_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => k8_bhsp_1(A,k1_bhsp_4(A,B),k1_bhsp_4(A,C)) = k1_bhsp_4(A,k8_bhsp_1(A,B,C)) ) ) ) )).

fof(t2_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => k4_normsp_1(A,k1_bhsp_4(A,B),k1_bhsp_4(A,C)) = k1_bhsp_4(A,k4_normsp_1(A,B,C)) ) ) ) )).

fof(t3_bhsp_4,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_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => k1_bhsp_4(A,k6_normsp_1(A,C,B)) = k6_normsp_1(A,k1_bhsp_4(A,C),B) ) ) ) )).

fof(t4_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k1_bhsp_4(A,k6_bhsp_1(A,B)) = k6_bhsp_1(A,k1_bhsp_4(A,B)) ) ) )).

fof(t5_bhsp_4,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_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,u1_struct_0(A))
                        & m2_relset_1(E,k5_numbers,u1_struct_0(A)) )
                     => k8_bhsp_1(A,k6_normsp_1(A,k1_bhsp_4(A,D),B),k6_normsp_1(A,k1_bhsp_4(A,E),C)) = k1_bhsp_4(A,k8_bhsp_1(A,k6_normsp_1(A,D,B),k6_normsp_1(A,E,C))) ) ) ) ) ) )).

fof(d2_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v1_bhsp_4(B,A)
          <=> v1_bhsp_2(k1_bhsp_4(A,B),A) ) ) ) )).

fof(d3_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k2_bhsp_4(A,B) = k1_bhsp_2(A,k1_bhsp_4(A,B)) ) ) )).

fof(t6_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v1_bhsp_4(B,A)
                  & v1_bhsp_4(C,A) )
               => ( v1_bhsp_4(k8_bhsp_1(A,B,C),A)
                  & k2_bhsp_4(A,k8_bhsp_1(A,B,C)) = k4_rlvect_1(A,k2_bhsp_4(A,B),k2_bhsp_4(A,C)) ) ) ) ) ) )).

fof(t7_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v1_bhsp_4(B,A)
                  & v1_bhsp_4(C,A) )
               => ( v1_bhsp_4(k4_normsp_1(A,B,C),A)
                  & k2_bhsp_4(A,k4_normsp_1(A,B,C)) = k6_rlvect_1(A,k2_bhsp_4(A,B),k2_bhsp_4(A,C)) ) ) ) ) ) )).

fof(t8_bhsp_4,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_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( v1_bhsp_4(C,A)
               => ( v1_bhsp_4(k6_normsp_1(A,C,B),A)
                  & k2_bhsp_4(A,k6_normsp_1(A,C,B)) = k3_rlvect_1(A,k2_bhsp_4(A,C),B) ) ) ) ) ) )).

fof(t9_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v1_bhsp_4(B,A)
           => ( v1_bhsp_2(B,A)
              & k1_bhsp_2(A,B) = k1_rlvect_1(A) ) ) ) ) )).

fof(t10_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_bhsp_3(A)
           => ( v1_bhsp_4(B,A)
            <=> ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ? [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                              & ? [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(D,E)
                                  & r1_xreal_0(D,F)
                                  & r1_xreal_0(C,k3_bhsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,k1_bhsp_4(A,B),E),k2_normsp_1(A,k1_bhsp_4(A,B),F)))) ) ) ) ) ) ) ) ) ) )).

fof(t11_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v1_bhsp_4(B,A)
           => v2_bhsp_3(k1_bhsp_4(A,B),A) ) ) ) )).

fof(t12_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(A,C,D) = k2_normsp_1(A,B,np__0) )
               => k1_bhsp_4(A,k2_bhsp_3(A,B,np__1)) = k4_normsp_1(A,k2_bhsp_3(A,k1_bhsp_4(A,B),np__1),C) ) ) ) ) )).

fof(t13_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v1_bhsp_4(B,A)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => v1_bhsp_4(k2_bhsp_3(A,B,C),A) ) ) ) ) )).

fof(t14_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & v1_bhsp_4(k2_bhsp_3(A,B,C),A) )
           => v1_bhsp_4(B,A) ) ) ) )).

fof(d4_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_bhsp_4(A,B,C) = k2_normsp_1(A,k1_bhsp_4(A,B),C) ) ) ) )).

fof(t15_bhsp_4,axiom,(
    $true )).

fof(t16_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k3_bhsp_4(A,B,np__0) = k2_normsp_1(A,B,np__0) ) ) )).

fof(t17_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k3_bhsp_4(A,B,np__1) = k4_rlvect_1(A,k3_bhsp_4(A,B,np__0),k2_normsp_1(A,B,np__1)) ) ) )).

fof(t18_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k3_bhsp_4(A,B,np__1) = k4_rlvect_1(A,k2_normsp_1(A,B,np__0),k2_normsp_1(A,B,np__1)) ) ) )).

fof(t19_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_bhsp_4(A,B,k1_nat_1(C,np__1)) = k4_rlvect_1(A,k3_bhsp_4(A,B,C),k2_normsp_1(A,B,k1_nat_1(C,np__1))) ) ) ) )).

fof(t20_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_normsp_1(A,B,k1_nat_1(C,np__1)) = k6_rlvect_1(A,k3_bhsp_4(A,B,k1_nat_1(C,np__1)),k3_bhsp_4(A,B,C)) ) ) ) )).

fof(t21_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k2_normsp_1(A,B,np__1) = k6_rlvect_1(A,k3_bhsp_4(A,B,np__1),k3_bhsp_4(A,B,np__0)) ) ) )).

fof(d5_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k4_bhsp_4(A,B,C,D) = k6_rlvect_1(A,k3_bhsp_4(A,B,C),k3_bhsp_4(A,B,D)) ) ) ) ) )).

fof(t22_bhsp_4,axiom,(
    $true )).

fof(t23_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => k4_bhsp_4(A,B,np__1,np__0) = k2_normsp_1(A,B,np__1) ) ) )).

fof(t24_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k4_bhsp_4(A,B,k1_nat_1(C,np__1),C) = k2_normsp_1(A,B,k1_nat_1(C,np__1)) ) ) ) )).

fof(t25_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_bhsp_3(A)
           => ( v1_bhsp_4(B,A)
            <=> ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ? [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                              & ? [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(D,E)
                                  & r1_xreal_0(D,F)
                                  & r1_xreal_0(C,k3_bhsp_1(A,k6_rlvect_1(A,k3_bhsp_4(A,B,E),k3_bhsp_4(A,B,F)))) ) ) ) ) ) ) ) ) ) )).

fof(t26_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v4_bhsp_3(A)
           => ( v1_bhsp_4(B,A)
            <=> ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ? [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                              & ? [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(D,E)
                                  & r1_xreal_0(D,F)
                                  & r1_xreal_0(C,k3_bhsp_1(A,k4_bhsp_4(A,B,E,F))) ) ) ) ) ) ) ) ) ) )).

fof(d6_bhsp_4,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)
         => k5_bhsp_4(A,B) = k2_seq_1(k5_numbers,k1_numbers,k1_series_1(A),B) ) ) )).

fof(d7_bhsp_4,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)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k6_bhsp_4(A,B,C) = k5_real_1(k5_bhsp_4(A,B),k5_bhsp_4(A,C)) ) ) ) )).

fof(d8_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v2_bhsp_4(B,A)
          <=> v1_series_1(k2_bhsp_2(A,B)) ) ) ) )).

fof(t27_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v2_bhsp_4(B,A)
                  & v2_bhsp_4(C,A) )
               => v2_bhsp_4(k8_bhsp_1(A,B,C),A) ) ) ) ) )).

fof(t28_bhsp_4,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_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( v2_bhsp_4(C,A)
               => v2_bhsp_4(k6_normsp_1(A,C,B),A) ) ) ) ) )).

fof(t29_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k2_bhsp_2(A,B),D),k2_seq_1(k5_numbers,k1_numbers,C,D)) )
                  & v1_series_1(C) )
               => v2_bhsp_4(B,A) ) ) ) ) )).

fof(t30_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( k2_normsp_1(A,B,D) != k1_rlvect_1(A)
                        & k2_seq_1(k5_numbers,k1_numbers,C,D) = k6_real_1(k3_bhsp_1(A,k2_normsp_1(A,B,k1_nat_1(D,np__1))),k3_bhsp_1(A,k2_normsp_1(A,B,D))) ) )
                  & v4_seq_2(C) )
               => ( r1_xreal_0(np__1,k2_seq_2(C))
                  | v2_bhsp_4(B,A) ) ) ) ) ) )).

fof(t31_bhsp_4,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_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ~ ( ~ r1_xreal_0(B,np__0)
                  & ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r1_xreal_0(D,E)
                           => r1_xreal_0(B,k3_bhsp_1(A,k2_normsp_1(A,C,E))) ) ) )
                  & v1_bhsp_2(C,A)
                  & k1_bhsp_2(A,C) = k1_rlvect_1(A) ) ) ) ) )).

fof(t32_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ~ ( ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => k2_normsp_1(A,B,C) != k1_rlvect_1(A) )
              & ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(C,D)
                       => r1_xreal_0(np__1,k6_real_1(k3_bhsp_1(A,k2_normsp_1(A,B,k1_nat_1(D,np__1))),k3_bhsp_1(A,k2_normsp_1(A,B,D)))) ) ) )
              & v1_bhsp_4(B,A) ) ) ) )).

fof(t33_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ~ ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_normsp_1(A,B,D) != k1_rlvect_1(A) )
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k6_real_1(k3_bhsp_1(A,k2_normsp_1(A,B,k1_nat_1(D,np__1))),k3_bhsp_1(A,k2_normsp_1(A,B,D))) )
                  & v4_seq_2(C)
                  & ~ r1_xreal_0(k2_seq_2(C),np__1)
                  & v1_bhsp_4(B,A) ) ) ) ) )).

fof(t34_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_power(D,k3_bhsp_1(A,k2_normsp_1(A,B,D))) )
                  & v4_seq_2(C) )
               => ( r1_xreal_0(np__1,k2_seq_2(C))
                  | v2_bhsp_4(B,A) ) ) ) ) ) )).

fof(t35_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ~ ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_power(D,k2_seq_1(k5_numbers,k1_numbers,k2_bhsp_2(A,B),D)) )
                  & ? [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r1_xreal_0(D,E)
                           => r1_xreal_0(np__1,k2_seq_1(k5_numbers,k1_numbers,C,E)) ) ) )
                  & v1_bhsp_4(B,A) ) ) ) ) )).

fof(t36_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ~ ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_power(D,k2_seq_1(k5_numbers,k1_numbers,k2_bhsp_2(A,B),D)) )
                  & v4_seq_2(C)
                  & ~ r1_xreal_0(k2_seq_2(C),np__1)
                  & v1_bhsp_4(B,A) ) ) ) ) )).

fof(t37_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => v3_seqm_3(k1_series_1(k2_bhsp_2(A,B))) ) ) )).

fof(t38_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k2_bhsp_2(A,B)),C)) ) ) ) )).

fof(t39_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_xreal_0(k3_bhsp_1(A,k2_normsp_1(A,k1_bhsp_4(A,B),C)),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k2_bhsp_2(A,B)),C)) ) ) ) )).

fof(t40_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_xreal_0(k3_bhsp_1(A,k3_bhsp_4(A,B,C)),k5_bhsp_4(k2_bhsp_2(A,B),C)) ) ) ) )).

fof(t41_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => r1_xreal_0(k3_bhsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,k1_bhsp_4(A,B),D),k2_normsp_1(A,k1_bhsp_4(A,B),C))),k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k2_bhsp_2(A,B)),D),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k2_bhsp_2(A,B)),C)))) ) ) ) ) )).

fof(t42_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => r1_xreal_0(k3_bhsp_1(A,k6_rlvect_1(A,k3_bhsp_4(A,B,D),k3_bhsp_4(A,B,C))),k18_complex1(k5_real_1(k5_bhsp_4(k2_bhsp_2(A,B),D),k5_bhsp_4(k2_bhsp_2(A,B),C)))) ) ) ) ) )).

fof(t43_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => r1_xreal_0(k3_bhsp_1(A,k4_bhsp_4(A,B,D,C)),k18_complex1(k6_bhsp_4(k2_bhsp_2(A,B),D,C))) ) ) ) ) )).

fof(t44_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( ( v4_bhsp_3(A)
              & v2_bhsp_4(B,A) )
           => v1_bhsp_4(B,A) ) ) ) )).

fof(d9_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k7_bhsp_4(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k3_rlvect_1(A,k2_normsp_1(A,B,E),k2_seq_1(k5_numbers,k1_numbers,C,E)) ) ) ) ) ) ) )).

fof(t45_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => k7_bhsp_4(A,k8_bhsp_1(A,B,C),D) = k8_bhsp_1(A,k7_bhsp_4(A,B,D),k7_bhsp_4(A,C,D)) ) ) ) ) )).

fof(t46_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => k7_bhsp_4(A,B,k9_seq_1(C,D)) = k8_bhsp_1(A,k7_bhsp_4(A,B,C),k7_bhsp_4(A,B,D)) ) ) ) ) )).

fof(t47_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => k7_bhsp_4(A,B,k11_seq_1(C,D)) = k7_bhsp_4(A,k7_bhsp_4(A,B,D),C) ) ) ) ) )).

fof(t48_bhsp_4,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_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & m2_relset_1(D,k5_numbers,k1_numbers) )
                 => k7_bhsp_4(A,C,k14_seq_1(D,B)) = k6_normsp_1(A,k7_bhsp_4(A,C,D),B) ) ) ) ) )).

fof(t49_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k7_bhsp_4(A,k6_bhsp_1(A,B),C) = k7_bhsp_4(A,B,k17_seq_1(C)) ) ) ) )).

fof(t50_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_seq_2(C)
                  & v1_bhsp_2(B,A) )
               => v1_bhsp_2(k7_bhsp_4(A,B,C),A) ) ) ) ) )).

fof(t51_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v3_seq_2(C)
                  & v2_bhsp_3(B,A) )
               => v2_bhsp_3(k7_bhsp_4(A,B,C),A) ) ) ) ) )).

fof(t52_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_seq_2(C)
                  & v1_bhsp_2(B,A) )
               => ( v1_bhsp_2(k7_bhsp_4(A,B,C),A)
                  & k1_bhsp_2(A,k7_bhsp_4(A,B,C)) = k3_rlvect_1(A,k1_bhsp_2(A,B),k2_seq_2(C)) ) ) ) ) ) )).

fof(d10_bhsp_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v3_bhsp_4(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ~ ( ~ r1_xreal_0(B,np__0)
                & ! [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ? [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                        & ? [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & r1_xreal_0(C,D)
                            & r1_xreal_0(C,E)
                            & r1_xreal_0(B,k18_complex1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,A,E)))) ) ) ) ) ) ) ) )).

fof(t53_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( v4_bhsp_3(A)
                  & v1_bhsp_3(B,A)
                  & v3_bhsp_4(C) )
               => v1_bhsp_3(k7_bhsp_4(A,B,C),A) ) ) ) ) )).

fof(t54_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_normsp_1(A,k1_bhsp_4(A,k7_bhsp_4(A,k1_bhsp_4(A,B),k10_seq_1(C,k1_seqm_3(C,np__1)))),D) = k6_rlvect_1(A,k2_normsp_1(A,k1_bhsp_4(A,k7_bhsp_4(A,B,C)),k1_nat_1(D,np__1)),k2_normsp_1(A,k7_bhsp_4(A,k1_bhsp_4(A,B),C),k1_nat_1(D,np__1))) ) ) ) ) )).

fof(t55_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_normsp_1(A,k1_bhsp_4(A,k7_bhsp_4(A,B,C)),k1_nat_1(D,np__1)) = k6_rlvect_1(A,k2_normsp_1(A,k7_bhsp_4(A,k1_bhsp_4(A,B),C),k1_nat_1(D,np__1)),k2_normsp_1(A,k1_bhsp_4(A,k7_bhsp_4(A,k1_bhsp_4(A,B),k10_seq_1(k1_seqm_3(C,np__1),C))),D)) ) ) ) ) )).

fof(t56_bhsp_4,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,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k3_bhsp_4(A,k7_bhsp_4(A,B,C),k1_nat_1(D,np__1)) = k6_rlvect_1(A,k2_normsp_1(A,k7_bhsp_4(A,k1_bhsp_4(A,B),C),k1_nat_1(D,np__1)),k3_bhsp_4(A,k7_bhsp_4(A,k1_bhsp_4(A,B),k10_seq_1(k1_seqm_3(C,np__1),C)),D)) ) ) ) ) )).

fof(s1_bhsp_4,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,u1_struct_0(f1_s1_bhsp_4))
      & m2_relset_1(A,k5_numbers,u1_struct_0(f1_s1_bhsp_4))
      & k2_normsp_1(f1_s1_bhsp_4,A,np__0) = f2_s1_bhsp_4
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(f1_s1_bhsp_4))
             => ( C = k2_normsp_1(f1_s1_bhsp_4,A,B)
               => k2_normsp_1(f1_s1_bhsp_4,A,k1_nat_1(B,np__1)) = f3_s1_bhsp_4(B,C) ) ) ) ) )).

fof(dt_k1_bhsp_4,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_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => ( v1_funct_1(k1_bhsp_4(A,B))
        & v1_funct_2(k1_bhsp_4(A,B),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k1_bhsp_4(A,B),k5_numbers,u1_struct_0(A)) ) ) )).

fof(dt_k2_bhsp_4,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_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A)) )
     => m1_subset_1(k2_bhsp_4(A,B),u1_struct_0(A)) ) )).

fof(dt_k3_bhsp_4,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)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k3_bhsp_4(A,B,C),u1_struct_0(A)) ) )).

fof(dt_k4_bhsp_4,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & v2_bhsp_1(A)
        & l1_bhsp_1(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k5_numbers) )
     => m1_subset_1(k4_bhsp_4(A,B,C,D),u1_struct_0(A)) ) )).

fof(dt_k5_bhsp_4,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_subset_1(k5_bhsp_4(A,B),k1_numbers) ) )).

fof(dt_k6_bhsp_4,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k6_bhsp_4(A,B,C),k1_numbers) ) )).

fof(dt_k7_bhsp_4,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)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k1_numbers)
        & m1_relset_1(C,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k7_bhsp_4(A,B,C))
        & v1_funct_2(k7_bhsp_4(A,B,C),k5_numbers,u1_struct_0(A))
        & m2_relset_1(k7_bhsp_4(A,B,C),k5_numbers,u1_struct_0(A)) ) ) )).
%------------------------------------------------------------------------------