fof(t27_bhsp_3,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)) )
         => m1_bhsp_3(B,A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_funct_1,t38_nat_1,d11_seqm_3,t35_funct_1,t29_seqm_3,d4_bhsp_3,t26_bhsp_3,t35_funct_1,t113_funct_2]),
    [file(bhsp_3,t27_bhsp_3),interesting(1.00)]).

fof(t6_bhsp_3,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_3(B,A)
           => v1_bhsp_3(k6_bhsp_1(A,B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_bhsp_3,t77_bhsp_1]),
    [file(bhsp_3,t6_bhsp_3),interesting(0.99)]).

fof(t24_bhsp_3,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)
           => v2_bhsp_3(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_bhsp_2,t34_bhsp_1,t10_xreal_1,t38_bhsp_1,t2_xreal_1,t21_xreal_1,t23_bhsp_3,t10_xreal_1,t10_xreal_1,t10_xreal_1,t2_xreal_1,t2_xreal_1,d3_bhsp_3]),
    [file(bhsp_3,t24_bhsp_3),interesting(0.98)]).

fof(t51_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( v5_seqm_3(B)
               => v5_seqm_3(k2_bhsp_3(A,B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_bhsp_3,t29_bhsp_3]),
    [file(bhsp_3,t51_bhsp_3),interesting(0.93)]).

fof(t48_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( v1_bhsp_3(B,A)
               => v1_bhsp_3(k2_bhsp_3(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_bhsp_3,t34_bhsp_3]),
    [file(bhsp_3,t48_bhsp_3),interesting(0.88)]).

fof(t53_bhsp_3,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)) )
         => ( ( v3_bhsp_3(A)
              & v1_bhsp_3(B,A) )
           => v2_bhsp_3(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_bhsp_3,t24_bhsp_3]),
    [file(bhsp_3,t53_bhsp_3),interesting(0.88)]).

fof(t43_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => m1_bhsp_3(k2_bhsp_3(A,B,C),A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_seq_1,t38_nat_1,d11_seqm_3,t29_seqm_3,d4_bhsp_3,t26_bhsp_3,d5_bhsp_3,t113_funct_2]),
    [file(bhsp_3,t43_bhsp_3),interesting(0.85)]).

fof(t50_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( v2_bhsp_3(B,A)
               => v2_bhsp_3(k2_bhsp_3(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_bhsp_3,t31_bhsp_3]),
    [file(bhsp_3,t50_bhsp_3),interesting(0.85)]).

fof(t44_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( v1_bhsp_2(B,A)
               => ( v1_bhsp_2(k2_bhsp_3(A,B,C),A)
                  & k1_bhsp_2(A,k2_bhsp_3(A,B,C)) = k1_bhsp_2(A,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_bhsp_3,t32_bhsp_3,t33_bhsp_3]),
    [file(bhsp_3,t44_bhsp_3),interesting(0.82)]).

fof(t8_bhsp_3,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_3(C,A)
               => v1_bhsp_3(k5_normsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_bhsp_3,t78_bhsp_1]),
    [file(bhsp_3,t8_bhsp_3),interesting(0.82)]).

fof(t20_bhsp_3,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)) )
             => ( ( v2_bhsp_3(B,A)
                  & v2_bhsp_3(C,A) )
               => v2_bhsp_3(k4_normsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t19_bhsp_3,t71_bhsp_1,t18_bhsp_3]),
    [file(bhsp_3,t20_bhsp_3),interesting(0.80)]).

fof(t25_bhsp_3,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)) )
             => ( ( v2_bhsp_3(B,A)
                  & r2_bhsp_3(A,B,C) )
               => v2_bhsp_3(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_bhsp_3,d2_bhsp_3,t10_xreal_1,d5_bhsp_1,t38_bhsp_1,t2_xreal_1,t21_xreal_1,t8_xreal_1,t2_xreal_1,t23_bhsp_3,t10_xreal_1,t10_xreal_1,t10_xreal_1,t2_xreal_1,t2_xreal_1,d3_bhsp_3]),
    [file(bhsp_3,t25_bhsp_3),interesting(0.78)]).

fof(t16_bhsp_3,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)
                  & r2_bhsp_3(A,B,C) )
               => v1_bhsp_2(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_seq_2,d2_bhsp_2,d2_bhsp_3,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,t42_bhsp_1,t2_xreal_1,d1_bhsp_2]),
    [file(bhsp_3,t16_bhsp_3),interesting(0.75)]).

fof(t33_bhsp_3,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)) )
             => ( ( m1_bhsp_3(B,A,C)
                  & v1_bhsp_2(C,A) )
               => k1_bhsp_2(A,B) = k1_bhsp_2(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t32_bhsp_3,d4_bhsp_3,d2_bhsp_2,t33_seqm_3,t2_xreal_1,t26_bhsp_3,d2_bhsp_2]),
    [file(bhsp_3,t33_bhsp_3),interesting(0.74)]).

fof(t21_bhsp_3,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)) )
             => ( v2_bhsp_3(C,A)
               => v2_bhsp_3(k6_normsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_bhsp_3,d8_normsp_1,t23_rlvect_1,t32_bhsp_1,d3_bhsp_3,t133_complex1,t122_real_2,d8_normsp_1,t33_bhsp_1,t132_complex1,t66_xreal_1,d3_bhsp_3]),
    [file(bhsp_3,t21_bhsp_3),interesting(0.73)]).

fof(t39_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k2_bhsp_3(A,k6_bhsp_1(A,B),C) = k6_bhsp_1(A,k2_bhsp_3(A,B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_3,d10_bhsp_1,d5_bhsp_3,d10_bhsp_1,t113_funct_2]),
    [file(bhsp_3,t39_bhsp_3),interesting(0.67)]).

fof(t38_bhsp_3,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)) )
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_bhsp_3(A,k8_bhsp_1(A,B,C),D) = k8_bhsp_1(A,k2_bhsp_3(A,B,D),k2_bhsp_3(A,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_3,d5_normsp_1,d5_bhsp_3,d5_bhsp_3,d5_normsp_1,t113_funct_2]),
    [file(bhsp_3,t38_bhsp_3),interesting(0.56)]).

fof(t40_bhsp_3,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)) )
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_bhsp_3(A,k4_normsp_1(A,B,C),D) = k4_normsp_1(A,k2_bhsp_3(A,B,D),k2_bhsp_3(A,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t71_bhsp_1,t38_bhsp_3,t39_bhsp_3,t71_bhsp_1]),
    [file(bhsp_3,t40_bhsp_3),interesting(0.54)]).

fof(t17_bhsp_3,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)) )
             => ! [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(C,A)
                      & k1_bhsp_2(A,C) = B
                      & r2_bhsp_3(A,C,D) )
                   => ( v1_bhsp_2(D,A)
                      & k1_bhsp_2(A,D) = B ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_seq_2,d2_bhsp_2,d2_bhsp_3,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,t42_bhsp_1,t2_xreal_1,d1_bhsp_2,d2_bhsp_2]),
    [file(bhsp_3,t17_bhsp_3),interesting(0.48)]).

fof(t23_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ? [D] : 
                  ( m1_subset_1(D,k1_numbers)
                  & ~ r1_xreal_0(D,0)
                  & ! [E] : 
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( r1_xreal_0(E,C)
                          & r1_xreal_0(D,k3_bhsp_1(A,k2_normsp_1(A,B,E))) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t34_bhsp_1,t10_xreal_1,t18_nat_1,t10_xreal_1,t10_xreal_1,t10_xreal_1,t2_xreal_1,t10_xreal_1,t2_xreal_1,t26_nat_1,t34_bhsp_1,t10_xreal_1,t2_xreal_1,t10_xreal_1,t10_xreal_1,t26_nat_1,s1_nat_1]),
    [file(bhsp_3,t23_bhsp_3),interesting(0.45)]).

fof(t41_bhsp_3,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)) )
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_bhsp_3(A,k6_normsp_1(A,C,B),D) = k6_normsp_1(A,k2_bhsp_3(A,C,D),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_3,d8_normsp_1,d5_bhsp_3,d8_normsp_1,t113_funct_2]),
    [file(bhsp_3,t41_bhsp_3),interesting(0.44)]).

fof(t14_bhsp_3,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)) )
             => ( ? [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,B,E) = k2_normsp_1(A,C,E) ) ) )
               => r2_bhsp_3(A,B,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_3,t41_bhsp_1]),
    [file(bhsp_3,t14_bhsp_3),interesting(0.30)]).

fof(t26_bhsp_3,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,k1_numbers)
                & v1_seqm_3(C)
                & v7_seqm_3(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_normsp_1(A,k1_bhsp_3(A,C,B),D) = k2_normsp_1(A,B,k2_seqm_3(C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t22_funct_1]),
    [file(bhsp_3,t26_bhsp_3),interesting(0.27)]).

fof(t42_bhsp_3,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,k1_numbers)
                & v1_seqm_3(C)
                & v7_seqm_3(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_bhsp_3(A,k1_bhsp_3(A,C,B),D) = k1_bhsp_3(A,k1_seqm_3(C,D),B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_3,t26_bhsp_3,d9_seqm_3,t26_bhsp_3,t113_funct_2]),
    [file(bhsp_3,t42_bhsp_3),interesting(0.24)]).

fof(d2_bhsp_3,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,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( r1_bhsp_3(A,B,C)
              <=> ! [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),k2_normsp_1(A,C,F))) ) ) ) ) ) ) ) ) ),
    file(bhsp_3,d2_bhsp_3),
    [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(t10_bhsp_3,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)) )
         => r1_bhsp_3(A,B,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_3,t41_bhsp_1]),
    [file(bhsp_3,t10_bhsp_3),interesting(0.00)]).

fof(t11_bhsp_3,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)) )
             => ( r1_bhsp_3(A,B,C)
               => r1_bhsp_3(A,C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_3,d2_bhsp_3]),
    [file(bhsp_3,t11_bhsp_3),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(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(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(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(t12_bhsp_3,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)) )
             => ! [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)) )
                 => ( ( r2_bhsp_3(A,B,C)
                      & r2_bhsp_3(A,C,D) )
                   => r2_bhsp_3(A,B,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_3,t3_seq_2,d2_bhsp_3,d2_bhsp_3,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,t42_bhsp_1,t2_xreal_1]),
    [file(bhsp_3,t12_bhsp_3),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(t13_bhsp_3,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)) )
             => ( r2_bhsp_3(A,B,C)
              <=> ! [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),k2_normsp_1(A,C,F)))) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_3,d5_bhsp_1,d2_bhsp_3,d5_bhsp_1]),
    [file(bhsp_3,t13_bhsp_3),interesting(0.00)]).

fof(d1_bhsp_3,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_3(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,0)
                    & ! [D] : 
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & ? [F] : 
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(D,E)
                                & r1_xreal_0(D,F)
                                & r1_xreal_0(C,k5_bhsp_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F))) ) ) ) ) ) ) ) ) ),
    file(bhsp_3,d1_bhsp_3),
    [interesting(0.00)]).

fof(t224_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ~ ( ~ r1_xreal_0(A,0)
          & r1_xreal_0(k7_xcmplx_0(A,3),0) ) ) ),
    file(xreal_1,t224_xreal_1),
    [interesting(0.00)]).

fof(t15_bhsp_3,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_3(B,A)
                  & r2_bhsp_3(A,B,C) )
               => v1_bhsp_3(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,t224_xreal_1,d1_bhsp_3,d2_bhsp_3,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t2_xreal_1,t10_xreal_1,t42_bhsp_1,t2_xreal_1,t10_xreal_1,t42_bhsp_1,t2_xreal_1]),
    [file(bhsp_3,t15_bhsp_3),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(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(t70_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)) )
         => ( v5_seqm_3(B)
          <=> ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(A,B,C) = k2_normsp_1(A,B,D) ) ) ) ) ) ),
    file(bhsp_1,t70_bhsp_1),
    [interesting(0.00)]).

fof(t1_bhsp_3,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_3(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,t70_bhsp_1,t41_bhsp_1]),
    [file(bhsp_3,t1_bhsp_3),interesting(0.00)]).

fof(d3_bhsp_3,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)) )
         => ( v2_bhsp_3(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,k1_numbers)
                & ~ r1_xreal_0(C,0)
                & ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => r1_xreal_0(k3_bhsp_1(A,k2_normsp_1(A,B,D)),C) ) ) ) ) ) ),
    file(bhsp_3,d3_bhsp_3),
    [interesting(0.00)]).

fof(d10_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] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => ( C = k6_bhsp_1(A,B)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_normsp_1(A,C,D) = k5_rlvect_1(A,k2_normsp_1(A,B,D)) ) ) ) ) ) ),
    file(bhsp_1,d10_bhsp_1),
    [interesting(0.00)]).

fof(t37_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,k5_rlvect_1(A,B)) = k3_bhsp_1(A,B) ) ) ),
    file(bhsp_1,t37_bhsp_1),
    [interesting(0.00)]).

fof(t19_bhsp_3,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)) )
         => ( v2_bhsp_3(B,A)
           => v2_bhsp_3(k6_bhsp_1(A,B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_bhsp_3,d10_bhsp_1,t37_bhsp_1,d3_bhsp_3]),
    [file(bhsp_3,t19_bhsp_3),interesting(0.00)]).

fof(t71_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)) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(A))
                & m2_relset_1(C,k5_numbers,u1_struct_0(A)) )
             => k4_normsp_1(A,B,C) = k8_bhsp_1(A,B,k6_bhsp_1(A,C)) ) ) ) ),
    file(bhsp_1,t71_bhsp_1),
    [interesting(0.00)]).

fof(t9_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(A,B)
                      & r1_xreal_0(C,D) )
                   => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ),
    file(xreal_1,t9_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(t36_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(k3_bhsp_1(A,k4_rlvect_1(A,B,C)),k3_real_1(k3_bhsp_1(A,B),k3_bhsp_1(A,C))) ) ) ) ),
    file(bhsp_1,t36_bhsp_1),
    [interesting(0.00)]).

fof(t18_bhsp_3,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)) )
             => ( ( v2_bhsp_3(B,A)
                  & v2_bhsp_3(C,A) )
               => v2_bhsp_3(k8_bhsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_bhsp_3,d3_bhsp_3,t10_xreal_1,t9_xreal_1,d5_normsp_1,t36_bhsp_1,t2_xreal_1,d3_bhsp_3]),
    [file(bhsp_3,t18_bhsp_3),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(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(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(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(t122_real_2,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ( ( ~ r1_xreal_0(0,A)
                  & ~ r1_xreal_0(0,B) )
                | ( ~ r1_xreal_0(A,0)
                  & ~ r1_xreal_0(B,0) ) )
              & r1_xreal_0(k3_xcmplx_0(A,B),0) ) ) ) ),
    file(real_2,t122_real_2),
    [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(t132_complex1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => r1_xreal_0(0,k17_complex1(A)) ) ),
    file(complex1,t132_complex1),
    [interesting(0.00)]).

fof(t66_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(0,C) )
               => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t66_xreal_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(t3_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ~ ! [B] : 
            ( v1_xreal_0(B)
           => r1_xreal_0(B,A) ) ) ),
    file(xreal_1,t3_xreal_1),
    [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(t34_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))
         => r1_xreal_0(0,k3_bhsp_1(A,B)) ) ) ),
    file(bhsp_1,t34_bhsp_1),
    [interesting(0.00)]).

fof(t22_bhsp_3,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)
           => v2_bhsp_3(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_normsp_1,t3_xreal_1,d1_xreal_0,t32_bhsp_1,d3_bhsp_3,t32_bhsp_1,t34_bhsp_1,t3_xreal_1,d1_xreal_0,d3_bhsp_3]),
    [file(bhsp_3,t22_bhsp_3),interesting(0.00)]).

fof(t38_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(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,t38_bhsp_1),
    [interesting(0.00)]).

fof(t21_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(k2_xcmplx_0(A,B),C)
              <=> r1_xreal_0(A,k6_xcmplx_0(C,B)) ) ) ) ) ),
    file(xreal_1,t21_xreal_1),
    [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(t18_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => r1_xreal_0(0,A) ) ),
    file(nat_1,t18_nat_1),
    [interesting(0.00)]).

fof(t26_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,k2_xcmplx_0(B,1))
              & ~ r1_xreal_0(A,B)
              & A != k2_xcmplx_0(B,1) ) ) ) ),
    file(nat_1,t26_nat_1),
    [interesting(0.00)]).

fof(s1_nat_1,theorem,
    ( ( p1_s1_nat_1(0)
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s1_nat_1(A)
           => p1_s1_nat_1(k1_nat_1(A,1)) ) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => p1_s1_nat_1(A) ) ),
    file(nat_1,s1_nat_1),
    [interesting(0.00)]).

fof(t35_funct_1,theorem,(
    ! [A,B] : 
      ( r2_hidden(B,A)
     => k1_funct_1(k6_relat_1(A),B) = B ) ),
    file(funct_1,t35_funct_1),
    [interesting(0.00)]).

fof(t38_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A)
          <=> r1_xreal_0(A,B) ) ) ) ),
    file(nat_1,t38_nat_1),
    [interesting(0.00)]).

fof(d11_seqm_3,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v1_seqm_3(A)
      <=> ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,1)),k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ),
    file(seqm_3,d11_seqm_3),
    [interesting(0.00)]).

fof(t29_seqm_3,theorem,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v1_seqm_3(A)
          & v7_seqm_3(A)
          & m2_relset_1(A,k5_numbers,k1_numbers) )
      <=> ( v1_seqm_3(A)
          & ! [B] : 
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => m2_subset_1(k2_seq_1(k5_numbers,k1_numbers,A,B),k1_numbers,k5_numbers) ) ) ) ) ),
    file(seqm_3,t29_seqm_3),
    [interesting(0.00)]).

fof(d4_bhsp_3,definition,(
    ! [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)) )
         => ! [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)) )
             => ( m1_bhsp_3(C,A,B)
              <=> ? [D] : 
                    ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_numbers)
                    & v1_seqm_3(D)
                    & v7_seqm_3(D)
                    & m2_relset_1(D,k5_numbers,k1_numbers)
                    & C = k5_relat_1(D,B) ) ) ) ) ) ),
    file(bhsp_3,d4_bhsp_3),
    [interesting(0.00)]).

fof(d1_funct_2,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
         => ( v1_funct_2(C,A,B)
          <=> A = k4_relset_1(A,B,C) ) )
        & ( B = k1_xboole_0
         => ( A = k1_xboole_0
            | ( v1_funct_2(C,A,B)
            <=> C = k1_xboole_0 ) ) ) ) ) ),
    file(funct_2,d1_funct_2),
    [interesting(0.00)]).

fof(t22_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B)))
           => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ),
    file(funct_1,t22_funct_1),
    [interesting(0.00)]).

fof(t113_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) )
         => ( ! [E] : 
                ( m1_subset_1(E,A)
               => k1_funct_1(C,E) = k1_funct_1(D,E) )
           => C = D ) ) ) ),
    file(funct_2,t113_funct_2),
    [interesting(0.00)]).

fof(t55_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ! [C] : 
              ( v1_relat_1(C)
             => k5_relat_1(k5_relat_1(A,B),C) = k5_relat_1(A,k5_relat_1(B,C)) ) ) ) ),
    file(relat_1,t55_relat_1),
    [interesting(0.00)]).

fof(t28_bhsp_3,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)) )
             => ! [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)) )
                 => ( ( m1_bhsp_3(B,A,C)
                      & m1_bhsp_3(C,A,D) )
                   => m1_bhsp_3(B,A,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_bhsp_3,d4_bhsp_3,d4_bhsp_3,t55_relat_1]),
    [file(bhsp_3,t28_bhsp_3),interesting(0.00)]).

fof(t2_bhsp_3,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_3(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,0)
                    & ! [D] : 
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ? [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & ? [F] : 
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(D,E)
                                & r1_xreal_0(D,F)
                                & r1_xreal_0(C,k3_bhsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,E),k2_normsp_1(A,B,F)))) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,d5_bhsp_1,d1_bhsp_3,d5_bhsp_1]),
    [file(bhsp_3,t2_bhsp_3),interesting(0.00)]).

fof(t30_bhsp_3,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)) )
             => ( ( v5_seqm_3(B)
                  & m1_bhsp_3(C,A,B) )
               => B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_bhsp_3,t26_bhsp_3,t70_bhsp_1,t113_funct_2]),
    [file(bhsp_3,t30_bhsp_3),interesting(0.00)]).

fof(d5_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_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 = k2_bhsp_3(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(A,D,E) = k2_normsp_1(A,B,k1_nat_1(E,C)) ) ) ) ) ) ) ),
    file(bhsp_3,d5_bhsp_3),
    [interesting(0.00)]).

fof(t35_bhsp_3,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)) )
         => k2_bhsp_3(A,B,0) = B ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_3,t113_funct_2]),
    [file(bhsp_3,t35_bhsp_3),interesting(0.00)]).

fof(t36_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_bhsp_3(A,k2_bhsp_3(A,B,C),D) = k2_bhsp_3(A,k2_bhsp_3(A,B,D),C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_3,d5_bhsp_3,d5_bhsp_3,d5_bhsp_3,t113_funct_2]),
    [file(bhsp_3,t36_bhsp_3),interesting(0.00)]).

fof(t37_bhsp_3,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] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k2_bhsp_3(A,k2_bhsp_3(A,B,C),D) = k2_bhsp_3(A,B,k1_nat_1(C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_3,d5_bhsp_3,d5_bhsp_3,t113_funct_2]),
    [file(bhsp_3,t37_bhsp_3),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(t3_bhsp_3,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_3(B,A)
                  & v1_bhsp_3(C,A) )
               => v1_bhsp_3(k8_bhsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,t3_seq_2,d1_bhsp_3,d1_bhsp_3,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d5_normsp_1,d5_normsp_1,t47_bhsp_1,t2_xreal_1]),
    [file(bhsp_3,t3_bhsp_3),interesting(0.00)]).

fof(d9_seqm_3,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k1_seqm_3(A,B)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(D,B)) ) ) ) ) ) ),
    file(seqm_3,d9_seqm_3),
    [interesting(0.00)]).

fof(s1_seq_1,theorem,(
    ? [A] : 
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & m2_relset_1(A,k5_numbers,k1_numbers)
      & ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_seq_1(k5_numbers,k1_numbers,A,B) = f1_s1_seq_1(B) ) ) ),
    file(seq_1,s1_seq_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))) ) ) ) ) ) ) ) ) ),
    file(bhsp_2,t9_bhsp_2),
    [interesting(0.00)]).

fof(t33_seqm_3,theorem,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v1_seqm_3(A)
        & v7_seqm_3(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_xreal_0(B,k2_seqm_3(A,B)) ) ) ),
    file(seqm_3,t33_seqm_3),
    [interesting(0.00)]).

fof(t32_bhsp_3,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)
                  & m1_bhsp_3(C,A,B) )
               => v1_bhsp_2(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_bhsp_2,d4_bhsp_3,t33_seqm_3,t2_xreal_1,t26_bhsp_3,t9_bhsp_2]),
    [file(bhsp_3,t32_bhsp_3),interesting(0.00)]).

fof(t45_bhsp_3,theorem,(
    $true ),
    file(bhsp_3,t45_bhsp_3),
    [interesting(0.00)]).

fof(t28_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,B)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => B != k2_xcmplx_0(A,C) ) ) ) ) ),
    file(nat_1,t28_nat_1),
    [interesting(0.00)]).

fof(t46_bhsp_3,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)
                     => B != k2_bhsp_3(A,C,D) )
                  | v1_bhsp_2(C,A) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_bhsp_2,t28_nat_1,t8_xreal_1,d5_bhsp_3,t9_bhsp_2]),
    [file(bhsp_3,t46_bhsp_3),interesting(0.00)]).

fof(t47_bhsp_3,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_3(B,A)
               => ( ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => B != k2_bhsp_3(A,C,D) )
                  | v1_bhsp_3(C,A) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,d1_bhsp_3,t28_nat_1,t28_nat_1,t8_xreal_1,t8_xreal_1,d5_bhsp_3,d5_bhsp_3]),
    [file(bhsp_3,t47_bhsp_3),interesting(0.00)]).

fof(t34_bhsp_3,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_3(B,A)
                  & m1_bhsp_3(C,A,B) )
               => v1_bhsp_3(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_bhsp_3,d1_bhsp_3,t33_seqm_3,t2_xreal_1,t26_bhsp_3,d1_bhsp_3]),
    [file(bhsp_3,t34_bhsp_3),interesting(0.00)]).

fof(t29_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ),
    file(nat_1,t29_nat_1),
    [interesting(0.00)]).

fof(t49_bhsp_3,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)) )
             => ! [D] : 
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r2_bhsp_3(A,B,C)
                   => r2_bhsp_3(A,k2_bhsp_3(A,B,D),k2_bhsp_3(A,C,D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_3,d2_bhsp_3,t29_nat_1,t2_xreal_1,d5_bhsp_3,d5_bhsp_3]),
    [file(bhsp_3,t49_bhsp_3),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_3,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_3(B,A)
                  & v1_bhsp_3(C,A) )
               => v1_bhsp_3(k4_normsp_1(A,B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,t3_seq_2,d1_bhsp_3,d1_bhsp_3,t37_nat_1,t2_xreal_1,t37_nat_1,t2_xreal_1,t10_xreal_1,d6_normsp_1,d6_normsp_1,t48_bhsp_1,t2_xreal_1]),
    [file(bhsp_3,t4_bhsp_3),interesting(0.00)]).

fof(t31_bhsp_3,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)) )
             => ( ( v2_bhsp_3(B,A)
                  & m1_bhsp_3(C,A,B) )
               => v2_bhsp_3(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_bhsp_3,d4_bhsp_3,d3_bhsp_3,t26_bhsp_3]),
    [file(bhsp_3,t31_bhsp_3),interesting(0.00)]).

fof(t68_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)) )
         => ( v5_seqm_3(B)
          <=> ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_normsp_1(A,B,C) = k1_funct_1(B,k3_real_1(C,1)) ) ) ) ) ),
    file(bhsp_1,t68_bhsp_1),
    [interesting(0.00)]).

fof(t29_bhsp_3,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)) )
             => ( ( v5_seqm_3(B)
                  & m1_bhsp_3(C,A,B) )
               => v5_seqm_3(C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_bhsp_3,t70_bhsp_1,t26_bhsp_3,t26_bhsp_3,t68_bhsp_1]),
    [file(bhsp_3,t29_bhsp_3),interesting(0.00)]).

fof(t52_bhsp_3,theorem,(
    $true ),
    file(bhsp_3,t52_bhsp_3),
    [interesting(0.00)]).

fof(d6_bhsp_3,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) )
     => ( v3_bhsp_3(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_3(B,A)
             => v1_bhsp_2(B,A) ) ) ) ) ),
    file(bhsp_3,d6_bhsp_3),
    [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(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(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_3,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_3(C,A)
               => v1_bhsp_3(k6_normsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,t23_rlvect_1,t23_rlvect_1,t41_bhsp_1,t44_bhsp_1,d8_normsp_1,d8_normsp_1,t133_complex1,t133_complex1,t73_real_1,d1_bhsp_3,d5_bhsp_1,t48_rlvect_1,t33_bhsp_1,d5_bhsp_1,d9_xcmplx_0,d7_xcmplx_0,t70_xreal_1,d8_normsp_1,d8_normsp_1,d1_bhsp_3]),
    [file(bhsp_3,t5_bhsp_3),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(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_3,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_3(C,A)
               => v1_bhsp_3(k7_bhsp_1(A,C,B),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_3,d1_bhsp_3,t47_bhsp_1,t41_bhsp_1,t2_xreal_1,d12_bhsp_1,d12_bhsp_1]),
    [file(bhsp_3,t7_bhsp_3),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(t9_bhsp_3,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_3(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_2,d1_bhsp_3,t3_seq_2,t42_bhsp_1,t10_xreal_1,t2_xreal_1]),
    [file(bhsp_3,t9_bhsp_3),interesting(0.00)]).