SET007 Axioms: SET007+759.ax


%------------------------------------------------------------------------------
% File     : SET007+759 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Bessel's Inequality
% 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_5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   36 (   2 unit)
%            Number of atoms       :  515 (  58 equality)
%            Maximal formula depth :   21 (  12 average)
%            Number of connectives :  524 (  45 ~  ;   3  |; 309  &)
%                                         (   6 <=>; 161 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   34 (   1 propositional; 0-3 arity)
%            Number of functors    :   38 (   7 constant; 0-6 arity)
%            Number of variables   :  136 (   0 singleton; 127 !;   9 ?)
%            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_bhsp_5,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_bhsp_5(B,A)
          & v1_finset_1(B) ) ) )).

fof(rc2_bhsp_5,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] :
          ( m2_bhsp_5(B,A)
          & v1_finset_1(B) ) ) )).

fof(t1_bhsp_5,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_finseq_1(C,A)
         => ( ( v2_funct_1(B)
              & v2_funct_1(C)
              & k2_relat_1(B) = k2_relat_1(C) )
           => ( k5_finsop_1(B) = k5_finsop_1(C)
              & ? [D] :
                  ( v1_funct_1(D)
                  & v1_funct_2(D,k5_finsop_1(B),k5_finsop_1(B))
                  & v3_funct_2(D,k5_finsop_1(B),k5_finsop_1(B))
                  & m2_relset_1(D,k5_finsop_1(B),k5_finsop_1(B))
                  & C = k5_relat_1(D,B)
                  & k1_relat_1(D) = k5_finsop_1(B)
                  & k2_relat_1(D) = k5_finsop_1(B) ) ) ) ) ) )).

fof(d1_bhsp_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),A)
            & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
         => ( ( v1_binop_1(B,A)
              & v2_binop_1(B,A)
              & v1_setwiseo(B,A) )
           => ! [C] :
                ( ( v1_finset_1(C)
                  & m1_subset_1(C,k1_zfmisc_1(A)) )
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ( D = k1_bhsp_5(A,B,C)
                    <=> ? [E] :
                          ( m2_finseq_1(E,A)
                          & v2_funct_1(E)
                          & k2_relat_1(E) = C
                          & D = k2_finsop_1(A,E,B) ) ) ) ) ) ) ) )).

fof(d2_bhsp_5,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))) )
         => ( ( B != k1_xboole_0
             => k2_bhsp_5(A,B) = k1_bhsp_5(u1_struct_0(A),u1_rlvect_1(A),B) )
            & ( B = k1_xboole_0
             => k2_bhsp_5(A,B) = k1_rlvect_1(A) ) ) ) ) )).

fof(d3_bhsp_5,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,u1_struct_0(A))
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k3_bhsp_5(A,B,C,D) = k1_funct_1(u1_bhsp_1(A),k4_tarski(B,k1_funct_1(C,D))) ) ) ) ) )).

fof(d4_bhsp_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,A)
                & m2_relset_1(C,B,A) )
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => k4_bhsp_5(A,B,C,D) = k5_relat_1(D,C) ) ) ) ) )).

fof(d5_bhsp_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( ( v1_binop_1(C,A)
                  & v2_binop_1(C,A)
                  & v1_setwiseo(C,A) )
               => ! [D] :
                    ( ( v1_finset_1(D)
                      & m1_subset_1(D,k1_zfmisc_1(B)) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,B,A)
                          & m2_relset_1(E,B,A) )
                       => ( r1_tarski(D,k1_relat_1(E))
                         => ! [F] :
                              ( m1_subset_1(F,A)
                             => ( F = k5_bhsp_5(A,B,C,D,E)
                              <=> ? [G] :
                                    ( m2_finseq_1(G,B)
                                    & v2_funct_1(G)
                                    & k2_relat_1(G) = D
                                    & F = k2_finsop_1(A,k4_bhsp_5(A,B,E,G),C) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_bhsp_5,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,u1_struct_0(A))
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( D = k6_bhsp_5(A,B,C)
                  <=> ? [E] :
                        ( m2_finseq_1(E,u1_struct_0(A))
                        & v2_funct_1(E)
                        & k2_relat_1(E) = C
                        & ? [F] :
                            ( m2_finseq_1(F,k1_numbers)
                            & k5_finsop_1(F) = k5_finsop_1(E)
                            & ! [G] :
                                ( m2_subset_1(G,k1_numbers,k5_numbers)
                               => ( r2_hidden(G,k5_finsop_1(F))
                                 => k1_funct_1(F,G) = k3_bhsp_5(A,B,E,G) ) )
                            & D = k2_finsop_1(k1_numbers,F,k33_binop_2) ) ) ) ) ) ) ) )).

fof(d7_bhsp_5,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,u1_struct_0(A))
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( ( C != k1_xboole_0
                 => k7_bhsp_5(A,B,C) = k6_bhsp_5(A,B,C) )
                & ( C = k1_xboole_0
                 => k7_bhsp_5(A,B,C) = np__0 ) ) ) ) ) )).

fof(d8_bhsp_5,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)))
         => ( m1_bhsp_5(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( ( r2_hidden(C,B)
                        & r2_hidden(D,B) )
                     => ( C = D
                        | k2_bhsp_1(A,C,D) = np__0 ) ) ) ) ) ) ) )).

fof(t2_bhsp_5,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) )
     => m1_bhsp_5(k1_xboole_0,A) ) )).

fof(d9_bhsp_5,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)))
         => ( m2_bhsp_5(B,A)
          <=> ( m1_bhsp_5(B,A)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ( r2_hidden(C,B)
                   => k2_bhsp_1(A,C,C) = np__1 ) ) ) ) ) ) )).

fof(t3_bhsp_5,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) )
     => m2_bhsp_5(k1_xboole_0,A) ) )).

fof(t4_bhsp_5,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,u1_struct_0(A))
         => ( B = k1_rlvect_1(A)
          <=> ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => k2_bhsp_1(A,B,C) = np__0 ) ) ) ) )).

fof(t5_bhsp_5,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,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k9_binop_2(k7_square_1(k3_bhsp_1(A,k4_rlvect_1(A,B,C))),k7_square_1(k3_bhsp_1(A,k6_rlvect_1(A,B,C)))) = k9_binop_2(k11_binop_2(np__2,k7_square_1(k3_bhsp_1(A,B))),k11_binop_2(np__2,k7_square_1(k3_bhsp_1(A,C)))) ) ) ) )).

fof(t6_bhsp_5,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,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_bhsp_1(A,B,C)
               => k7_square_1(k3_bhsp_1(A,k4_rlvect_1(A,B,C))) = k9_binop_2(k7_square_1(k3_bhsp_1(A,B)),k7_square_1(k3_bhsp_1(A,C))) ) ) ) ) )).

fof(t7_bhsp_5,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] :
          ( m2_finseq_1(B,u1_struct_0(A))
         => ( ( r1_xreal_0(np__1,k3_finseq_1(B))
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r2_hidden(C,k5_finsop_1(B))
                          & r2_hidden(D,k5_finsop_1(B)) )
                       => ( C = D
                          | k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(B,C),k1_funct_1(B,D))) = np__0 ) ) ) ) )
           => ! [C] :
                ( m2_finseq_1(C,k1_numbers)
               => ( ( k5_finsop_1(B) = k5_finsop_1(C)
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ( r2_hidden(D,k5_finsop_1(C))
                         => k1_funct_1(C,D) = k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(B,D),k1_funct_1(B,D))) ) ) )
                 => k2_bhsp_1(A,k2_finsop_1(u1_struct_0(A),B,u1_rlvect_1(A)),k2_finsop_1(u1_struct_0(A),B,u1_rlvect_1(A))) = k2_finsop_1(k1_numbers,C,k33_binop_2) ) ) ) ) ) )).

fof(t8_bhsp_5,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,u1_struct_0(A))
         => ! [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(u1_bhsp_1(A),k4_tarski(B,k1_funct_1(C,E))) ) ) )
                     => k2_bhsp_1(A,B,k2_finsop_1(u1_struct_0(A),C,u1_rlvect_1(A))) = k2_finsop_1(k1_numbers,D,k33_binop_2) ) ) ) ) ) ) )).

fof(t9_bhsp_5,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_xboole_0(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,k1_relat_1(C))
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ! [E] :
                          ( m1_subset_1(E,u1_struct_0(A))
                         => ( ( r2_hidden(D,B)
                              & r2_hidden(E,B) )
                           => ( D = E
                              | k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E))) = np__0 ) ) ) ) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                      & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                   => ( ( r1_tarski(B,k1_relat_1(D))
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ( r2_hidden(E,B)
                             => k1_funct_1(D,E) = k1_funct_1(u1_bhsp_1(A),k4_tarski(k1_funct_1(C,E),k1_funct_1(C,E))) ) ) )
                     => ! [E] :
                          ( m2_finseq_1(E,u1_struct_0(A))
                         => ( ( v2_funct_1(E)
                              & k2_relat_1(E) = B )
                           => k1_funct_1(u1_bhsp_1(A),k4_tarski(k2_finsop_1(u1_struct_0(A),k4_bhsp_5(u1_struct_0(A),u1_struct_0(A),C,E),u1_rlvect_1(A)),k2_finsop_1(u1_struct_0(A),k4_bhsp_5(u1_struct_0(A),u1_struct_0(A),C,E),u1_rlvect_1(A)))) = k2_finsop_1(k1_numbers,k4_bhsp_5(k1_numbers,u1_struct_0(A),D,E),k33_binop_2) ) ) ) ) ) ) ) ) )).

fof(t10_bhsp_5,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,u1_struct_0(A))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
                 => ( r1_tarski(C,k1_relat_1(D))
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),k1_numbers)
                          & m2_relset_1(E,u1_struct_0(A),k1_numbers) )
                       => ( ( r1_tarski(C,k1_relat_1(E))
                            & ! [F] :
                                ( m1_subset_1(F,u1_struct_0(A))
                               => ( r2_hidden(F,C)
                                 => k1_funct_1(E,F) = k1_funct_1(u1_bhsp_1(A),k4_tarski(B,k1_funct_1(D,F))) ) ) )
                         => ! [F] :
                              ( m2_finseq_1(F,u1_struct_0(A))
                             => ( ( v2_funct_1(F)
                                  & k2_relat_1(F) = C )
                               => k1_funct_1(u1_bhsp_1(A),k4_tarski(B,k2_finsop_1(u1_struct_0(A),k4_bhsp_5(u1_struct_0(A),u1_struct_0(A),D,F),u1_rlvect_1(A)))) = k2_finsop_1(k1_numbers,k4_bhsp_5(k1_numbers,u1_struct_0(A),E,F),k33_binop_2) ) ) ) ) ) ) ) ) ) )).

fof(t11_bhsp_5,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,u1_struct_0(A))
           => ! [C] :
                ( ( v1_finset_1(C)
                  & m2_bhsp_5(C,A) )
               => ( ~ v1_xboole_0(C)
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                        & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                     => ( ( r1_tarski(C,k1_relat_1(D))
                          & ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => ( r2_hidden(E,C)
                               => k1_funct_1(D,E) = k7_square_1(k2_bhsp_1(A,B,E)) ) ) )
                       => ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,u1_struct_0(A),u1_struct_0(A))
                              & m2_relset_1(E,u1_struct_0(A),u1_struct_0(A)) )
                           => ( ( r1_tarski(C,k1_relat_1(E))
                                & ! [F] :
                                    ( m1_subset_1(F,u1_struct_0(A))
                                   => ( r2_hidden(F,C)
                                     => k1_funct_1(E,F) = k3_rlvect_1(A,F,k2_bhsp_1(A,B,F)) ) ) )
                             => k2_bhsp_1(A,B,k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),C,E)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,C,D) ) ) ) ) ) ) ) ) ) )).

fof(t12_bhsp_5,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,u1_struct_0(A))
           => ! [C] :
                ( ( v1_finset_1(C)
                  & m2_bhsp_5(C,A) )
               => ( ~ v1_xboole_0(C)
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),u1_struct_0(A))
                        & m2_relset_1(D,u1_struct_0(A),u1_struct_0(A)) )
                     => ( ( r1_tarski(C,k1_relat_1(D))
                          & ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => ( r2_hidden(E,C)
                               => k1_funct_1(D,E) = k3_rlvect_1(A,E,k2_bhsp_1(A,B,E)) ) ) )
                       => ! [E] :
                            ( ( v1_funct_1(E)
                              & v1_funct_2(E,u1_struct_0(A),k1_numbers)
                              & m2_relset_1(E,u1_struct_0(A),k1_numbers) )
                           => ( ( r1_tarski(C,k1_relat_1(E))
                                & ! [F] :
                                    ( m1_subset_1(F,u1_struct_0(A))
                                   => ( r2_hidden(F,C)
                                     => k1_funct_1(E,F) = k7_square_1(k2_bhsp_1(A,B,F)) ) ) )
                             => k2_bhsp_1(A,k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),C,D),k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),C,D)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,C,E) ) ) ) ) ) ) ) ) ) )).

fof(t13_bhsp_5,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,u1_struct_0(A))
           => ! [C] :
                ( ( v1_finset_1(C)
                  & m2_bhsp_5(C,A) )
               => ( ~ v1_xboole_0(C)
                 => ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),k1_numbers)
                        & m2_relset_1(D,u1_struct_0(A),k1_numbers) )
                     => ( ( r1_tarski(C,k1_relat_1(D))
                          & ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => ( r2_hidden(E,C)
                               => k1_funct_1(D,E) = k7_square_1(k2_bhsp_1(A,B,E)) ) ) )
                       => r1_xreal_0(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,C,D),k7_square_1(k3_bhsp_1(A,B))) ) ) ) ) ) ) ) )).

fof(t14_bhsp_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( ( v1_binop_1(C,A)
                  & v2_binop_1(C,A)
                  & v1_setwiseo(C,A) )
               => ! [D] :
                    ( ( v1_finset_1(D)
                      & m1_subset_1(D,k1_zfmisc_1(B)) )
                   => ! [E] :
                        ( ( v1_finset_1(E)
                          & m1_subset_1(E,k1_zfmisc_1(B)) )
                       => ( r1_xboole_0(D,E)
                         => ! [F] :
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,B,A)
                                & m2_relset_1(F,B,A) )
                             => ( ( r1_tarski(D,k1_relat_1(F))
                                  & r1_tarski(E,k1_relat_1(F)) )
                               => ! [G] :
                                    ( ( v1_finset_1(G)
                                      & m1_subset_1(G,k1_zfmisc_1(B)) )
                                   => ( G = k2_xboole_0(D,E)
                                     => k5_bhsp_5(A,B,C,G,F) = k2_binop_1(A,A,A,C,k5_bhsp_5(A,B,C,D,F),k5_bhsp_5(A,B,C,E,F)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_m1_bhsp_5,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_bhsp_5(B,A)
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(existence_m1_bhsp_5,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_bhsp_5(B,A) ) )).

fof(dt_m2_bhsp_5,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] :
          ( m2_bhsp_5(B,A)
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(existence_m2_bhsp_5,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] : m2_bhsp_5(B,A) ) )).

fof(dt_k1_bhsp_5,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m1_relset_1(B,k2_zfmisc_1(A,A),A)
        & v1_finset_1(C)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k1_bhsp_5(A,B,C),A) ) )).

fof(dt_k2_bhsp_5,axiom,(
    $true )).

fof(dt_k3_bhsp_5,axiom,(
    $true )).

fof(dt_k4_bhsp_5,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,B,A)
        & m1_relset_1(C,B,A)
        & m1_finseq_1(D,B) )
     => m2_finseq_1(k4_bhsp_5(A,B,C,D),A) ) )).

fof(dt_k5_bhsp_5,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A)
        & v1_finset_1(D)
        & m1_subset_1(D,k1_zfmisc_1(B))
        & v1_funct_1(E)
        & v1_funct_2(E,B,A)
        & m1_relset_1(E,B,A) )
     => m1_subset_1(k5_bhsp_5(A,B,C,D,E),A) ) )).

fof(dt_k6_bhsp_5,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,u1_struct_0(A))
        & v1_finset_1(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k6_bhsp_5(A,B,C),k1_numbers) ) )).

fof(dt_k7_bhsp_5,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,u1_struct_0(A))
        & v1_finset_1(C)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k7_bhsp_5(A,B,C),k1_numbers) ) )).
%------------------------------------------------------------------------------