fof(t19_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m1_topmetr(B,A) )
         => m1_pre_topc(k5_pcomps_1(B),k5_pcomps_1(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t16_topmetr,d1_topmetr,d9_pre_topc,t16_topmetr,t94_zfmisc_1,d4_tarski,t30_pcomps_1,t92_zfmisc_1,t1_xboole_1,d5_pcomps_1,t16_topmetr,t16_topmetr,d3_tarski,t16_topmetr,t16_topmetr,d5_pcomps_1,t12_topmetr,t13_topmetr,l11_topmetr,d4_tarski,d3_xboole_0,d3_tarski,d3_xboole_0,d4_tarski,d3_xboole_0,d3_xboole_0,d10_xboole_0,t16_topmetr,t16_topmetr,t12_topmetr,t16_topmetr,d3_xboole_0,d5_pcomps_1,t13_topmetr,t26_xboole_1,t16_topmetr,d5_pcomps_1,t16_topmetr]),
    [file(topmetr,t19_topmetr),interesting(1.00)]).

fof(l38_topmetr,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(0,C)
                  & r1_xreal_0(k2_xcmplx_0(A,C),B) )
               => r1_xreal_0(A,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t11_xreal_1,t173_real_2,t2_xreal_1]),
    [file(topmetr,l38_topmetr),interesting(0.86)]).

fof(t27_topmetr,theorem,(
    k4_topmetr(0,1) = k22_borsuk_1 ),
    inference(mizar_proof,[status(thm)],[t24_topmetr,t26_topmetr,d16_borsuk_1]),
    [file(topmetr,t27_topmetr),interesting(0.82)]).

fof(t3_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m1_pre_topc(B,A) )
         => ( v3_compts_1(A)
           => v3_compts_1(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_topmetr,d4_compts_1,d7_xboole_0,t17_xboole_1,d5_pre_topc,d9_pre_topc,d5_pre_topc,d3_xboole_0,t16_xboole_1,t16_xboole_1,d7_xboole_0,d4_compts_1]),
    [file(topmetr,t3_topmetr),interesting(0.80)]).

fof(t24_topmetr,theorem,(
    u1_struct_0(k3_topmetr) = k1_numbers ),
    inference(mizar_proof,[status(thm)],[d14_metric_1,d6_pcomps_1]),
    [file(topmetr,t24_topmetr),interesting(0.76)]).

fof(t25_topmetr,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => u1_struct_0(k4_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t16_topmetr,t14_topmetr]),
    [file(topmetr,t25_topmetr),interesting(0.75)]).

fof(t4_topmetr,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_pre_topc(B,A)
         => ! [C] : 
              ( m1_pre_topc(C,A)
             => ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
               => m1_pre_topc(B,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_pre_topc,t17_xboole_1,d9_pre_topc,t28_xboole_1,t16_xboole_1,d9_pre_topc,t16_xboole_1,t28_xboole_1,d9_pre_topc,d9_pre_topc]),
    [file(topmetr,t4_topmetr),interesting(0.74)]).

fof(t14_topmetr,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => u1_struct_0(k2_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d14_metric_1,t15_rcomp_1,d3_topmetr,d2_topmetr]),
    [file(topmetr,t14_topmetr),interesting(0.73)]).

fof(t2_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m1_pre_topc(B,A) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => m1_subset_1(C,u1_struct_0(A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_pre_topc,d3_tarski]),
    [file(topmetr,t2_topmetr),interesting(0.68)]).

fof(t20_topmetr,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] : 
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) )
             => ( B = C
               => k3_pre_topc(k15_euclid(A),B) = k5_pcomps_1(k1_topmetr(k14_euclid(A),C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t19_topmetr,d2_topmetr,t16_topmetr,d10_pre_topc,d10_pre_topc]),
    [file(topmetr,t20_topmetr),interesting(0.67)]).

fof(t16_topmetr,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ( u1_struct_0(A) = u1_struct_0(k5_pcomps_1(A))
        & u1_pre_topc(k5_pcomps_1(A)) = k4_pcomps_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_pcomps_1]),
    [file(topmetr,t16_topmetr),interesting(0.62)]).

fof(t1_topmetr,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_pre_topc(A,B)
          <=> r1_tarski(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_pre_topc,d10_xboole_0,d8_pre_topc]),
    [file(topmetr,t1_topmetr),interesting(0.61)]).

fof(t12_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & m1_topmetr(B,A) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => m1_subset_1(C,u1_struct_0(A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_topmetr]),
    [file(topmetr,t12_topmetr),interesting(0.56)]).

fof(t26_topmetr,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ! [C] : 
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
               => ( C = k1_rcomp_1(A,B)
                 => k4_topmetr(A,B) = k3_pre_topc(k3_topmetr,C) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_topmetr,d10_pre_topc,d10_pre_topc]),
    [file(topmetr,t26_topmetr),interesting(0.55)]).

fof(t28_topmetr,theorem,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr))
        & m2_relset_1(A,u1_struct_0(k3_topmetr),u1_struct_0(k3_topmetr)) )
     => ( ? [B] : 
            ( m1_subset_1(B,k1_numbers)
            & ? [C] : 
                ( m1_subset_1(C,k1_numbers)
                & ! [D] : 
                    ( m1_subset_1(D,k1_numbers)
                   => k1_funct_1(A,D) = k3_real_1(k4_real_1(B,D),C) ) ) )
       => v5_pre_topc(A,k3_topmetr,k3_topmetr) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_connsp_2,t54_tops_1,t24_topmetr,d14_metric_1,t22_topmetr,t55_tops_1,d1_connsp_2,t24_topmetr,d1_funct_2,d1_tarski,t24_topmetr,d1_tarski,d12_funct_1,t44_tops_1,t24_topmetr,t37_zfmisc_1,t133_complex1,t122_real_2,t127_real_2,t24_topmetr,d14_metric_1,t24_topmetr,d14_metric_1,t21_topmetr,t55_tops_1,l11_topmetr,d1_connsp_2,d3_tarski,d12_funct_1,t24_topmetr,t7_funct_2,d14_metric_1,t24_topmetr,d14_metric_1,t12_metric_1,t15_topmetr,t133_complex1,t70_xreal_1,t151_complex1,t7_absvalue,t93_xcmplx_1,d14_metric_1,t125_real_2,l38_topmetr,t15_topmetr,t12_metric_1,t1_xboole_1,t1_xboole_1,d2_borsuk_1]),
    [file(topmetr,t28_topmetr),interesting(0.53)]).

fof(t21_topmetr,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( ( v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B))))
                 => ( D = k9_metric_1(B,C,A)
                   => v3_pre_topc(D,k5_pcomps_1(B)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t16_topmetr,t33_pcomps_1,d5_pre_topc]),
    [file(topmetr,t21_topmetr),interesting(0.48)]).

fof(t23_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ( v2_topmetr(A)
      <=> ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( v1_topmetr(B,A)
                & r1_topmetr(A,B)
                & ! [C] : 
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( r1_tarski(C,B)
                        & r1_topmetr(A,C)
                        & v1_finset_1(C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_topmetr,t16_topmetr,d5_topmetr,t16_topmetr,t1_topmetr,d1_tops_2,t16_topmetr,d4_topmetr,t21_topmetr,d3_compts_1,t16_topmetr,t1_topmetr,t16_topmetr,d5_topmetr,d3_tarski,d4_topmetr,d3_tarski,t1_topmetr,t16_topmetr,d3_tarski,d4_tarski,d1_tops_2,t22_topmetr,d1_xreal_0,l11_topmetr,d4_tarski,d5_topmetr,t5_tops_2,s1_funct_2,t12_relset_1,t3_tops_2,t12_relset_1,d3_tarski,d5_topmetr,t16_topmetr,d4_tarski,d1_funct_2,d5_funct_1,d4_tarski,t1_topmetr,d1_funct_2,t26_finset_1,d3_compts_1,d6_topmetr]),
    [file(topmetr,t23_topmetr),interesting(0.48)]).

fof(t13_topmetr,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( ( v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] : 
              ( m1_topmetr(C,B)
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_struct_0(C))
                     => ( D = E
                       => k9_metric_1(C,E,A) = k3_xboole_0(k9_metric_1(B,D,A),u1_struct_0(C)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_struct_0,d1_topmetr,d1_struct_0,t12_topmetr,t12_metric_1,d1_metric_1,d1_topmetr,d1_metric_1,t12_metric_1,d3_xboole_0,d3_tarski,d3_xboole_0,d1_struct_0,d1_struct_0,d3_xboole_0,t12_topmetr,t12_metric_1,d1_metric_1,d1_topmetr,d1_metric_1,t12_metric_1,d3_tarski,d10_xboole_0]),
    [file(topmetr,t13_topmetr),interesting(0.45)]).

fof(l11_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(C,0)
               => r2_hidden(B,k9_metric_1(A,B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t16_tbsp_1]),
    [file(topmetr,l11_topmetr),interesting(0.34)]).

fof(t5_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( m1_pre_topc(k3_pre_topc(A,B),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)))
                & m1_pre_topc(k3_pre_topc(A,C),k3_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C))) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d10_pre_topc,d10_pre_topc,t7_xboole_1,t4_topmetr,d10_pre_topc,d10_pre_topc,t7_xboole_1,t4_topmetr]),
    [file(topmetr,t5_topmetr),interesting(0.34)]).

fof(l12_topmetr,theorem,(
    ! [A] : 
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ( v6_metric_1(g1_metric_1(u1_struct_0(A),u1_metric_1(A)))
        & v7_metric_1(g1_metric_1(u1_struct_0(A),u1_metric_1(A)))
        & v8_metric_1(g1_metric_1(u1_struct_0(A),u1_metric_1(A)))
        & v9_metric_1(g1_metric_1(u1_struct_0(A),u1_metric_1(A)))
        & l1_metric_1(g1_metric_1(u1_struct_0(A),u1_metric_1(A))) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_metric_1,d1_metric_1,t1_metric_1,t2_metric_1,d1_metric_1,d1_metric_1,d1_metric_1,d1_metric_1,d1_metric_1,d1_metric_1,t4_metric_1,t6_metric_1]),
    [file(topmetr,l12_topmetr),interesting(0.30)]).

fof(t8_topmetr,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] : 
                  ( m1_pre_topc(D,B)
                 => ( v5_pre_topc(C,A,B)
                   => ! [E] : 
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),u1_struct_0(D))
                          & m2_relset_1(E,u1_struct_0(A),u1_struct_0(D)) )
                       => ( E = C
                         => v5_pre_topc(E,A,D) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_relset_1,t43_pre_topc,d1_funct_2,t167_relat_1,t137_funct_1,t168_relat_1,t28_xboole_1,t169_relat_1,t28_xboole_1,d12_pre_topc,d12_pre_topc]),
    [file(topmetr,t8_topmetr),interesting(0.23)]).

fof(t7_topmetr,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] : 
              ( ( v2_pre_topc(C)
                & l1_pre_topc(C) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) )
                 => ( ( v5_pre_topc(D,A,C)
                      & m1_pre_topc(C,B) )
                   => ! [E] : 
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
                          & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
                       => ( E = D
                         => v5_pre_topc(E,A,B) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_relset_1,t168_relat_1,t28_xboole_1,t16_xboole_1,t168_relat_1,t17_xboole_1,t43_pre_topc,d12_pre_topc,d12_pre_topc]),
    [file(topmetr,t7_topmetr),interesting(0.21)]).

fof(t22_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
         => ( v3_pre_topc(B,k5_pcomps_1(A))
          <=> ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( r2_hidden(C,B)
                    & ! [D] : 
                        ( v1_xreal_0(D)
                       => ~ ( ~ r1_xreal_0(D,0)
                            & r1_tarski(k9_metric_1(A,C,D),B) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t16_topmetr,d5_pre_topc,t16_topmetr,d5_pcomps_1,d1_xreal_0,d5_pcomps_1,t16_topmetr,d5_pre_topc]),
    [file(topmetr,t22_topmetr),interesting(0.16)]).

fof(t6_topmetr,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( r2_hidden(B,C)
               => ! [D] : 
                    ( m1_connsp_2(D,A,B)
                   => ! [E] : 
                        ( m1_subset_1(E,u1_struct_0(k3_pre_topc(A,C)))
                       => ! [F] : 
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C))))
                           => ( ( F = k5_subset_1(u1_struct_0(A),D,C)
                                & E = B )
                             => m1_connsp_2(F,k3_pre_topc(A,C),E) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_connsp_2,t54_tops_1,t38_tops_2,d10_pre_topc,t32_tops_2,d3_xboole_0,t26_xboole_1,t54_tops_1,d1_connsp_2]),
    [file(topmetr,t6_topmetr),interesting(0.16)]).

fof(t9_topmetr,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ( v5_pre_topc(C,A,B)
                   => ! [E] : 
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D)))
                          & m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) )
                       => ( E = C
                         => v5_pre_topc(E,A,k3_pre_topc(B,D)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d10_pre_topc,t12_relset_1,t43_pre_topc,t167_relat_1,d10_pre_topc,t137_funct_1,t168_relat_1,t28_xboole_1,t169_relat_1,t28_xboole_1,d12_pre_topc,d12_pre_topc]),
    [file(topmetr,t9_topmetr),interesting(0.12)]).

fof(d1_metric_1,definition,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => k2_metric_1(A,B,C) = k1_metric_1(u1_struct_0(A),u1_struct_0(A),u1_metric_1(A),B,C) ) ) ) ),
    file(metric_1,d1_metric_1),
    [interesting(0.00)]).

fof(t1_metric_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ( ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => k2_metric_1(A,B,B) = 0 )
      <=> v6_metric_1(A) ) ) ),
    file(metric_1,t1_metric_1),
    [interesting(0.00)]).

fof(t2_metric_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ( ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k2_metric_1(A,B,C) = 0
                 => B = C ) ) )
      <=> v7_metric_1(A) ) ) ),
    file(metric_1,t2_metric_1),
    [interesting(0.00)]).

fof(t4_metric_1,theorem,(
    ! [A] : 
      ( l1_metric_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(k2_metric_1(A,B,D),k3_real_1(k2_metric_1(A,B,C),k2_metric_1(A,C,D))) ) ) )
      <=> v9_metric_1(A) ) ) ),
    file(metric_1,t4_metric_1),
    [interesting(0.00)]).

fof(t6_metric_1,theorem,(
    ! [A] : 
      ( l1_metric_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_metric_1(A,B,C) = 0
                       => B = C )
                      & ( B = C
                       => k2_metric_1(A,B,C) = 0 )
                      & k2_metric_1(A,B,C) = k2_metric_1(A,C,B)
                      & r1_xreal_0(k2_metric_1(A,B,D),k3_real_1(k2_metric_1(A,B,C),k2_metric_1(A,C,D))) ) ) ) )
       => ( v6_metric_1(A)
          & v7_metric_1(A)
          & v8_metric_1(A)
          & v9_metric_1(A)
          & l1_metric_1(A) ) ) ) ),
    file(metric_1,t6_metric_1),
    [interesting(0.00)]).

fof(t55_tops_2,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
              <=> ! [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                   => ( v3_pre_topc(D,B)
                     => v3_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ),
    file(tops_2,t55_tops_2),
    [interesting(0.00)]).

fof(d10_pre_topc,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( ( v1_pre_topc(C)
                & m1_pre_topc(C,A) )
             => ( C = k3_pre_topc(A,B)
              <=> k2_pre_topc(C) = B ) ) ) ) ),
    file(pre_topc,d10_pre_topc),
    [interesting(0.00)]).

fof(t139_funct_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => k10_relat_1(k7_relat_1(C,A),B) = k3_xboole_0(A,k10_relat_1(C,B)) ) ),
    file(funct_1,t139_funct_1),
    [interesting(0.00)]).

fof(t32_tops_2,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_pre_topc(B,A)
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( v3_pre_topc(C,B)
              <=> ? [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & v3_pre_topc(D,A)
                    & k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) ),
    file(tops_2,t32_tops_2),
    [interesting(0.00)]).

fof(t10_topmetr,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & l1_pre_topc(B) )
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ! [E] : 
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) )
                     => ( ( v5_pre_topc(D,A,B)
                          & E = k2_partfun1(u1_struct_0(A),u1_struct_0(B),D,C) )
                       => v5_pre_topc(E,k3_pre_topc(A,C),B) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t55_tops_2,d10_pre_topc,t139_funct_1,t32_tops_2,t55_tops_2]),
    [file(topmetr,t10_topmetr),interesting(0.00)]).

fof(t11_topmetr,theorem,(
    $true ),
    file(topmetr,t11_topmetr),
    [interesting(0.00)]).

fof(t17_topmetr,theorem,(
    $true ),
    file(topmetr,t17_topmetr),
    [interesting(0.00)]).

fof(t18_topmetr,theorem,(
    $true ),
    file(topmetr,t18_topmetr),
    [interesting(0.00)]).

fof(d6_pcomps_1,definition,(
    ! [A] : 
      ( l1_metric_1(A)
     => k5_pcomps_1(A) = g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A)) ) ),
    file(pcomps_1,d6_pcomps_1),
    [interesting(0.00)]).

fof(d1_topmetr,definition,(
    ! [A] : 
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( ( v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ( m1_topmetr(B,A)
          <=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
              & ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(B))
                 => ! [D] : 
                      ( m1_subset_1(D,u1_struct_0(B))
                     => k1_metric_1(u1_struct_0(B),u1_struct_0(B),u1_metric_1(B),C,D) = k1_binop_1(u1_metric_1(A),C,D) ) ) ) ) ) ) ),
    file(topmetr,d1_topmetr),
    [interesting(0.00)]).

fof(d9_pre_topc,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( l1_pre_topc(B)
         => ( m1_pre_topc(B,A)
          <=> ( r1_tarski(k2_pre_topc(B),k2_pre_topc(A))
              & ! [C] : 
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
                 => ( r2_hidden(C,u1_pre_topc(B))
                  <=> ? [D] : 
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                        & r2_hidden(D,u1_pre_topc(A))
                        & C = k3_xboole_0(D,k2_pre_topc(B)) ) ) ) ) ) ) ) ),
    file(pre_topc,d9_pre_topc),
    [interesting(0.00)]).

fof(t94_zfmisc_1,theorem,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
         => r1_tarski(C,B) )
     => r1_tarski(k3_tarski(A),B) ) ),
    file(zfmisc_1,t94_zfmisc_1),
    [interesting(0.00)]).

fof(d4_tarski,definition,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    [interesting(0.00)]).

fof(t30_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ~ ( v9_metric_1(A)
                      & r2_hidden(B,k9_metric_1(A,C,D))
                      & ! [E] : 
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,0)
                              & r1_tarski(k9_metric_1(A,B,E),k9_metric_1(A,C,D)) ) ) ) ) ) ) ) ),
    file(pcomps_1,t30_pcomps_1),
    [interesting(0.00)]).

fof(t92_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => r1_tarski(A,k3_tarski(B)) ) ),
    file(zfmisc_1,t92_zfmisc_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(d5_pcomps_1,definition,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( B = k4_pcomps_1(A)
          <=> ! [C] : 
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                <=> ! [D] : 
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(D,C)
                          & ! [E] : 
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,0)
                                  & r1_tarski(k9_metric_1(A,D,E),C) ) ) ) ) ) ) ) ) ) ),
    file(pcomps_1,d5_pcomps_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(d1_struct_0,definition,(
    ! [A] : 
      ( l1_struct_0(A)
     => ( v3_struct_0(A)
      <=> v1_xboole_0(u1_struct_0(A)) ) ) ),
    file(struct_0,d1_struct_0),
    [interesting(0.00)]).

fof(t12_metric_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( l1_metric_1(B)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(B))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(B))
                 => ( r2_hidden(D,k9_metric_1(B,C,A))
                  <=> ( ~ v3_struct_0(B)
                      & ~ r1_xreal_0(A,k2_metric_1(B,C,D)) ) ) ) ) ) ) ),
    file(metric_1,t12_metric_1),
    [interesting(0.00)]).

fof(d3_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    [interesting(0.00)]).

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

fof(t16_tbsp_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( v6_metric_1(A)
               => ( r1_xreal_0(C,0)
                  | r2_hidden(B,k9_metric_1(A,B,C)) ) ) ) ) ) ),
    file(tbsp_1,t16_tbsp_1),
    [interesting(0.00)]).

fof(t26_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,C)) ) ),
    file(xboole_1,t26_xboole_1),
    [interesting(0.00)]).

fof(d2_topmetr,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] : 
              ( ( v1_metric_1(C)
                & m1_topmetr(C,A) )
             => ( C = k1_topmetr(A,B)
              <=> u1_struct_0(C) = B ) ) ) ) ),
    file(topmetr,d2_topmetr),
    [interesting(0.00)]).

fof(d6_topmetr,definition,(
    ! [A] : 
      ( l1_metric_1(A)
     => ( v2_topmetr(A)
      <=> v2_compts_1(k5_pcomps_1(A)) ) ) ),
    file(topmetr,d6_topmetr),
    [interesting(0.00)]).

fof(d5_topmetr,definition,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_topmetr(A,B)
          <=> r1_tarski(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ) ),
    file(topmetr,d5_topmetr),
    [interesting(0.00)]).

fof(d8_pre_topc,definition,(
    ! [A] : 
      ( l1_struct_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_pre_topc(A,B)
          <=> k2_pre_topc(A) = k5_setfam_1(u1_struct_0(A),B) ) ) ) ),
    file(pre_topc,d8_pre_topc),
    [interesting(0.00)]).

fof(d1_tops_2,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_tops_2(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                 => v3_pre_topc(C,A) ) ) ) ) ) ),
    file(tops_2,d1_tops_2),
    [interesting(0.00)]).

fof(d4_topmetr,definition,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_topmetr(B,A)
          <=> ! [C] : ~ ( r2_hidden(C,B)
                & ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] : 
                        ( m1_subset_1(E,k1_numbers)
                       => C != k9_metric_1(A,D,E) ) ) ) ) ) ) ),
    file(topmetr,d4_topmetr),
    [interesting(0.00)]).

fof(t33_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( v9_metric_1(A)
               => r2_hidden(k9_metric_1(A,B,C),k4_pcomps_1(A)) ) ) ) ) ),
    file(pcomps_1,t33_pcomps_1),
    [interesting(0.00)]).

fof(d5_pre_topc,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_pre_topc(B,A)
          <=> r2_hidden(B,u1_pre_topc(A)) ) ) ) ),
    file(pre_topc,d5_pre_topc),
    [interesting(0.00)]).

fof(d3_compts_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v2_compts_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( r1_pre_topc(A,B)
                & v1_tops_2(B,A)
                & ! [C] : 
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( r1_tarski(C,B)
                        & r1_pre_topc(A,C)
                        & v1_finset_1(C) ) ) ) ) ) ) ),
    file(compts_1,d3_compts_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(t5_tops_2,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ~ ( r1_pre_topc(A,B)
              & B = k1_xboole_0 ) ) ) ),
    file(tops_2,t5_tops_2),
    [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(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(t3_tops_2,theorem,(
    ! [A] : 
      ( l1_struct_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] : 
              ( r1_tarski(C,B)
             => m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ) ),
    file(tops_2,t3_tops_2),
    [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(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(t26_finset_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_finset_1(k1_relat_1(A))
       => v1_finset_1(k2_relat_1(A)) ) ) ),
    file(finset_1,t26_finset_1),
    [interesting(0.00)]).

fof(d14_metric_1,definition,(
    k8_metric_1 = g1_metric_1(k1_numbers,k7_metric_1) ),
    file(metric_1,d14_metric_1),
    [interesting(0.00)]).

fof(t15_rcomp_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ~ ( ~ r1_xreal_0(B,A)
                & k2_rcomp_1(A,B) = k1_xboole_0 )
            & ( r1_xreal_0(A,B)
             => ( r2_hidden(A,k1_rcomp_1(A,B))
                & r2_hidden(B,k1_rcomp_1(A,B)) ) )
            & r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(A,B)) ) ) ) ),
    file(rcomp_1,t15_rcomp_1),
    [interesting(0.00)]).

fof(d3_topmetr,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ! [C] : 
                ( ( ~ v3_struct_0(C)
                  & v1_metric_1(C)
                  & m1_topmetr(C,k8_metric_1) )
               => ( C = k2_topmetr(A,B)
                <=> ! [D] : 
                      ( ( ~ v1_xboole_0(D)
                        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k8_metric_1))) )
                     => ( D = k1_rcomp_1(A,B)
                       => C = k1_topmetr(k8_metric_1,D) ) ) ) ) ) ) ) ),
    file(topmetr,d3_topmetr),
    [interesting(0.00)]).

fof(d16_borsuk_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( A = k22_borsuk_1
      <=> ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(k8_metric_1))))
           => ( B = k1_rcomp_1(0,1)
             => A = k3_pre_topc(k5_pcomps_1(k8_metric_1),B) ) ) ) ) ),
    file(borsuk_1,d16_borsuk_1),
    [interesting(0.00)]).

fof(d1_connsp_2,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( m1_connsp_2(C,A,B)
              <=> r2_hidden(B,k1_tops_1(A,C)) ) ) ) ) ),
    file(connsp_2,d1_connsp_2),
    [interesting(0.00)]).

fof(t54_tops_1,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B,C] : 
          ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
         => ( r2_hidden(B,k1_tops_1(A,C))
          <=> ? [D] : 
                ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                & v3_pre_topc(D,A)
                & r1_tarski(D,C)
                & r2_hidden(B,D) ) ) ) ) ),
    file(tops_1,t54_tops_1),
    [interesting(0.00)]).

fof(t55_tops_1,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( l1_pre_topc(B)
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                 => ( ( v3_pre_topc(D,B)
                     => k1_tops_1(B,D) = D )
                    & ( k1_tops_1(A,C) = C
                     => v3_pre_topc(C,A) ) ) ) ) ) ) ),
    file(tops_1,t55_tops_1),
    [interesting(0.00)]).

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

fof(d12_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( C = k9_relat_1(A,B)
        <=> ! [D] : 
              ( r2_hidden(D,C)
            <=> ? [E] : 
                  ( r2_hidden(E,k1_relat_1(A))
                  & r2_hidden(E,B)
                  & D = k1_funct_1(A,E) ) ) ) ) ),
    file(funct_1,d12_funct_1),
    [interesting(0.00)]).

fof(t44_tops_1,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(k1_tops_1(A,B),B) ) ) ),
    file(tops_1,t44_tops_1),
    [interesting(0.00)]).

fof(t37_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(k1_tarski(A),B)
    <=> r2_hidden(A,B) ) ),
    file(zfmisc_1,t37_zfmisc_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(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(t7_funct_2,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m2_relset_1(D,A,B) )
     => ( r2_hidden(C,A)
       => ( B = k1_xboole_0
          | r2_hidden(k1_funct_1(D,C),B) ) ) ) ),
    file(funct_2,t7_funct_2),
    [interesting(0.00)]).

fof(d13_metric_1,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 = k7_metric_1
      <=> ! [B] : 
            ( m1_subset_1(B,k1_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => k1_metric_1(k1_numbers,k1_numbers,A,B,C) = k18_complex1(k5_real_1(B,C)) ) ) ) ) ),
    file(metric_1,d13_metric_1),
    [interesting(0.00)]).

fof(t15_topmetr,theorem,(
    ! [A] : 
      ( m1_subset_1(A,u1_struct_0(k8_metric_1))
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(k8_metric_1))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ( ( C = A
                      & D = B )
                   => k4_metric_1(k8_metric_1,A,B) = k18_complex1(k6_xcmplx_0(C,D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_xreal_0,d1_metric_1,d14_metric_1,d13_metric_1]),
    [file(topmetr,t15_topmetr),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(t151_complex1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => k17_complex1(k3_xcmplx_0(A,B)) = k4_real_1(k17_complex1(A),k17_complex1(B)) ) ) ),
    file(complex1,t151_complex1),
    [interesting(0.00)]).

fof(t7_absvalue,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( A = 0
      <=> k18_complex1(A) = 0 ) ) ),
    file(absvalue,t7_absvalue),
    [interesting(0.00)]).

fof(t93_xcmplx_1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => ! [C] : 
              ( v1_xcmplx_0(C)
             => ( A != 0
               => k7_xcmplx_0(B,C) = k3_xcmplx_0(k7_xcmplx_0(B,k3_xcmplx_0(C,A)),A) ) ) ) ) ),
    file(xcmplx_1,t93_xcmplx_1),
    [interesting(0.00)]).

fof(t125_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(0,k7_xcmplx_0(A,B)) ) ) ) ),
    file(real_2,t125_real_2),
    [interesting(0.00)]).

fof(t11_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(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ),
    file(xreal_1,t11_xreal_1),
    [interesting(0.00)]).

fof(t173_real_2,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( ~ r1_xreal_0(0,A)
             => ( ~ r1_xreal_0(B,k2_xcmplx_0(A,B))
                & ~ r1_xreal_0(k6_xcmplx_0(B,A),B) ) )
            & ~ ( ~ ( r1_xreal_0(B,k2_xcmplx_0(A,B))
                    & r1_xreal_0(k6_xcmplx_0(B,A),B) )
                & r1_xreal_0(0,A) ) ) ) ) ),
    file(real_2,t173_real_2),
    [interesting(0.00)]).

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

fof(d2_borsuk_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
              <=> ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] : 
                        ( m1_connsp_2(E,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D))
                       => ? [F] : 
                            ( m1_connsp_2(F,A,D)
                            & r1_tarski(k4_pre_topc(A,B,C,F),E) ) ) ) ) ) ) ) ),
    file(borsuk_1,d2_borsuk_1),
    [interesting(0.00)]).

fof(d4_compts_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v3_compts_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( B != C
                    & ! [D] : 
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ! [E] : 
                            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                           => ~ ( v3_pre_topc(D,A)
                                & v3_pre_topc(E,A)
                                & r2_hidden(B,D)
                                & r2_hidden(C,E)
                                & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ),
    file(compts_1,d4_compts_1),
    [interesting(0.00)]).

fof(d7_xboole_0,definition,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) ),
    file(xboole_0,d7_xboole_0),
    [interesting(0.00)]).

fof(t17_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ),
    file(xboole_1,t17_xboole_1),
    [interesting(0.00)]).

fof(t16_xboole_1,theorem,(
    ! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) ),
    file(xboole_1,t16_xboole_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(t28_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k3_xboole_0(A,B) = A ) ),
    file(xboole_1,t28_xboole_1),
    [interesting(0.00)]).

fof(t38_tops_2,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => m1_subset_1(k5_subset_1(u1_struct_0(A),B,C),k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,C)))) ) ) ) ),
    file(tops_2,t38_tops_2),
    [interesting(0.00)]).

fof(t168_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k10_relat_1(B,A) = k10_relat_1(B,k3_xboole_0(k2_relat_1(B),A)) ) ),
    file(relat_1,t168_relat_1),
    [interesting(0.00)]).

fof(t43_pre_topc,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_pre_topc(B,A)
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( v4_pre_topc(C,B)
              <=> ? [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & v4_pre_topc(D,A)
                    & k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) ),
    file(pre_topc,t43_pre_topc),
    [interesting(0.00)]).

fof(d12_pre_topc,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( l1_pre_topc(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v5_pre_topc(C,A,B)
              <=> ! [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                   => ( v4_pre_topc(D,B)
                     => v4_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ),
    file(pre_topc,d12_pre_topc),
    [interesting(0.00)]).

fof(t167_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => r1_tarski(k10_relat_1(B,A),k1_relat_1(B)) ) ),
    file(relat_1,t167_relat_1),
    [interesting(0.00)]).

fof(t137_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => k10_relat_1(C,k3_xboole_0(A,B)) = k3_xboole_0(k10_relat_1(C,A),k10_relat_1(C,B)) ) ),
    file(funct_1,t137_funct_1),
    [interesting(0.00)]).

fof(t169_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => k10_relat_1(A,k2_relat_1(A)) = k1_relat_1(A) ) ),
    file(relat_1,t169_relat_1),
    [interesting(0.00)]).