fof(t3_bhsp_6,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_zfmisc_1(u1_struct_0(A)))
         => ( v1_bhsp_6(B,A)
           => v3_bhsp_6(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_6,d4_bhsp_6,t127_real_2,t88_xcmplx_1,t70_xreal_1,t10_xreal_1,t92_real_1,d5_bhsp_6]),
    [file(bhsp_6,t3_bhsp_6),interesting(1.00)]).

fof(t10_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A))
          & v4_bhsp_3(A) )
       => ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v1_bhsp_6(B,A)
            <=> ! [C] : 
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,0)
                      & ! [D] : 
                          ( ( v1_finset_1(D)
                            & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                         => ~ ( ~ v1_xboole_0(D)
                              & r1_tarski(D,B)
                              & ! [E] : 
                                  ( ( v1_finset_1(E)
                                    & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                                 => ~ ( ~ v1_xboole_0(E)
                                      & r1_tarski(E,B)
                                      & r1_xboole_0(D,E)
                                      & r1_xreal_0(C,k3_bhsp_1(A,k1_bhsp_6(A,E))) ) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_bhsp_6,t127_real_2,t2_bhsp_6,d6_rlvect_1,t16_rlvect_1,t10_rlvect_1,t37_bhsp_1,t47_rlvect_1,t10_rlvect_1,t16_rlvect_1,t43_rlvect_1,t43_rlvect_1,t41_rlvect_1,t36_bhsp_1,t37_bhsp_1,t7_xboole_1,t8_xboole_1,t10_xreal_1,t2_xreal_1,t73_real_1,s1_funct_2,s3_recdef_1,t7_xboole_1,t1_xboole_1,s1_nat_1,d1_xboole_0,t7_xboole_1,t63_xboole_1,t8_xreal_1,d5_real_1,t7_xboole_1,t38_nat_1,t1_xboole_1,s1_nat_1,t14_finset_1,d2_xboole_0,t8_xboole_1,s1_nat_1,d5_funct_1,d3_tarski,t4_funct_2,s1_funct_2,t73_real_1,t124_xreal_1,t10_seq_4,t10_xreal_1,t78_xreal_1,d9_xcmplx_0,t16_rlvect_1,t32_bhsp_1,t36_xboole_1,t39_xboole_1,t12_xboole_1,t1_xboole_1,t79_xboole_1,t2_bhsp_6,t2_xreal_1,t70_xreal_1,t41_rlvect_1,t28_rlvect_1,t27_rlvect_1,t37_bhsp_1,t2_xreal_1,t36_xboole_1,t39_xboole_1,t12_xboole_1,t1_xboole_1,t79_xboole_1,t2_bhsp_6,d6_rlvect_1,t47_rlvect_1,t43_rlvect_1,t16_rlvect_1,t10_rlvect_1,t16_rlvect_1,t10_rlvect_1,t16_rlvect_1,t43_rlvect_1,t43_rlvect_1,t47_rlvect_1,t43_rlvect_1,t36_bhsp_1,t32_bhsp_1,t2_xreal_1,t2_xreal_1,t10_xreal_1,t36_xboole_1,t39_xboole_1,t12_xboole_1,t1_xboole_1,t79_xboole_1,t36_xboole_1,t39_xboole_1,t12_xboole_1,t1_xboole_1,t79_xboole_1,t2_xreal_1,t2_xreal_1,t10_xreal_1,t2_bhsp_6,t2_bhsp_6,d6_rlvect_1,t16_rlvect_1,t10_rlvect_1,d6_rlvect_1,t16_rlvect_1,t10_rlvect_1,t10_rlvect_1,t16_rlvect_1,t43_rlvect_1,t43_rlvect_1,t47_rlvect_1,t43_rlvect_1,t47_rlvect_1,t36_bhsp_1,t37_bhsp_1,t2_xreal_1,t2_bhsp_3,d7_bhsp_3,d6_bhsp_3,t9_bhsp_2,t73_real_1,t124_xreal_1,t10_seq_4,t29_nat_1,t2_xreal_1,t10_xreal_1,t78_xreal_1,d9_xcmplx_0,t29_nat_1,t37_bhsp_1,t47_rlvect_1,t10_xreal_1,t36_xboole_1,t39_xboole_1,t12_xboole_1,t1_xboole_1,t79_xboole_1,t2_xreal_1,t2_bhsp_6,d6_rlvect_1,t36_bhsp_1,t37_bhsp_1,t47_rlvect_1,t10_xreal_1,t2_xreal_1,d2_bhsp_6]),
    [file(bhsp_6,t10_bhsp_6),interesting(0.77)]).

fof(t9_bhsp_6,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_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ( ~ v1_xboole_0(B)
           => v1_bhsp_6(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d10_xboole_0,t28_rlvect_1,t32_bhsp_1,d2_bhsp_6]),
    [file(bhsp_6,t9_bhsp_6),interesting(0.71)]).

fof(t7_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v3_bhsp_6(B,A)
             => ! [C] : 
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & v2_hahnban(C,A)
                    & v3_hahnban(C,A)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => ( v2_bhsp_6(C,A)
                   => r1_bhsp_6(A,B,C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t6_bhsp_6,d6_bhsp_6]),
    [file(bhsp_6,t7_bhsp_6),interesting(0.63)]).

fof(t8_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v1_bhsp_6(B,A)
             => ! [C] : 
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & v2_hahnban(C,A)
                    & v3_hahnban(C,A)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => ( v2_bhsp_6(C,A)
                   => r1_bhsp_6(A,B,C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_bhsp_6,t7_bhsp_6]),
    [file(bhsp_6,t8_bhsp_6),interesting(0.56)]).

fof(t2_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] : 
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] : 
                ( ( v1_finset_1(C)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
               => ( r1_xboole_0(B,C)
                 => ! [D] : 
                      ( ( v1_finset_1(D)
                        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) )
                     => ( D = k4_subset_1(u1_struct_0(A),B,C)
                       => k1_bhsp_6(A,D) = k4_rlvect_1(A,k1_bhsp_6(A,B),k1_bhsp_6(A,C)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_funct_1,d1_funct_2,t1_bhsp_6,t1_bhsp_6,t14_bhsp_5,t5_rlvect_1,t1_bhsp_6]),
    [file(bhsp_6,t2_bhsp_6),interesting(0.50)]).

fof(t5_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] : 
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ( ~ v1_xboole_0(B)
             => ! [C] : 
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                    & v2_hahnban(C,A)
                    & v3_hahnban(C,A)
                    & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
                 => k8_funct_2(u1_struct_0(A),k6_supinf_1,C,k1_bhsp_6(A,B)) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,B,C) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,d1_bhsp_6,d4_bhsp_5,t12_funct_1,t21_funct_1,t2_tarski,d4_bhsp_5,t23_funct_1,t59_card_2,t20_int_1,t77_finseq_4,t4_bhsp_6,d5_bhsp_5]),
    [file(bhsp_6,t5_bhsp_6),interesting(0.31)]).

fof(t1_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] : 
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] : 
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,u1_struct_0(A),u1_struct_0(A))
                  & m2_relset_1(C,u1_struct_0(A),u1_struct_0(A)) )
               => ( ( r1_tarski(B,k4_relset_1(u1_struct_0(A),u1_struct_0(A),C))
                    & ! [D] : 
                        ( r2_hidden(D,k4_relset_1(u1_struct_0(A),u1_struct_0(A),C))
                       => k1_funct_1(C,D) = D ) )
                 => k1_bhsp_6(A,B) = k5_bhsp_5(u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_bhsp_6,d4_bhsp_5,t12_funct_1,t21_funct_1,t2_tarski,t12_relset_1,t12_funct_1,d4_bhsp_5,t23_funct_1,t9_funct_1,d5_bhsp_5]),
    [file(bhsp_6,t1_bhsp_6),interesting(0.25)]).

fof(t4_bhsp_6,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,u1_struct_0(A),k6_supinf_1)
            & v2_hahnban(B,A)
            & v3_hahnban(B,A)
            & m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
         => ! [C] : 
              ( m2_finseq_1(C,u1_struct_0(A))
             => ( r1_xreal_0(1,k3_finseq_1(C))
               => ! [D] : 
                    ( m2_finseq_1(D,k1_numbers)
                   => ( ( k5_finsop_1(C) = k5_finsop_1(D)
                        & ! [E] : 
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( r2_hidden(E,k5_finsop_1(D))
                             => k1_funct_1(D,E) = k1_funct_1(B,k1_funct_1(C,E)) ) ) )
                     => k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k2_finsop_1(u1_struct_0(A),C,u1_rlvect_1(A))) = k2_finsop_1(k1_numbers,D,k33_binop_2) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_finsop_1,d3_finseq_1,d3_finseq_1,t8_finseq_1,t2_finsop_1,t29_nat_1,t27_finseq_3,t20_int_1,t92_real_1,t27_finseq_3,t12_funct_1,t2_xreal_1,d9_binop_2,d4_hahnban,t5_rlvect_1,s1_nat_1]),
    [file(bhsp_6,t4_bhsp_6),interesting(0.13)]).

fof(d2_bhsp_6,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,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_bhsp_6(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,0)
                        & ! [E] : 
                            ( ( v1_finset_1(E)
                              & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) )
                           => ~ ( ~ v1_xboole_0(E)
                                & r1_tarski(E,B)
                                & ! [F] : 
                                    ( ( v1_finset_1(F)
                                      & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                                   => ~ ( r1_tarski(E,F)
                                        & r1_tarski(F,B)
                                        & r1_xreal_0(D,k3_bhsp_1(A,k6_rlvect_1(A,C,k1_bhsp_6(A,F)))) ) ) ) ) ) ) ) ) ) ) ),
    file(bhsp_6,d2_bhsp_6),
    [interesting(0.00)]).

fof(t127_real_2,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ( ( ~ r1_xreal_0(A,0)
                  & ~ r1_xreal_0(B,0) )
                | ( ~ r1_xreal_0(0,A)
                  & ~ r1_xreal_0(0,B) ) )
              & r1_xreal_0(k7_xcmplx_0(A,B),0) ) ) ) ),
    file(real_2,t127_real_2),
    [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(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(d1_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] : 
            ( ( v1_finset_1(B)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( C = k1_bhsp_6(A,B)
                <=> ? [D] : 
                      ( m2_finseq_1(D,u1_struct_0(A))
                      & v2_funct_1(D)
                      & k5_relset_1(k5_numbers,u1_struct_0(A),D) = B
                      & C = k2_finsop_1(u1_struct_0(A),D,u1_rlvect_1(A)) ) ) ) ) ) ) ),
    file(bhsp_6,d1_bhsp_6),
    [interesting(0.00)]).

fof(d4_bhsp_5,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,A)
                & m2_relset_1(C,B,A) )
             => ! [D] : 
                  ( m2_finseq_1(D,B)
                 => k4_bhsp_5(A,B,C,D) = k5_relat_1(D,C) ) ) ) ) ),
    file(bhsp_5,d4_bhsp_5),
    [interesting(0.00)]).

fof(t12_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => r2_hidden(k1_funct_1(B,A),k2_relat_1(B)) ) ) ),
    file(funct_1,t12_funct_1),
    [interesting(0.00)]).

fof(t21_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)))
          <=> ( r2_hidden(A,k1_relat_1(C))
              & r2_hidden(k1_funct_1(C,A),k1_relat_1(B)) ) ) ) ) ),
    file(funct_1,t21_funct_1),
    [interesting(0.00)]).

fof(t2_tarski,theorem,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    [interesting(0.00)]).

fof(t12_relset_1,theorem,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( r1_tarski(k1_relat_1(C),A)
        & r1_tarski(k2_relat_1(C),B) ) ) ),
    file(relset_1,t12_relset_1),
    [interesting(0.00)]).

fof(t23_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(B))
           => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ),
    file(funct_1,t23_funct_1),
    [interesting(0.00)]).

fof(t9_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_relat_1(A) = k1_relat_1(B)
              & ! [C] : 
                  ( r2_hidden(C,k1_relat_1(A))
                 => k1_funct_1(A,C) = k1_funct_1(B,C) ) )
           => A = B ) ) ) ),
    file(funct_1,t9_funct_1),
    [interesting(0.00)]).

fof(d5_bhsp_5,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( ( v1_binop_1(C,A)
                  & v2_binop_1(C,A)
                  & v1_setwiseo(C,A) )
               => ! [D] : 
                    ( ( v1_finset_1(D)
                      & m1_subset_1(D,k1_zfmisc_1(B)) )
                   => ! [E] : 
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,B,A)
                          & m2_relset_1(E,B,A) )
                       => ( r1_tarski(D,k1_relat_1(E))
                         => ! [F] : 
                              ( m1_subset_1(F,A)
                             => ( F = k5_bhsp_5(A,B,C,D,E)
                              <=> ? [G] : 
                                    ( m2_finseq_1(G,B)
                                    & v2_funct_1(G)
                                    & k2_relat_1(G) = D
                                    & F = k2_finsop_1(A,k4_bhsp_5(A,B,E,G),C) ) ) ) ) ) ) ) ) ) ) ),
    file(bhsp_5,d5_bhsp_5),
    [interesting(0.00)]).

fof(t14_bhsp_5,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( ( v1_binop_1(C,A)
                  & v2_binop_1(C,A)
                  & v1_setwiseo(C,A) )
               => ! [D] : 
                    ( ( v1_finset_1(D)
                      & m1_subset_1(D,k1_zfmisc_1(B)) )
                   => ! [E] : 
                        ( ( v1_finset_1(E)
                          & m1_subset_1(E,k1_zfmisc_1(B)) )
                       => ( r1_xboole_0(D,E)
                         => ! [F] : 
                              ( ( v1_funct_1(F)
                                & v1_funct_2(F,B,A)
                                & m2_relset_1(F,B,A) )
                             => ( ( r1_tarski(D,k1_relat_1(F))
                                  & r1_tarski(E,k1_relat_1(F)) )
                               => ! [G] : 
                                    ( ( v1_finset_1(G)
                                      & m1_subset_1(G,k1_zfmisc_1(B)) )
                                   => ( G = k2_xboole_0(D,E)
                                     => k5_bhsp_5(A,B,C,G,F) = k2_binop_1(A,A,A,C,k5_bhsp_5(A,B,C,D,F),k5_bhsp_5(A,B,C,E,F)) ) ) ) ) ) ) ) ) ) ) ) ),
    file(bhsp_5,t14_bhsp_5),
    [interesting(0.00)]).

fof(t5_rlvect_1,theorem,(
    ! [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))
             => k2_rlvect_1(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_rlvect_1(A),B,C) ) ) ) ),
    file(rlvect_1,t5_rlvect_1),
    [interesting(0.00)]).

fof(d6_rlvect_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_rlvect_1(A) )
     => ( v4_rlvect_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))
                   => k2_rlvect_1(A,k2_rlvect_1(A,B,C),D) = k2_rlvect_1(A,B,k2_rlvect_1(A,C,D)) ) ) ) ) ) ),
    file(rlvect_1,d6_rlvect_1),
    [interesting(0.00)]).

fof(t16_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))
         => ( k2_rlvect_1(A,B,k5_rlvect_1(A,B)) = k1_rlvect_1(A)
            & k2_rlvect_1(A,k5_rlvect_1(A,B),B) = k1_rlvect_1(A) ) ) ) ),
    file(rlvect_1,t16_rlvect_1),
    [interesting(0.00)]).

fof(t10_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))
         => ( k2_rlvect_1(A,B,k1_rlvect_1(A)) = B
            & k2_rlvect_1(A,k1_rlvect_1(A),B) = B ) ) ) ),
    file(rlvect_1,t10_rlvect_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(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(t43_rlvect_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v3_rlvect_1(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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k6_rlvect_1(A,B,k6_rlvect_1(A,C,D)) = k4_rlvect_1(A,k6_rlvect_1(A,B,C),D) ) ) ) ) ),
    file(rlvect_1,t43_rlvect_1),
    [interesting(0.00)]).

fof(t41_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))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k6_rlvect_1(A,B,k2_rlvect_1(A,C,D)) = k6_rlvect_1(A,k6_rlvect_1(A,B,D),C) ) ) ) ) ),
    file(rlvect_1,t41_rlvect_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(t7_xboole_1,theorem,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    [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(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(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(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(s1_funct_2,theorem,
    ( ! [A] : ~ ( r2_hidden(A,f1_s1_funct_2)
        & ! [B] : ~ ( r2_hidden(B,f2_s1_funct_2)
            & p1_s1_funct_2(A,B) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s1_funct_2,f2_s1_funct_2)
        & m2_relset_1(A,f1_s1_funct_2,f2_s1_funct_2)
        & ! [B] : 
            ( r2_hidden(B,f1_s1_funct_2)
           => p1_s1_funct_2(B,k1_funct_1(A,B)) ) ) ),
    file(funct_2,s1_funct_2),
    [interesting(0.00)]).

fof(s3_recdef_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = k5_numbers
      & k1_funct_1(A,0) = f1_s3_recdef_1
      & ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_funct_1(A,k1_nat_1(B,1)) = f2_s3_recdef_1(B,k1_funct_1(A,B)) ) ) ),
    file(recdef_1,s3_recdef_1),
    [interesting(0.00)]).

fof(t1_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_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(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.00)]).

fof(t63_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_xboole_0(B,C) )
     => r1_xboole_0(A,C) ) ),
    file(xboole_1,t63_xboole_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(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(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(t14_finset_1,theorem,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_xboole_0(A,B)) ) ),
    file(finset_1,t14_finset_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(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(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.00)]).

fof(t4_funct_2,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(k2_relat_1(B),A)
       => ( v1_funct_1(B)
          & v1_funct_2(B,k1_relat_1(B),A)
          & m2_relset_1(B,k1_relat_1(B),A) ) ) ) ),
    file(funct_2,t4_funct_2),
    [interesting(0.00)]).

fof(t124_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( ~ ( ~ r1_xreal_0(A,0)
            & r1_xreal_0(k5_xcmplx_0(A),0) )
        & ~ ( ~ r1_xreal_0(k5_xcmplx_0(A),0)
            & r1_xreal_0(A,0) ) ) ) ),
    file(xreal_1,t124_xreal_1),
    [interesting(0.00)]).

fof(t10_seq_4,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ~ ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r1_xreal_0(B,A) ) ) ),
    file(seq_4,t10_seq_4),
    [interesting(0.00)]).

fof(t78_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(B,0)
                  & ~ r1_xreal_0(C,B)
                  & r1_xreal_0(k7_xcmplx_0(A,B),k7_xcmplx_0(A,C)) ) ) ) ) ),
    file(xreal_1,t78_xreal_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(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(t36_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ),
    file(xboole_1,t36_xboole_1),
    [interesting(0.00)]).

fof(t39_xboole_1,theorem,(
    ! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) ),
    file(xboole_1,t39_xboole_1),
    [interesting(0.00)]).

fof(t12_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k2_xboole_0(A,B) = B ) ),
    file(xboole_1,t12_xboole_1),
    [interesting(0.00)]).

fof(t79_xboole_1,theorem,(
    ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),B) ),
    file(xboole_1,t79_xboole_1),
    [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(t28_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,B) = k1_rlvect_1(A) ) ) ),
    file(rlvect_1,t28_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(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)))) ) ) ) ) ) ) ) ) ),
    file(bhsp_3,t2_bhsp_3),
    [interesting(0.00)]).

fof(d7_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) )
     => ( v4_bhsp_3(A)
      <=> ( ~ v3_struct_0(A)
          & v3_rlvect_1(A)
          & 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) ) ) ) ),
    file(bhsp_3,d7_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(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(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(d4_bhsp_6,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,u1_struct_0(A),k6_supinf_1)
            & v2_hahnban(B,A)
            & v3_hahnban(B,A)
            & m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
         => ( v2_bhsp_6(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,k1_numbers)
                & ~ r1_xreal_0(C,0)
                & ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => r1_xreal_0(k18_complex1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,D)),k11_binop_2(C,k3_bhsp_1(A,D))) ) ) ) ) ) ),
    file(bhsp_6,d4_bhsp_6),
    [interesting(0.00)]).

fof(t88_xcmplx_1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => ( A != 0
           => k3_xcmplx_0(k7_xcmplx_0(B,A),A) = B ) ) ) ),
    file(xcmplx_1,t88_xcmplx_1),
    [interesting(0.00)]).

fof(t92_real_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(k6_xcmplx_0(A,D),k6_xcmplx_0(B,C)) )
                    & ~ ( ( ( ~ r1_xreal_0(B,A)
                            & r1_xreal_0(C,D) )
                          | ( r1_xreal_0(A,B)
                            & ~ r1_xreal_0(D,C) ) )
                        & r1_xreal_0(k6_xcmplx_0(B,C),k6_xcmplx_0(A,D)) ) ) ) ) ) ) ),
    file(real_1,t92_real_1),
    [interesting(0.00)]).

fof(d5_bhsp_6,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,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_bhsp_6(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & ! [D] : 
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),k6_supinf_1)
                      & v2_hahnban(D,A)
                      & v3_hahnban(D,A)
                      & m2_relset_1(D,u1_struct_0(A),k6_supinf_1) )
                   => ( v2_bhsp_6(D,A)
                     => ! [E] : 
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,0)
                              & ! [F] : 
                                  ( ( v1_finset_1(F)
                                    & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                                 => ~ ( ~ v1_xboole_0(F)
                                      & r1_tarski(F,B)
                                      & ! [G] : 
                                          ( ( v1_finset_1(G)
                                            & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                                         => ~ ( r1_tarski(F,G)
                                              & r1_tarski(G,B)
                                              & r1_xreal_0(E,k18_complex1(k8_funct_2(u1_struct_0(A),k6_supinf_1,D,k6_rlvect_1(A,C,k1_bhsp_6(A,G))))) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file(bhsp_6,d5_bhsp_6),
    [interesting(0.00)]).

fof(t32_hahnban,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] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),k6_supinf_1)
            & v2_hahnban(B,A)
            & v3_hahnban(B,A)
            & m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k6_rlvect_1(A,C,D)) = k5_real_1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C),k8_funct_2(u1_struct_0(A),k6_supinf_1,B,D)) ) ) ) ) ),
    file(hahnban,t32_hahnban),
    [interesting(0.00)]).

fof(t3_xboole_1,theorem,(
    ! [A] : 
      ( r1_tarski(A,k1_xboole_0)
     => A = k1_xboole_0 ) ),
    file(xboole_1,t3_xboole_1),
    [interesting(0.00)]).

fof(t59_card_2,theorem,(
    ! [A] : 
      ( k1_card_1(A) = 0
     => A = k1_xboole_0 ) ),
    file(card_2,t59_card_2),
    [interesting(0.00)]).

fof(t20_int_1,theorem,(
    ! [A] : 
      ( v1_int_1(A)
     => ! [B] : 
          ( v1_int_1(B)
         => ( ~ r1_xreal_0(B,A)
           => r1_xreal_0(k2_xcmplx_0(A,1),B) ) ) ) ),
    file(int_1,t20_int_1),
    [interesting(0.00)]).

fof(t77_finseq_4,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v2_funct_1(A)
      <=> k4_card_1(k2_relat_1(A)) = k3_finseq_1(A) ) ) ),
    file(finseq_4,t77_finseq_4),
    [interesting(0.00)]).

fof(t2_finsop_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m2_finseq_1(B,A)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ~ ( r1_xreal_0(1,k3_finseq_1(B))
                  & ! [D] : 
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,A)
                        & m2_relset_1(D,k5_numbers,A) )
                     => ~ ( k8_funct_2(k5_numbers,A,D,1) = k1_funct_1(B,1)
                          & ! [E] : 
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ ( 0 != E
                                  & ~ r1_xreal_0(k3_finseq_1(B),E)
                                  & k8_funct_2(k5_numbers,A,D,k1_nat_1(E,1)) != k1_binop_1(C,k8_funct_2(k5_numbers,A,D,E),k1_funct_1(B,k1_nat_1(E,1))) ) )
                          & k2_finsop_1(A,B,C) = k8_funct_2(k5_numbers,A,D,k3_finseq_1(B)) ) ) ) ) ) ) ),
    file(finsop_1,t2_finsop_1),
    [interesting(0.00)]).

fof(d3_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k3_finseq_1(A)
          <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ),
    file(finseq_1,d3_finseq_1),
    [interesting(0.00)]).

fof(t8_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( k2_finseq_1(A) = k2_finseq_1(B)
           => A = B ) ) ) ),
    file(finseq_1,t8_finseq_1),
    [interesting(0.00)]).

fof(t27_finseq_3,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r2_hidden(B,k4_finseq_1(A))
          <=> ( r1_xreal_0(1,B)
              & r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) ),
    file(finseq_3,t27_finseq_3),
    [interesting(0.00)]).

fof(d9_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )
     => ( A = k33_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k1_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => k2_binop_1(k1_numbers,k1_numbers,k1_numbers,A,B,C) = k9_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d9_binop_2),
    [interesting(0.00)]).

fof(d4_hahnban,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,u1_struct_0(A),k6_supinf_1)
            & m2_relset_1(B,u1_struct_0(A),k6_supinf_1) )
         => ( v2_hahnban(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => k8_funct_2(u1_struct_0(A),k6_supinf_1,B,k4_rlvect_1(A,C,D)) = k3_real_1(k8_funct_2(u1_struct_0(A),k6_supinf_1,B,C),k8_funct_2(u1_struct_0(A),k6_supinf_1,B,D)) ) ) ) ) ) ),
    file(hahnban,d4_hahnban),
    [interesting(0.00)]).

fof(t6_bhsp_6,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) )
     => ( ( v1_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v2_binop_1(u1_rlvect_1(A),u1_struct_0(A))
          & v1_setwiseo(u1_rlvect_1(A),u1_struct_0(A)) )
       => ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ~ ( v3_bhsp_6(B,A)
                & ! [C] : 
                    ( m1_subset_1(C,u1_struct_0(A))
                   => ? [D] : 
                        ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(A),k6_supinf_1)
                        & v2_hahnban(D,A)
                        & v3_hahnban(D,A)
                        & m2_relset_1(D,u1_struct_0(A),k6_supinf_1)
                        & v2_bhsp_6(D,A)
                        & ? [E] : 
                            ( m1_subset_1(E,k1_numbers)
                            & ~ r1_xreal_0(E,0)
                            & ! [F] : 
                                ( ( v1_finset_1(F)
                                  & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                               => ~ ( ~ v1_xboole_0(F)
                                    & r1_tarski(F,B)
                                    & ! [G] : 
                                        ( ( v1_finset_1(G)
                                          & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                                       => ~ ( r1_tarski(F,G)
                                            & r1_tarski(G,B)
                                            & r1_xreal_0(E,k18_complex1(k10_binop_2(k8_funct_2(u1_struct_0(A),k6_supinf_1,D,C),k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,D)))) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_bhsp_6,t32_hahnban,t3_xboole_1,t5_bhsp_6]),
    [file(bhsp_6,t6_bhsp_6),interesting(0.00)]).

fof(d6_bhsp_6,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,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),k6_supinf_1)
                & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) )
             => ( r1_bhsp_6(A,B,C)
              <=> ? [D] : 
                    ( m1_subset_1(D,k1_numbers)
                    & ! [E] : 
                        ( m1_subset_1(E,k1_numbers)
                       => ~ ( ~ r1_xreal_0(E,0)
                            & ! [F] : 
                                ( ( v1_finset_1(F)
                                  & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) )
                               => ~ ( ~ v1_xboole_0(F)
                                    & r1_tarski(F,B)
                                    & ! [G] : 
                                        ( ( v1_finset_1(G)
                                          & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) )
                                       => ~ ( r1_tarski(F,G)
                                            & r1_tarski(G,B)
                                            & r1_xreal_0(E,k18_complex1(k10_binop_2(D,k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,C)))) ) ) ) ) ) ) ) ) ) ) ) ),
    file(bhsp_6,d6_bhsp_6),
    [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)]).