fof(t6_bhsp_2,theorem,(
    ! [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_2(B,A)
           => v1_bhsp_2(k6_bhsp_1(A,B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_bhsp_2,t77_bhsp_1]),
    [file(bhsp_2,t6_bhsp_2),interesting(1.00)]).

fof(t16_bhsp_2,theorem,(
    ! [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_2(B,A)
           => k1_bhsp_2(A,k6_bhsp_1(A,B)) = k5_rlvect_1(A,k1_bhsp_2(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t15_bhsp_2,t77_bhsp_1,t29_rlvect_1]),
    [file(bhsp_2,t16_bhsp_2),interesting(0.90)]).

fof(t15_bhsp_2,theorem,(
    ! [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_2(C,A)
               => k1_bhsp_2(A,k6_normsp_1(A,C,B)) = k3_rlvect_1(A,k1_bhsp_2(A,C),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_bhsp_2,d2_bhsp_2,t23_rlvect_1,t23_rlvect_1,t41_bhsp_1,t44_bhsp_1,d8_normsp_1,t133_complex1,t133_complex1,t73_real_1,d2_bhsp_2,d5_bhsp_1,t48_rlvect_1,t33_bhsp_1,d5_bhsp_1,d9_xcmplx_0,d7_xcmplx_0,t70_xreal_1,d8_normsp_1,d2_bhsp_2]),
    [file(bhsp_2,t15_bhsp_2),interesting(0.73)]).

fof(t22_bhsp_2,theorem,(
    ! [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_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_2(C,A)
                  & k1_bhsp_2(A,C) = B )
               => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,C,B)))
                  & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,C,B))) = 0 ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t8_bhsp_2,t20_bhsp_2,d1_xreal_0,t19_bhsp_2,t26_rlvect_1,t39_bhsp_1,t2_xreal_1,t32_bhsp_1,d7_normsp_1,d3_bhsp_2,d7_seq_2]),
    [file(bhsp_2,t22_bhsp_2),interesting(0.70)]).

fof(t18_bhsp_2,theorem,(
    ! [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_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_2(C,A)
               => k1_bhsp_2(A,k5_normsp_1(A,C,B)) = k6_rlvect_1(A,k1_bhsp_2(A,C),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t17_bhsp_2,t78_bhsp_1,d11_rlvect_1]),
    [file(bhsp_2,t18_bhsp_2),interesting(0.64)]).

fof(t32_bhsp_2,theorem,(
    ! [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_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_2(C,A)
                  & k1_bhsp_2(A,C) = B )
               => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k6_bhsp_1(A,C),k5_rlvect_1(A,B))))
                  & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k6_bhsp_1(A,C),k5_rlvect_1(A,B)))) = 0 ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t6_bhsp_2,t16_bhsp_2,t22_bhsp_2]),
    [file(bhsp_2,t32_bhsp_2),interesting(0.64)]).

fof(t31_bhsp_2,theorem,(
    ! [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_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_2(C,A)
                  & k1_bhsp_2(A,C) = B )
               => ( v4_seq_2(k2_bhsp_2(A,k6_bhsp_1(A,C)))
                  & k2_seq_2(k2_bhsp_2(A,k6_bhsp_1(A,C))) = k3_bhsp_1(A,k5_rlvect_1(A,B)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t6_bhsp_2,t16_bhsp_2,t21_bhsp_2]),
    [file(bhsp_2,t31_bhsp_2),interesting(0.61)]).

fof(l37_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k2_bhsp_2(A,k7_bhsp_1(A,D,C)))
                      & k2_seq_2(k2_bhsp_2(A,k7_bhsp_1(A,D,C))) = k3_bhsp_1(A,k4_rlvect_1(A,B,C)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_bhsp_2,t17_bhsp_2,t21_bhsp_2]),
    [file(bhsp_2,l37_bhsp_2),interesting(0.60)]).

fof(t33_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k7_bhsp_1(A,D,C),k4_rlvect_1(A,B,C))))
                      & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k7_bhsp_1(A,D,C),k4_rlvect_1(A,B,C)))) = 0 ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_bhsp_2,t17_bhsp_2,t22_bhsp_2]),
    [file(bhsp_2,t33_bhsp_2),interesting(0.60)]).

fof(t35_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k5_normsp_1(A,D,C),k6_rlvect_1(A,B,C))))
                      & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k5_normsp_1(A,D,C),k6_rlvect_1(A,B,C)))) = 0 ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t33_bhsp_2,t78_bhsp_1,d11_rlvect_1]),
    [file(bhsp_2,t35_bhsp_2),interesting(0.52)]).

fof(t34_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,D,C)))
                      & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,D,C))) = k3_bhsp_1(A,k6_rlvect_1(A,B,C)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[l37_bhsp_2,t78_bhsp_1,d11_rlvect_1]),
    [file(bhsp_2,t34_bhsp_2),interesting(0.48)]).

fof(t30_bhsp_2,theorem,(
    ! [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,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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k6_normsp_1(A,D,C),k3_rlvect_1(A,B,C))))
                      & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k6_normsp_1(A,D,C),k3_rlvect_1(A,B,C)))) = 0 ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_bhsp_2,t15_bhsp_2,t22_bhsp_2]),
    [file(bhsp_2,t30_bhsp_2),interesting(0.44)]).

fof(t29_bhsp_2,theorem,(
    ! [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,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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k2_bhsp_2(A,k6_normsp_1(A,D,C)))
                      & k2_seq_2(k2_bhsp_2(A,k6_normsp_1(A,D,C))) = k3_bhsp_1(A,k3_rlvect_1(A,B,C)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_bhsp_2,t15_bhsp_2,t21_bhsp_2]),
    [file(bhsp_2,t29_bhsp_2),interesting(0.41)]).

fof(t39_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k3_bhsp_2(A,k7_bhsp_1(A,D,C),k4_rlvect_1(A,B,C)))
                      & k2_seq_2(k3_bhsp_2(A,k7_bhsp_1(A,D,C),k4_rlvect_1(A,B,C))) = 0 ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_bhsp_2,t17_bhsp_2,t24_bhsp_2]),
    [file(bhsp_2,t39_bhsp_2),interesting(0.31)]).

fof(t38_bhsp_2,theorem,(
    ! [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,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)) )
                 => ( ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B )
                   => ( v4_seq_2(k3_bhsp_2(A,k6_normsp_1(A,D,C),k3_rlvect_1(A,B,C)))
                      & k2_seq_2(k3_bhsp_2(A,k6_normsp_1(A,D,C),k3_rlvect_1(A,B,C))) = 0 ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_bhsp_2,t15_bhsp_2,t24_bhsp_2]),
    [file(bhsp_2,t38_bhsp_2),interesting(0.27)]).

fof(t28_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                     => ( ( v1_bhsp_2(D,A)
                          & k1_bhsp_2(A,D) = B
                          & v1_bhsp_2(E,A)
                          & k1_bhsp_2(A,E) = C )
                       => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k4_normsp_1(A,D,E),k6_rlvect_1(A,B,C))))
                          & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k4_normsp_1(A,D,E),k6_rlvect_1(A,B,C)))) = 0 ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_bhsp_2,t14_bhsp_2,t22_bhsp_2]),
    [file(bhsp_2,t28_bhsp_2),interesting(0.16)]).

fof(t26_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                     => ( ( v1_bhsp_2(D,A)
                          & k1_bhsp_2(A,D) = B
                          & v1_bhsp_2(E,A)
                          & k1_bhsp_2(A,E) = C )
                       => ( v4_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k3_normsp_1(A,D,E),k4_rlvect_1(A,B,C))))
                          & k2_seq_2(k2_bhsp_2(A,k5_normsp_1(A,k3_normsp_1(A,D,E),k4_rlvect_1(A,B,C)))) = 0 ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_bhsp_2,t13_bhsp_2,t22_bhsp_2]),
    [file(bhsp_2,t26_bhsp_2),interesting(0.12)]).

fof(t25_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                     => ( ( v1_bhsp_2(D,A)
                          & k1_bhsp_2(A,D) = B
                          & v1_bhsp_2(E,A)
                          & k1_bhsp_2(A,E) = C )
                       => ( v4_seq_2(k2_bhsp_2(A,k3_normsp_1(A,D,E)))
                          & k2_seq_2(k2_bhsp_2(A,k3_normsp_1(A,D,E))) = k3_bhsp_1(A,k4_rlvect_1(A,B,C)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_bhsp_2,t13_bhsp_2,t21_bhsp_2]),
    [file(bhsp_2,t25_bhsp_2),interesting(0.09)]).

fof(t37_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                     => ( ( v1_bhsp_2(D,A)
                          & k1_bhsp_2(A,D) = B
                          & v1_bhsp_2(E,A)
                          & k1_bhsp_2(A,E) = C )
                       => ( v4_seq_2(k3_bhsp_2(A,k4_normsp_1(A,D,E),k6_rlvect_1(A,B,C)))
                          & k2_seq_2(k3_bhsp_2(A,k4_normsp_1(A,D,E),k6_rlvect_1(A,B,C))) = 0 ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_bhsp_2,t14_bhsp_2,t24_bhsp_2]),
    [file(bhsp_2,t37_bhsp_2),interesting(0.01)]).

fof(t17_normsp_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & l1_struct_0(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ( ( v1_funct_1(A)
                  & v1_funct_2(A,k5_numbers,u1_struct_0(B))
                  & m2_relset_1(A,k5_numbers,u1_struct_0(B)) )
              <=> ( k1_relat_1(A) = k5_numbers
                  & ! [D] : 
                      ( r2_hidden(D,k5_numbers)
                     => m1_subset_1(k1_funct_1(A,D),u1_struct_0(B)) ) ) ) ) ) ) ),
    file(normsp_1,t17_normsp_1),
    [interesting(0.00)]).

fof(d5_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( B = k2_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : 
                  ( r2_hidden(D,k1_relat_1(A))
                  & C = k1_funct_1(A,D) ) ) ) ) ),
    file(funct_1,d5_funct_1),
    [interesting(0.00)]).

fof(t27_normsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(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)) )
         => ( v5_seqm_3(B)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & k2_relat_1(B) = k1_struct_0(A,C) ) ) ) ) ),
    file(normsp_1,t27_normsp_1),
    [interesting(0.00)]).

fof(d4_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(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)) )
         => ( v5_seqm_3(B)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,u1_struct_0(A),B,D) = C ) ) ) ) ) ),
    file(normsp_1,d4_normsp_1),
    [interesting(0.00)]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.00)]).

fof(d1_bhsp_2,definition,(
    ! [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_2(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,0)
                        & ! [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] : 
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & r1_xreal_0(D,k5_bhsp_1(A,k2_normsp_1(A,B,F),C)) ) ) ) ) ) ) ) ) ),
    file(bhsp_2,d1_bhsp_2),
    [interesting(0.00)]).

fof(t41_bhsp_1,theorem,(
    ! [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))
         => k5_bhsp_1(A,B,B) = 0 ) ) ),
    file(bhsp_1,t41_bhsp_1),
    [interesting(0.00)]).

fof(t1_bhsp_2,theorem,(
    ! [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)) )
         => ( v5_seqm_3(B)
           => v1_bhsp_2(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_normsp_1,d1_bhsp_2,t41_bhsp_1]),
    [file(bhsp_2,t1_bhsp_2),interesting(0.00)]).

fof(d2_bhsp_2,definition,(
    ! [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_2(B,A)
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k1_bhsp_2(A,B)
                <=> ! [D] : 
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,0)
                          & ! [E] : 
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ? [F] : 
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(E,F)
                                  & r1_xreal_0(D,k5_bhsp_1(A,k2_normsp_1(A,B,F),C)) ) ) ) ) ) ) ) ) ) ),
    file(bhsp_2,d2_bhsp_2),
    [interesting(0.00)]).

fof(t10_bhsp_2,theorem,(
    ! [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_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( ( v5_seqm_3(C)
                  & r2_hidden(B,k2_relat_1(C)) )
               => k1_bhsp_2(A,C) = B ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_normsp_1,d4_normsp_1,d1_tarski,t1_bhsp_2,t17_normsp_1,d5_funct_1,d1_tarski,t41_bhsp_1,d2_bhsp_2]),
    [file(bhsp_2,t10_bhsp_2),interesting(0.00)]).

fof(t11_bhsp_2,theorem,(
    ! [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_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( v5_seqm_3(C)
               => ( ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => k2_normsp_1(A,C,D) != B )
                  | k1_bhsp_2(A,C) = B ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t17_normsp_1,d5_funct_1,t10_bhsp_2]),
    [file(bhsp_2,t11_bhsp_2),interesting(0.00)]).

fof(t2_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => r1_xreal_0(A,C) ) ) ) ) ),
    file(xreal_1,t2_xreal_1),
    [interesting(0.00)]).

fof(t2_bhsp_2,theorem,(
    ! [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_2(B,A)
               => ( ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ? [E] : 
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                          & r1_xreal_0(D,E)
                          & k2_normsp_1(A,C,E) != k2_normsp_1(A,B,E) ) )
                  | v1_bhsp_2(C,A) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_2,d1_bhsp_2,t2_xreal_1,t2_xreal_1]),
    [file(bhsp_2,t2_bhsp_2),interesting(0.00)]).

fof(t12_bhsp_2,theorem,(
    ! [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_2(B,A)
               => ( ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ? [E] : 
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                          & r1_xreal_0(D,E)
                          & k2_normsp_1(A,C,E) != k2_normsp_1(A,B,E) ) )
                  | k1_bhsp_2(A,B) = k1_bhsp_2(A,C) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_bhsp_2,d2_bhsp_2,t2_xreal_1,t2_xreal_1,d2_bhsp_2]),
    [file(bhsp_2,t12_bhsp_2),interesting(0.00)]).

fof(t47_bhsp_1,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_struct_0(A))
                     => r1_xreal_0(k5_bhsp_1(A,k4_rlvect_1(A,B,C),k4_rlvect_1(A,D,E)),k3_real_1(k5_bhsp_1(A,B,D),k5_bhsp_1(A,C,E))) ) ) ) ) ) ),
    file(bhsp_1,t47_bhsp_1),
    [interesting(0.00)]).

fof(d12_bhsp_1,definition,(
    ! [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] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [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_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k4_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ),
    file(bhsp_1,d12_bhsp_1),
    [interesting(0.00)]).

fof(t7_bhsp_2,theorem,(
    ! [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_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_2(C,A)
               => v1_bhsp_2(k7_bhsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_2,d1_bhsp_2,t47_bhsp_1,t41_bhsp_1,t2_xreal_1,d12_bhsp_1]),
    [file(bhsp_2,t7_bhsp_2),interesting(0.00)]).

fof(t49_bhsp_1,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k5_bhsp_1(A,k6_rlvect_1(A,B,C),k6_rlvect_1(A,D,C)) = k5_bhsp_1(A,B,D) ) ) ) ) ),
    file(bhsp_1,t49_bhsp_1),
    [interesting(0.00)]).

fof(d11_rlvect_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k6_rlvect_1(A,B,C) = k2_rlvect_1(A,B,k5_rlvect_1(A,C)) ) ) ) ),
    file(rlvect_1,d11_rlvect_1),
    [interesting(0.00)]).

fof(t30_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k5_rlvect_1(A,k5_rlvect_1(A,B)) = B ) ) ),
    file(rlvect_1,t30_rlvect_1),
    [interesting(0.00)]).

fof(t17_bhsp_2,theorem,(
    ! [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_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_2(C,A)
               => k1_bhsp_2(A,k7_bhsp_1(A,C,B)) = k4_rlvect_1(A,k1_bhsp_2(A,C),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_bhsp_2,d2_bhsp_2,t49_bhsp_1,d11_rlvect_1,t30_rlvect_1,d11_rlvect_1,t30_rlvect_1,d12_bhsp_1,d2_bhsp_2]),
    [file(bhsp_2,t17_bhsp_2),interesting(0.00)]).

fof(t78_bhsp_1,theorem,(
    ! [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_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => k5_normsp_1(A,C,B) = k7_bhsp_1(A,C,k5_rlvect_1(A,B)) ) ) ) ),
    file(bhsp_1,t78_bhsp_1),
    [interesting(0.00)]).

fof(t3_seq_2,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( ~ r1_xreal_0(A,0)
       => ( ~ r1_xreal_0(k7_xcmplx_0(A,2),0)
          & ~ r1_xreal_0(k7_xcmplx_0(A,4),0) ) ) ) ),
    file(seq_2,t3_seq_2),
    [interesting(0.00)]).

fof(t37_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ! [C] : 
              ( v4_ordinal2(C)
             => ( r1_xreal_0(A,B)
               => r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) ),
    file(nat_1,t37_nat_1),
    [interesting(0.00)]).

fof(t10_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ~ ( ~ r1_xreal_0(B,A)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ),
    file(xreal_1,t10_xreal_1),
    [interesting(0.00)]).

fof(d5_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( 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,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k3_normsp_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k4_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ),
    file(normsp_1,d5_normsp_1),
    [interesting(0.00)]).

fof(t3_bhsp_2,theorem,(
    ! [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_2(B,A)
                  & v1_bhsp_2(C,A) )
               => v1_bhsp_2(k3_normsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_2,d1_bhsp_2,d1_bhsp_2,t3_seq_2,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d5_normsp_1,t47_bhsp_1,t2_xreal_1]),
    [file(bhsp_2,t3_bhsp_2),interesting(0.00)]).

fof(t13_bhsp_2,theorem,(
    ! [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_2(B,A)
                  & v1_bhsp_2(C,A) )
               => k1_bhsp_2(A,k3_normsp_1(A,B,C)) = k4_rlvect_1(A,k1_bhsp_2(A,B),k1_bhsp_2(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_bhsp_2,t3_seq_2,d2_bhsp_2,d2_bhsp_2,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d5_normsp_1,t47_bhsp_1,t2_xreal_1,d2_bhsp_2]),
    [file(bhsp_2,t13_bhsp_2),interesting(0.00)]).

fof(d5_bhsp_1,definition,(
    ! [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))
             => k4_bhsp_1(A,B,C) = k3_bhsp_1(A,k6_rlvect_1(A,B,C)) ) ) ) ),
    file(bhsp_1,d5_bhsp_1),
    [interesting(0.00)]).

fof(t9_bhsp_2,theorem,(
    ! [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_2(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,0)
                        & ! [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] : 
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & r1_xreal_0(D,k3_bhsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_2,d5_bhsp_1,d1_bhsp_2,d5_bhsp_1]),
    [file(bhsp_2,t9_bhsp_2),interesting(0.00)]).

fof(d1_xreal_0,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
    <=> r2_hidden(A,k1_numbers) ) ),
    file(xreal_0,d1_xreal_0),
    [interesting(0.00)]).

fof(t39_bhsp_1,theorem,(
    ! [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_xreal_0(k18_complex1(k5_real_1(k3_bhsp_1(A,B),k3_bhsp_1(A,C))),k3_bhsp_1(A,k6_rlvect_1(A,B,C))) ) ) ) ),
    file(bhsp_1,t39_bhsp_1),
    [interesting(0.00)]).

fof(d3_bhsp_2,definition,(
    ! [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) )
             => ( C = k2_bhsp_2(A,B)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k3_bhsp_1(A,k2_normsp_1(A,B,D)) ) ) ) ) ) ),
    file(bhsp_2,d3_bhsp_2),
    [interesting(0.00)]).

fof(d6_seq_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
      <=> ? [B] : 
            ( v1_xreal_0(B)
            & ! [C] : 
                ( v1_xreal_0(C)
               => ~ ( ~ r1_xreal_0(C,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(C,k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ),
    file(seq_2,d6_seq_2),
    [interesting(0.00)]).

fof(t20_bhsp_2,theorem,(
    ! [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_2(B,A)
           => v4_seq_2(k2_bhsp_2(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_bhsp_2,d1_xreal_0,t39_bhsp_1,t2_xreal_1,d3_bhsp_2,d6_seq_2]),
    [file(bhsp_2,t20_bhsp_2),interesting(0.00)]).

fof(t19_bhsp_2,theorem,(
    ! [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_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_2(C,A)
               => ( k1_bhsp_2(A,C) = B
                <=> ! [D] : 
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,0)
                          & ! [E] : 
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ? [F] : 
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                  & r1_xreal_0(E,F)
                                  & r1_xreal_0(D,k3_bhsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,C,F),B))) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_2,d5_bhsp_1,d5_bhsp_1,d2_bhsp_2]),
    [file(bhsp_2,t19_bhsp_2),interesting(0.00)]).

fof(d7_seq_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
       => ! [B] : 
            ( v1_xreal_0(B)
           => ( B = k1_seq_2(A)
            <=> ! [C] : 
                  ( v1_xreal_0(C)
                 => ~ ( ~ r1_xreal_0(C,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(C,k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ),
    file(seq_2,d7_seq_2),
    [interesting(0.00)]).

fof(t21_bhsp_2,theorem,(
    ! [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_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_2(C,A)
                  & k1_bhsp_2(A,C) = B )
               => ( v4_seq_2(k2_bhsp_2(A,C))
                  & k2_seq_2(k2_bhsp_2(A,C)) = k3_bhsp_1(A,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_bhsp_2,d1_xreal_0,t19_bhsp_2,t39_bhsp_1,t2_xreal_1,d3_bhsp_2,d7_seq_2]),
    [file(bhsp_2,t21_bhsp_2),interesting(0.00)]).

fof(t8_bhsp_2,theorem,(
    ! [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_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_2(C,A)
               => v1_bhsp_2(k5_normsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_bhsp_2,t78_bhsp_1]),
    [file(bhsp_2,t8_bhsp_2),interesting(0.00)]).

fof(t26_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_rlvect_1(A,B,k1_rlvect_1(A)) = B ) ) ),
    file(rlvect_1,t26_rlvect_1),
    [interesting(0.00)]).

fof(t32_bhsp_1,theorem,(
    ! [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))
         => ( k3_bhsp_1(A,B) = 0
          <=> B = k1_rlvect_1(A) ) ) ) ),
    file(bhsp_1,t32_bhsp_1),
    [interesting(0.00)]).

fof(d7_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( 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] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [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 = k5_normsp_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ),
    file(normsp_1,d7_normsp_1),
    [interesting(0.00)]).

fof(d6_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( 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,u1_struct_0(A))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(A)) )
                 => ( D = k4_normsp_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,C,E)) ) ) ) ) ) ) ),
    file(normsp_1,d6_normsp_1),
    [interesting(0.00)]).

fof(t48_bhsp_1,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_struct_0(A))
                     => r1_xreal_0(k5_bhsp_1(A,k6_rlvect_1(A,B,C),k6_rlvect_1(A,D,E)),k3_real_1(k5_bhsp_1(A,B,D),k5_bhsp_1(A,C,E))) ) ) ) ) ) ),
    file(bhsp_1,t48_bhsp_1),
    [interesting(0.00)]).

fof(t4_bhsp_2,theorem,(
    ! [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_2(B,A)
                  & v1_bhsp_2(C,A) )
               => v1_bhsp_2(k4_normsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_2,d1_bhsp_2,d1_bhsp_2,t3_seq_2,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d6_normsp_1,t48_bhsp_1,t2_xreal_1]),
    [file(bhsp_2,t4_bhsp_2),interesting(0.00)]).

fof(t14_bhsp_2,theorem,(
    ! [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_2(B,A)
                  & v1_bhsp_2(C,A) )
               => k1_bhsp_2(A,k4_normsp_1(A,B,C)) = k6_rlvect_1(A,k1_bhsp_2(A,B),k1_bhsp_2(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_bhsp_2,t3_seq_2,d2_bhsp_2,d2_bhsp_2,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d6_normsp_1,t48_bhsp_1,t2_xreal_1,d2_bhsp_2]),
    [file(bhsp_2,t14_bhsp_2),interesting(0.00)]).

fof(t27_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                     => ( ( v1_bhsp_2(D,A)
                          & k1_bhsp_2(A,D) = B
                          & v1_bhsp_2(E,A)
                          & k1_bhsp_2(A,E) = C )
                       => ( v4_seq_2(k2_bhsp_2(A,k4_normsp_1(A,D,E)))
                          & k2_seq_2(k2_bhsp_2(A,k4_normsp_1(A,D,E))) = k3_bhsp_1(A,k6_rlvect_1(A,B,C)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_bhsp_2,t14_bhsp_2,t21_bhsp_2]),
    [file(bhsp_2,t27_bhsp_2),interesting(0.00)]).

fof(t23_rlvect_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & l2_rlvect_1(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ( ( A = 0
                  | C = k1_rlvect_1(B) )
               => k3_rlvect_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) ),
    file(rlvect_1,t23_rlvect_1),
    [interesting(0.00)]).

fof(t44_bhsp_1,theorem,(
    ! [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_xreal_0(0,k5_bhsp_1(A,B,C)) ) ) ) ),
    file(bhsp_1,t44_bhsp_1),
    [interesting(0.00)]).

fof(d8_normsp_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( ( 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] : 
              ( 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)) )
                 => ( D = k6_normsp_1(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),C) ) ) ) ) ) ) ),
    file(normsp_1,d8_normsp_1),
    [interesting(0.00)]).

fof(t133_complex1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ( ~ ( A != 0
            & r1_xreal_0(k17_complex1(A),0) )
        & ~ ( ~ r1_xreal_0(k17_complex1(A),0)
            & A = 0 ) ) ) ),
    file(complex1,t133_complex1),
    [interesting(0.00)]).

fof(t73_real_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,0)
               => ( ~ ( ~ r1_xreal_0(C,B)
                      & r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) )
                  & ~ ( ~ r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A))
                      & r1_xreal_0(C,B) ) ) ) ) ) ) ),
    file(real_1,t73_real_1),
    [interesting(0.00)]).

fof(t48_rlvect_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & l2_rlvect_1(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(B))
                 => k3_rlvect_1(B,k6_rlvect_1(B,C,D),A) = k6_rlvect_1(B,k3_rlvect_1(B,C,A),k3_rlvect_1(B,D,A)) ) ) ) ) ),
    file(rlvect_1,t48_rlvect_1),
    [interesting(0.00)]).

fof(t33_bhsp_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v3_rlvect_1(B)
            & v4_rlvect_1(B)
            & v5_rlvect_1(B)
            & v6_rlvect_1(B)
            & v7_rlvect_1(B)
            & v2_bhsp_1(B)
            & l1_bhsp_1(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => k3_bhsp_1(B,k3_rlvect_1(B,C,A)) = k4_real_1(k18_complex1(A),k3_bhsp_1(B,C)) ) ) ) ),
    file(bhsp_1,t33_bhsp_1),
    [interesting(0.00)]).

fof(d9_xcmplx_0,definition,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) ),
    file(xcmplx_0,d9_xcmplx_0),
    [interesting(0.00)]).

fof(d7_xcmplx_0,definition,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => ( ( A != 0
             => ( B = k5_xcmplx_0(A)
              <=> k3_xcmplx_0(A,B) = 1 ) )
            & ( A = 0
             => ( B = k5_xcmplx_0(A)
              <=> B = 0 ) ) ) ) ) ),
    file(xcmplx_0,d7_xcmplx_0),
    [interesting(0.00)]).

fof(t70_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ~ ( ~ r1_xreal_0(A,0)
                  & ~ r1_xreal_0(C,B)
                  & r1_xreal_0(k3_xcmplx_0(C,A),k3_xcmplx_0(B,A)) ) ) ) ) ),
    file(xreal_1,t70_xreal_1),
    [interesting(0.00)]).

fof(t5_bhsp_2,theorem,(
    ! [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_2(C,A)
               => v1_bhsp_2(k6_normsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_2,d1_bhsp_2,t23_rlvect_1,t23_rlvect_1,t41_bhsp_1,t44_bhsp_1,d8_normsp_1,t133_complex1,t133_complex1,t73_real_1,d5_bhsp_1,t48_rlvect_1,t33_bhsp_1,d5_bhsp_1,d9_xcmplx_0,d7_xcmplx_0,t70_xreal_1,d8_normsp_1]),
    [file(bhsp_2,t5_bhsp_2),interesting(0.00)]).

fof(t77_bhsp_1,theorem,(
    ! [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)) )
         => k6_normsp_1(A,B,k1_real_1(1)) = k6_bhsp_1(A,B) ) ) ),
    file(bhsp_1,t77_bhsp_1),
    [interesting(0.00)]).

fof(t29_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & v7_rlvect_1(A)
        & l2_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k5_rlvect_1(A,B) = k3_rlvect_1(A,B,k1_real_1(1)) ) ) ),
    file(rlvect_1,t29_rlvect_1),
    [interesting(0.00)]).

fof(d1_absvalue,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( ( r1_xreal_0(0,A)
         => k16_complex1(A) = A )
        & ( ~ r1_xreal_0(0,A)
         => k16_complex1(A) = k4_xcmplx_0(A) ) ) ) ),
    file(absvalue,d1_absvalue),
    [interesting(0.00)]).

fof(d4_bhsp_2,definition,(
    ! [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] : 
              ( m1_subset_1(C,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) )
                 => ( D = k3_bhsp_2(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_seq_1(k5_numbers,k1_numbers,D,E) = k5_bhsp_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ),
    file(bhsp_2,d4_bhsp_2),
    [interesting(0.00)]).

fof(t23_bhsp_2,theorem,(
    ! [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_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_2(C,A)
                  & k1_bhsp_2(A,C) = B )
               => v4_seq_2(k3_bhsp_2(A,C,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_xreal_0,d2_bhsp_2,t44_bhsp_1,d1_absvalue,d4_bhsp_2,d6_seq_2]),
    [file(bhsp_2,t23_bhsp_2),interesting(0.00)]).

fof(t24_bhsp_2,theorem,(
    ! [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_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_2(C,A)
                  & k1_bhsp_2(A,C) = B )
               => ( v4_seq_2(k3_bhsp_2(A,C,B))
                  & k2_seq_2(k3_bhsp_2(A,C,B)) = 0 ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t23_bhsp_2,d1_xreal_0,d2_bhsp_2,t44_bhsp_1,d1_absvalue,d4_bhsp_2,d7_seq_2]),
    [file(bhsp_2,t24_bhsp_2),interesting(0.00)]).

fof(t36_bhsp_2,theorem,(
    ! [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))
             => ! [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)) )
                     => ( ( v1_bhsp_2(D,A)
                          & k1_bhsp_2(A,D) = B
                          & v1_bhsp_2(E,A)
                          & k1_bhsp_2(A,E) = C )
                       => ( v4_seq_2(k3_bhsp_2(A,k3_normsp_1(A,D,E),k4_rlvect_1(A,B,C)))
                          & k2_seq_2(k3_bhsp_2(A,k3_normsp_1(A,D,E),k4_rlvect_1(A,B,C))) = 0 ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_bhsp_2,t13_bhsp_2,t24_bhsp_2]),
    [file(bhsp_2,t36_bhsp_2),interesting(0.00)]).

fof(t40_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k4_bhsp_2(A,C,D))
                  <=> ~ r1_xreal_0(D,k3_bhsp_1(A,k6_rlvect_1(A,C,B))) ) ) ) ) ) ),
    file(bhsp_2,t40_bhsp_2),
    [interesting(0.00)]).

fof(t41_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k4_bhsp_2(A,C,D))
                  <=> ~ r1_xreal_0(D,k5_bhsp_1(A,C,B)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t40_bhsp_2,d5_bhsp_1,d5_bhsp_1]),
    [file(bhsp_2,t41_bhsp_2),interesting(0.00)]).

fof(t42_bhsp_2,theorem,(
    ! [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,k1_numbers)
             => ( ~ r1_xreal_0(C,0)
               => r2_hidden(B,k4_bhsp_2(A,B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_bhsp_1,t41_bhsp_2]),
    [file(bhsp_2,t42_bhsp_2),interesting(0.00)]).

fof(t8_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,B)
              <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t8_xreal_1),
    [interesting(0.00)]).

fof(t42_bhsp_1,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => r1_xreal_0(k5_bhsp_1(A,B,C),k3_real_1(k5_bhsp_1(A,B,D),k5_bhsp_1(A,D,C))) ) ) ) ) ),
    file(bhsp_1,t42_bhsp_1),
    [interesting(0.00)]).

fof(t43_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] : 
                      ( m1_subset_1(E,k1_numbers)
                     => ~ ( r2_hidden(B,k4_bhsp_2(A,C,E))
                          & r2_hidden(D,k4_bhsp_2(A,C,E))
                          & r1_xreal_0(k4_real_1(2,E),k5_bhsp_1(A,B,D)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_bhsp_2,t41_bhsp_2,t8_xreal_1,t8_xreal_1,t2_xreal_1,t42_bhsp_1,t2_xreal_1]),
    [file(bhsp_2,t43_bhsp_2),interesting(0.00)]).

fof(t44_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] : 
                      ( m1_subset_1(E,k1_numbers)
                     => ( r2_hidden(B,k4_bhsp_2(A,C,E))
                       => r2_hidden(k6_rlvect_1(A,B,D),k4_bhsp_2(A,k6_rlvect_1(A,C,D),E)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_bhsp_2,t49_bhsp_1,t41_bhsp_2]),
    [file(bhsp_2,t44_bhsp_2),interesting(0.00)]).

fof(t47_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k5_rlvect_1(A,k6_rlvect_1(A,B,C)) = k2_rlvect_1(A,C,k5_rlvect_1(A,B)) ) ) ) ),
    file(rlvect_1,t47_rlvect_1),
    [interesting(0.00)]).

fof(t27_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v4_rlvect_1(A)
        & v5_rlvect_1(A)
        & v6_rlvect_1(A)
        & l1_rlvect_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => k6_rlvect_1(A,k1_rlvect_1(A),B) = k5_rlvect_1(A,B) ) ) ),
    file(rlvect_1,t27_rlvect_1),
    [interesting(0.00)]).

fof(t45_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k4_bhsp_2(A,C,D))
                   => r2_hidden(k6_rlvect_1(A,B,C),k4_bhsp_2(A,k1_rlvect_1(A),D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t40_bhsp_2,d11_rlvect_1,t47_rlvect_1,t27_rlvect_1]),
    [file(bhsp_2,t45_bhsp_2),interesting(0.00)]).

fof(t46_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] : 
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( r2_hidden(B,k4_bhsp_2(A,C,D))
                          & r1_xreal_0(D,E) )
                       => r2_hidden(B,k4_bhsp_2(A,C,E)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t40_bhsp_2,t2_xreal_1]),
    [file(bhsp_2,t46_bhsp_2),interesting(0.00)]).

fof(t47_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k5_bhsp_2(A,C,D))
                  <=> r1_xreal_0(k3_bhsp_1(A,k6_rlvect_1(A,C,B)),D) ) ) ) ) ) ),
    file(bhsp_2,t47_bhsp_2),
    [interesting(0.00)]).

fof(t48_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k5_bhsp_2(A,C,D))
                  <=> r1_xreal_0(k5_bhsp_1(A,C,B),D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t47_bhsp_2,d5_bhsp_1,d5_bhsp_1]),
    [file(bhsp_2,t48_bhsp_2),interesting(0.00)]).

fof(t49_bhsp_2,theorem,(
    ! [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,k1_numbers)
             => ( r1_xreal_0(0,C)
               => r2_hidden(B,k5_bhsp_2(A,B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_bhsp_1,t48_bhsp_2]),
    [file(bhsp_2,t49_bhsp_2),interesting(0.00)]).

fof(t51_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k6_bhsp_2(A,C,D))
                  <=> k3_bhsp_1(A,k6_rlvect_1(A,C,B)) = D ) ) ) ) ) ),
    file(bhsp_2,t51_bhsp_2),
    [interesting(0.00)]).

fof(t52_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k6_bhsp_2(A,C,D))
                  <=> k5_bhsp_1(A,C,B) = D ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t51_bhsp_2,d5_bhsp_1,d5_bhsp_1]),
    [file(bhsp_2,t52_bhsp_2),interesting(0.00)]).

fof(t53_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k6_bhsp_2(A,C,D))
                   => r2_hidden(B,k5_bhsp_2(A,C,D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t51_bhsp_2]),
    [file(bhsp_2,t53_bhsp_2),interesting(0.00)]).

fof(t50_bhsp_2,theorem,(
    ! [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))
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(B,k4_bhsp_2(A,C,D))
                   => r2_hidden(B,k5_bhsp_2(A,C,D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t40_bhsp_2]),
    [file(bhsp_2,t50_bhsp_2),interesting(0.00)]).

fof(t7_subset_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] : 
                ( m1_subset_1(D,A)
               => ( r2_hidden(D,B)
                 => r2_hidden(D,C) ) )
           => r1_tarski(B,C) ) ) ) ),
    file(subset_1,t7_subset_1),
    [interesting(0.00)]).

fof(t54_bhsp_2,theorem,(
    ! [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,k1_numbers)
             => r1_tarski(k4_bhsp_2(A,B,C),k5_bhsp_2(A,B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t50_bhsp_2,t7_subset_1]),
    [file(bhsp_2,t54_bhsp_2),interesting(0.00)]).

fof(t55_bhsp_2,theorem,(
    ! [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,k1_numbers)
             => r1_tarski(k6_bhsp_2(A,B,C),k5_bhsp_2(A,B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t51_bhsp_2,t7_subset_1]),
    [file(bhsp_2,t55_bhsp_2),interesting(0.00)]).

fof(t8_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    file(xboole_1,t8_xboole_1),
    [interesting(0.00)]).

fof(d5_real_1,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
          <=> ~ ( r1_xreal_0(B,A)
                & B != A ) ) ) ) ),
    file(real_1,d5_real_1),
    [interesting(0.00)]).

fof(d2_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    [interesting(0.00)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.00)]).

fof(t56_bhsp_2,theorem,(
    ! [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,k1_numbers)
             => k4_subset_1(u1_struct_0(A),k4_bhsp_2(A,B,C),k6_bhsp_2(A,B,C)) = k5_bhsp_2(A,B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t54_bhsp_2,t55_bhsp_2,t8_xboole_1,t47_bhsp_2,d5_real_1,d2_xboole_0,t7_subset_1,d10_xboole_0]),
    [file(bhsp_2,t56_bhsp_2),interesting(0.00)]).