fof(t28_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v3_compts_1(A)
          & v2_pcomps_1(A) )
       => v5_compts_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,t50_subset_1,t27_pcomps_1,d5_compts_1,t26_pcomps_1,d6_compts_1]),
    [file(pcomps_1,t28_pcomps_1),interesting(1.00)]).

fof(t27_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( ( v3_compts_1(A)
          & v2_pcomps_1(A) )
       => v4_compts_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t54_subset_1,t10_pcomps_1,d1_tarski,t3_xboole_0,d4_compts_1,t37_zfmisc_1,t26_pcomps_1,t37_zfmisc_1,d5_compts_1]),
    [file(pcomps_1,t27_pcomps_1),interesting(0.89)]).

fof(t34_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => r2_hidden(u1_struct_0(A),k4_pcomps_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[t29_pcomps_1,d5_pcomps_1]),
    [file(pcomps_1,t34_pcomps_1),interesting(0.68)]).

fof(t9_pcomps_1,theorem,(
    ! [A] : v2_compts_1(k2_pcomps_1(k1_tarski(A))) ),
    inference(mizar_proof,[status(thm)],[t24_finset_1,t13_finset_1,d3_compts_1]),
    [file(pcomps_1,t9_pcomps_1),interesting(0.62)]).

fof(t25_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v2_compts_1(A)
       => v2_pcomps_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_compts_1,d1_tops_2,d1_tops_2,t17_setfam_1,t13_pcomps_1,d4_pcomps_1]),
    [file(pcomps_1,t25_pcomps_1),interesting(0.58)]).

fof(t10_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v3_compts_1(A)
           => v4_pre_topc(k1_struct_0(A,B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t48_pre_topc,d1_tarski,d4_compts_1,d7_xboole_0,t37_zfmisc_1,t26_xboole_1,t3_xboole_1,d7_xboole_0,t54_pre_topc,t8_subset_1]),
    [file(pcomps_1,t10_pcomps_1),interesting(0.52)]).

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)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t30_pcomps_1,d5_pcomps_1]),
    [file(pcomps_1,t33_pcomps_1),interesting(0.51)]).

fof(t37_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ( v2_pre_topc(g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A)))
        & l1_pre_topc(g1_pre_topc(u1_struct_0(A),k4_pcomps_1(A))) ) ) ),
    inference(mizar_proof,[status(thm)],[t34_pcomps_1,t36_pcomps_1,t35_pcomps_1,d1_pre_topc]),
    [file(pcomps_1,t37_pcomps_1),interesting(0.50)]).

fof(t20_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => r1_setfam_1(B,k3_pcomps_1(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_pcomps_1,t48_pre_topc,d2_setfam_1]),
    [file(pcomps_1,t20_pcomps_1),interesting(0.48)]).

fof(t24_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( ( v1_pcomps_1(B,A)
              & v2_tops_2(B,A) )
           => v4_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_tarski,d3_pcomps_1,d2_tops_2,t52_pre_topc,d4_tarski,t7_subset_1,t23_pcomps_1,t22_pcomps_1,t23_pcomps_1,d10_xboole_0]),
    [file(pcomps_1,t24_pcomps_1),interesting(0.45)]).

fof(t13_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_finset_1(B)
           => v1_pcomps_1(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t12_pre_topc,t11_pcomps_1,t13_finset_1,d2_pcomps_1]),
    [file(pcomps_1,t13_pcomps_1),interesting(0.45)]).

fof(t38_pcomps_1,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) )
     => v3_compts_1(k5_pcomps_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[t2_metric_1,t5_metric_1,t127_real_2,t33_pcomps_1,t33_pcomps_1,t1_metric_1,t1_metric_1,d3_xboole_0,t12_metric_1,t10_xreal_1,t4_metric_1,t4_xboole_0,t12_metric_1,d5_pre_topc,d4_compts_1]),
    [file(pcomps_1,t38_pcomps_1),interesting(0.45)]).

fof(t15_pcomps_1,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( B = k1_xboole_0
           => k3_pcomps_1(A,B) = k1_xboole_0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_pcomps_1]),
    [file(pcomps_1,t15_pcomps_1),interesting(0.44)]).

fof(t1_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ( r1_xreal_0(C,D)
                   => r1_tarski(k9_metric_1(A,B,C),k9_metric_1(A,B,D)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_struct_0,t12_metric_1,t2_xreal_1,t12_metric_1,t7_subset_1]),
    [file(pcomps_1,t1_pcomps_1),interesting(0.43)]).

fof(t36_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( r1_tarski(B,k4_pcomps_1(A))
           => r2_hidden(k5_setfam_1(u1_struct_0(A),B),k4_pcomps_1(A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_tarski,t92_zfmisc_1,d5_pcomps_1,t1_xboole_1,d5_pcomps_1]),
    [file(pcomps_1,t36_pcomps_1),interesting(0.43)]).

fof(t14_pcomps_1,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => v2_tops_2(k3_pcomps_1(A,B),A) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_pcomps_1,d2_tops_2]),
    [file(pcomps_1,t14_pcomps_1),interesting(0.41)]).

fof(t11_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => r1_tarski(a_3_1_pcomps_1(A,B,C),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski]),
    [file(pcomps_1,t11_pcomps_1),interesting(0.41)]).

fof(t12_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( r1_tarski(B,C)
                  & v1_pcomps_1(C,A) )
               => v1_pcomps_1(B,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_pcomps_1,d3_tarski,t13_finset_1,d2_pcomps_1]),
    [file(pcomps_1,t12_pcomps_1),interesting(0.40)]).

fof(t21_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_pcomps_1(B,A)
           => v1_pcomps_1(k3_pcomps_1(A,B),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_pcomps_1,d3_pcomps_1,s1_pcomps_1,t20_pcomps_1,t21_setfam_1,d1_funct_2,d12_funct_1,d3_pcomps_1,t48_pre_topc,t63_xboole_1,d7_xboole_0,d3_pcomps_1,t2_pcomps_1,t41_tops_1,t2_pcomps_1,d7_xboole_0,d12_funct_1,t2_tarski,t17_finset_1,d2_pcomps_1]),
    [file(pcomps_1,t21_pcomps_1),interesting(0.39)]).

fof(t23_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_pcomps_1(B,A)
           => k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_pcomps_1,t11_pcomps_1,t3_tops_2,d4_xboole_0,t98_zfmisc_1,t39_tops_1,t11_pcomps_1,t39_xboole_1,t12_xboole_1,t96_zfmisc_1,t50_pre_topc,d2_xboole_0,t19_pcomps_1,t11_pcomps_1,t17_pcomps_1,t95_zfmisc_1,t7_subset_1,d4_tarski,d3_pcomps_1,t39_tops_1,t4_xboole_0,d3_xboole_0,d4_tarski,d3_xboole_0,t4_xboole_0,t39_tops_1,t7_subset_1,d10_xboole_0]),
    [file(pcomps_1,t23_pcomps_1),interesting(0.38)]).

fof(t22_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => r1_tarski(k5_setfam_1(u1_struct_0(A),B),k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B))) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_pcomps_1,t18_setfam_1]),
    [file(pcomps_1,t22_pcomps_1),interesting(0.38)]).

fof(t19_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_finset_1(B)
           => k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t73_finseq_1,d2_finseq_1,t4_finseq_1,t149_relat_1,t4_finseq_1,t149_relat_1,t2_zfmisc_1,t52_pre_topc,t15_pcomps_1,t2_zfmisc_1,t11_finseq_1,t153_relat_1,t144_relat_1,t3_tops_2,t144_relat_1,t3_tops_2,t18_nat_1,t9_xreal_1,t3_finseq_1,t96_zfmisc_1,t50_pre_topc,t117_funct_1,d5_funct_1,t31_zfmisc_1,t16_pcomps_1,t31_zfmisc_1,t31_zfmisc_1,t37_nat_1,t8_xreal_1,t96_zfmisc_1,t18_pcomps_1,s1_nat_1,t146_relat_1,t4_finseq_1,t146_relat_1,t149_relat_1,t149_relat_1,t2_zfmisc_1,t52_pre_topc,t15_pcomps_1,t2_zfmisc_1,t19_nat_1,t38_nat_1]),
    [file(pcomps_1,t19_pcomps_1),interesting(0.32)]).

fof(t18_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => k3_pcomps_1(A,k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C)) = k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),k3_pcomps_1(A,B),k3_pcomps_1(A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_pcomps_1,d3_pcomps_1,d2_xboole_0,d3_pcomps_1,d2_xboole_0,d2_xboole_0,d3_pcomps_1,d2_xboole_0,d3_pcomps_1,d3_pcomps_1,d2_xboole_0,d3_pcomps_1,d2_xboole_0,t2_tarski]),
    [file(pcomps_1,t18_pcomps_1),interesting(0.30)]).

fof(t16_pcomps_1,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(k1_zfmisc_1(u1_struct_0(A))))
             => ( C = k1_tarski(B)
               => k3_pcomps_1(A,C) = k1_tarski(k6_pre_topc(A,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_pcomps_1,d1_tarski,d1_tarski,d3_pcomps_1,d1_tarski]),
    [file(pcomps_1,t16_pcomps_1),interesting(0.29)]).

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)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_xreal_0,d1_struct_0,t12_metric_1,t12_metric_1,t22_xreal_1,t4_metric_1,t2_xreal_1,t12_metric_1,t52_xreal_1,t7_subset_1]),
    [file(pcomps_1,t30_pcomps_1),interesting(0.23)]).

fof(t26_pcomps_1,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)))
             => ~ ( v2_pcomps_1(A)
                  & v4_pre_topc(B,A)
                  & v4_pre_topc(C,A)
                  & r1_xboole_0(B,C)
                  & ! [D] : 
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(D,C)
                          & ! [E] : 
                              ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                             => ! [F] : 
                                  ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                                 => ~ ( v3_pre_topc(E,A)
                                      & v3_pre_topc(F,A)
                                      & r1_tarski(B,E)
                                      & r2_hidden(D,F)
                                      & r1_xboole_0(E,F) ) ) ) ) )
                  & ! [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)
                              & r1_tarski(B,D)
                              & r1_tarski(C,E)
                              & r1_xboole_0(D,E) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s3_subset_1,t18_pre_topc,d4_tarski,d4_tarski,d2_xboole_0,t7_subset_1,t14_pre_topc,d10_xboole_0,d8_pre_topc,t29_tops_1,d1_tops_2,d4_pcomps_1,t11_pcomps_1,t3_tops_2,d2_setfam_1,t63_xboole_1,t26_subset_1,t63_xboole_1,d7_xboole_0,t52_pre_topc,t41_tops_1,t2_pcomps_1,d7_xboole_0,t63_xboole_1,d7_xboole_0,t12_pcomps_1,t23_pcomps_1,t29_tops_1,t18_tops_2,t26_tops_2,t3_xboole_0,d4_tarski,d3_pcomps_1,d7_xboole_0,t3_xboole_0,t43_subset_1,t5_pcomps_1,t3_xboole_0,d4_tarski,t7_subset_1,t22_pcomps_1,t43_subset_1]),
    [file(pcomps_1,t26_pcomps_1),interesting(0.14)]).

fof(t31_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] : 
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] : 
                          ( m1_subset_1(F,u1_struct_0(A))
                         => ~ ( v9_metric_1(A)
                              & r2_hidden(D,k5_subset_1(u1_struct_0(A),k9_metric_1(A,E,B),k9_metric_1(A,F,C)))
                              & ! [G] : 
                                  ( m1_subset_1(G,k1_numbers)
                                 => ~ ( r1_tarski(k9_metric_1(A,D,G),k9_metric_1(A,E,B))
                                      & r1_tarski(k9_metric_1(A,D,G),k9_metric_1(A,F,C)) ) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_xboole_0,t30_pcomps_1,t30_pcomps_1,t35_square_1,t38_square_1,t1_pcomps_1,t1_xboole_1,t35_square_1,t38_square_1,t1_pcomps_1,t1_xboole_1]),
    [file(pcomps_1,t31_pcomps_1),interesting(0.11)]).

fof(s1_pcomps_1,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(f1_s1_pcomps_1)))
       => ( r2_hidden(A,f2_s1_pcomps_1)
         => r2_hidden(f4_s1_pcomps_1(A),f3_s1_pcomps_1) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f2_s1_pcomps_1,f3_s1_pcomps_1)
        & m2_relset_1(A,f2_s1_pcomps_1,f3_s1_pcomps_1)
        & ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(f1_s1_pcomps_1)))
           => ( r2_hidden(B,f2_s1_pcomps_1)
             => k1_funct_1(A,B) = f4_s1_pcomps_1(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s2_funct_2]),
    [file(pcomps_1,s1_pcomps_1),interesting(0.02)]).

fof(d3_pcomps_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( C = k3_pcomps_1(A,B)
              <=> ! [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( r2_hidden(D,C)
                    <=> ? [E] : 
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                          & D = k6_pre_topc(A,E)
                          & r2_hidden(E,B) ) ) ) ) ) ) ) ),
    file(pcomps_1,d3_pcomps_1),
    [interesting(0.00)]).

fof(d2_tops_2,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v2_tops_2(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ( r2_hidden(C,B)
                 => v4_pre_topc(C,A) ) ) ) ) ) ),
    file(tops_2,d2_tops_2),
    [interesting(0.00)]).

fof(d2_pcomps_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( v1_pcomps_1(B,A)
          <=> ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & r2_hidden(C,D)
                    & v3_pre_topc(D,A)
                    & v1_finset_1(a_3_0_pcomps_1(A,B,D)) ) ) ) ) ) ),
    file(pcomps_1,d2_pcomps_1),
    [interesting(0.00)]).

fof(s2_funct_2,theorem,
    ( ! [A] : 
        ( r2_hidden(A,f1_s2_funct_2)
       => r2_hidden(f3_s2_funct_2(A),f2_s2_funct_2) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s2_funct_2,f2_s2_funct_2)
        & m2_relset_1(A,f1_s2_funct_2,f2_s2_funct_2)
        & ! [B] : 
            ( r2_hidden(B,f1_s2_funct_2)
           => k1_funct_1(A,B) = f3_s2_funct_2(B) ) ) ),
    file(funct_2,s2_funct_2),
    [interesting(0.00)]).

fof(t48_pre_topc,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(B,k6_pre_topc(A,B)) ) ) ),
    file(pre_topc,t48_pre_topc),
    [interesting(0.00)]).

fof(d2_setfam_1,definition,(
    ! [A,B] : 
      ( r1_setfam_1(A,B)
    <=> ! [C] : ~ ( r2_hidden(C,A)
          & ! [D] : ~ ( r2_hidden(D,B)
              & r1_tarski(C,D) ) ) ) ),
    file(setfam_1,d2_setfam_1),
    [interesting(0.00)]).

fof(t21_setfam_1,theorem,(
    ! [A] : 
      ( r1_setfam_1(A,k1_xboole_0)
     => A = k1_xboole_0 ) ),
    file(setfam_1,t21_setfam_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(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(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(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(t52_pre_topc,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( v4_pre_topc(B,A)
             => k6_pre_topc(A,B) = B )
            & ( ( v2_pre_topc(A)
                & k6_pre_topc(A,B) = B )
             => v4_pre_topc(B,A) ) ) ) ) ),
    file(pre_topc,t52_pre_topc),
    [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(t2_pcomps_1,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ~ ( k6_pre_topc(A,B) != k1_xboole_0
                & B = k1_xboole_0 )
            & ~ ( B != k1_xboole_0
                & k6_pre_topc(A,B) = k1_xboole_0 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t52_pre_topc,t48_pre_topc,d3_tarski]),
    [file(pcomps_1,t2_pcomps_1),interesting(0.00)]).

fof(t41_tops_1,theorem,(
    ! [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)))
             => ( v3_pre_topc(B,A)
               => k6_pre_topc(A,k5_subset_1(u1_struct_0(A),B,k6_pre_topc(A,C))) = k6_pre_topc(A,k5_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ),
    file(tops_1,t41_tops_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(t17_finset_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_finset_1(A)
       => v1_finset_1(k9_relat_1(B,A)) ) ) ),
    file(finset_1,t17_finset_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(t7_subset_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] : 
                ( m1_subset_1(D,A)
               => ( r2_hidden(D,B)
                 => r2_hidden(D,C) ) )
           => r1_tarski(B,C) ) ) ) ),
    file(subset_1,t7_subset_1),
    [interesting(0.00)]).

fof(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(d4_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    [interesting(0.00)]).

fof(t98_zfmisc_1,theorem,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
         => r1_xboole_0(C,B) )
     => r1_xboole_0(k3_tarski(A),B) ) ),
    file(zfmisc_1,t98_zfmisc_1),
    [interesting(0.00)]).

fof(t39_tops_1,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,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(A,B))
              <=> ! [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ~ ( v3_pre_topc(D,A)
                        & r2_hidden(C,D)
                        & r1_xboole_0(B,D) ) ) ) ) ) ) ),
    file(tops_1,t39_tops_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(t96_zfmisc_1,theorem,(
    ! [A,B] : k3_tarski(k2_xboole_0(A,B)) = k2_xboole_0(k3_tarski(A),k3_tarski(B)) ),
    file(zfmisc_1,t96_zfmisc_1),
    [interesting(0.00)]).

fof(t50_pre_topc,theorem,(
    ! [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)))
             => k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C)) ) ) ) ),
    file(pre_topc,t50_pre_topc),
    [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(t73_finseq_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
    <=> ? [B] : 
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & v1_finseq_1(B)
          & A = k2_relat_1(B) ) ) ),
    file(finseq_1,t73_finseq_1),
    [interesting(0.00)]).

fof(d2_finseq_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( v1_finseq_1(A)
      <=> ? [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & k1_relat_1(A) = k2_finseq_1(B) ) ) ) ),
    file(finseq_1,d2_finseq_1),
    [interesting(0.00)]).

fof(t4_finseq_1,theorem,
    ( k2_finseq_1(0) = k1_xboole_0
    & k2_finseq_1(1) = k1_tarski(1)
    & k2_finseq_1(2) = k2_tarski(1,2) ),
    file(finseq_1,t4_finseq_1),
    [interesting(0.00)]).

fof(t149_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => k9_relat_1(A,k1_xboole_0) = k1_xboole_0 ) ),
    file(relat_1,t149_relat_1),
    [interesting(0.00)]).

fof(t2_zfmisc_1,theorem,(
    k3_tarski(k1_xboole_0) = k1_xboole_0 ),
    file(zfmisc_1,t2_zfmisc_1),
    [interesting(0.00)]).

fof(t11_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => k2_xboole_0(k2_finseq_1(A),k1_tarski(k2_xcmplx_0(A,1))) = k2_finseq_1(k2_xcmplx_0(A,1)) ) ),
    file(finseq_1,t11_finseq_1),
    [interesting(0.00)]).

fof(t153_relat_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => k9_relat_1(C,k2_xboole_0(A,B)) = k2_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ),
    file(relat_1,t153_relat_1),
    [interesting(0.00)]).

fof(t144_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => r1_tarski(k9_relat_1(B,A),k2_relat_1(B)) ) ),
    file(relat_1,t144_relat_1),
    [interesting(0.00)]).

fof(t18_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => r1_xreal_0(0,A) ) ),
    file(nat_1,t18_nat_1),
    [interesting(0.00)]).

fof(t9_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(C,D) )
                   => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ),
    file(xreal_1,t9_xreal_1),
    [interesting(0.00)]).

fof(t3_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r2_hidden(A,k2_finseq_1(B))
          <=> ( r1_xreal_0(1,A)
              & r1_xreal_0(A,B) ) ) ) ) ),
    file(finseq_1,t3_finseq_1),
    [interesting(0.00)]).

fof(t117_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => k9_relat_1(B,k1_tarski(A)) = k1_tarski(k1_funct_1(B,A)) ) ) ),
    file(funct_1,t117_funct_1),
    [interesting(0.00)]).

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

fof(t31_zfmisc_1,theorem,(
    ! [A] : k3_tarski(k1_tarski(A)) = A ),
    file(zfmisc_1,t31_zfmisc_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(t37_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ! [C] : 
              ( v4_ordinal2(C)
             => ( r1_xreal_0(A,B)
               => r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) ),
    file(nat_1,t37_nat_1),
    [interesting(0.00)]).

fof(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(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(t146_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => k9_relat_1(A,k1_relat_1(A)) = k2_relat_1(A) ) ),
    file(relat_1,t146_relat_1),
    [interesting(0.00)]).

fof(t19_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ~ ( 0 != A
          & r1_xreal_0(A,0) ) ) ),
    file(nat_1,t19_nat_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(t17_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( r1_tarski(B,C)
               => r1_tarski(k3_pcomps_1(A,B),k3_pcomps_1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_pcomps_1,d3_pcomps_1,d3_tarski]),
    [file(pcomps_1,t17_pcomps_1),interesting(0.00)]).

fof(t95_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k3_tarski(A),k3_tarski(B)) ) ),
    file(zfmisc_1,t95_zfmisc_1),
    [interesting(0.00)]).

fof(t4_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ r2_hidden(C,k3_xboole_0(A,B)) )
      & ~ ( ? [C] : r2_hidden(C,k3_xboole_0(A,B))
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t4_xboole_0),
    [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(t18_setfam_1,theorem,(
    ! [A,B] : 
      ( r1_setfam_1(A,B)
     => r1_tarski(k3_tarski(A),k3_tarski(B)) ) ),
    file(setfam_1,t18_setfam_1),
    [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_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(t17_setfam_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_setfam_1(A,B) ) ),
    file(setfam_1,t17_setfam_1),
    [interesting(0.00)]).

fof(t12_pre_topc,theorem,(
    ! [A] : 
      ( l1_struct_0(A)
     => k2_pre_topc(A) = u1_struct_0(A) ) ),
    file(pre_topc,t12_pre_topc),
    [interesting(0.00)]).

fof(t13_finset_1,theorem,(
    ! [A,B] : 
      ( ( r1_tarski(A,B)
        & v1_finset_1(B) )
     => v1_finset_1(A) ) ),
    file(finset_1,t13_finset_1),
    [interesting(0.00)]).

fof(d4_pcomps_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v2_pcomps_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))))
                   => ~ ( v1_tops_2(C,A)
                        & r1_pre_topc(A,C)
                        & r1_setfam_1(C,B)
                        & v1_pcomps_1(C,A) ) ) ) ) ) ) ),
    file(pcomps_1,d4_pcomps_1),
    [interesting(0.00)]).

fof(t3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ ( r2_hidden(C,A)
              & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    [interesting(0.00)]).

fof(t50_subset_1,theorem,(
    ! [A] : 
      ( A != k1_xboole_0
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] : 
              ( m1_subset_1(C,A)
             => ( ~ r2_hidden(C,B)
               => r2_hidden(C,k3_subset_1(A,B)) ) ) ) ) ),
    file(subset_1,t50_subset_1),
    [interesting(0.00)]).

fof(t54_subset_1,theorem,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ~ ( r2_hidden(B,k3_subset_1(A,C))
          & r2_hidden(B,C) ) ) ),
    file(subset_1,t54_subset_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(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(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(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(t54_pre_topc,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(C,k6_pre_topc(A,B))
              <=> ( ~ v3_struct_0(A)
                  & ! [D] : 
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( v3_pre_topc(D,A)
                          & r2_hidden(C,D)
                          & r1_xboole_0(B,D) ) ) ) ) ) ) ) ),
    file(pre_topc,t54_pre_topc),
    [interesting(0.00)]).

fof(t8_subset_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( ! [D] : 
                ( m1_subset_1(D,A)
               => ( r2_hidden(D,B)
                <=> r2_hidden(D,C) ) )
           => B = C ) ) ) ),
    file(subset_1,t8_subset_1),
    [interesting(0.00)]).

fof(s3_subset_1,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(f1_s3_subset_1))
      & ! [B] : 
          ( m1_subset_1(B,f1_s3_subset_1)
         => ( r2_hidden(B,A)
          <=> p1_s3_subset_1(B) ) ) ) ),
    file(subset_1,s3_subset_1),
    [interesting(0.00)]).

fof(t18_pre_topc,theorem,(
    ! [A] : 
      ( l1_struct_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => k4_subset_1(u1_struct_0(A),B,k3_subset_1(u1_struct_0(A),B)) = k2_pre_topc(A) ) ) ),
    file(pre_topc,t18_pre_topc),
    [interesting(0.00)]).

fof(t14_pre_topc,theorem,(
    ! [A] : 
      ( l1_struct_0(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => r1_tarski(B,k2_pre_topc(A)) ) ) ),
    file(pre_topc,t14_pre_topc),
    [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(t29_tops_1,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v4_pre_topc(B,A)
          <=> v3_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ),
    file(tops_1,t29_tops_1),
    [interesting(0.00)]).

fof(t26_subset_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => r1_xboole_0(B,k3_subset_1(A,B)) ) ),
    file(subset_1,t26_subset_1),
    [interesting(0.00)]).

fof(t18_tops_2,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( ( r1_tarski(B,C)
                  & v1_tops_2(C,A) )
               => v1_tops_2(B,A) ) ) ) ) ),
    file(tops_2,t18_tops_2),
    [interesting(0.00)]).

fof(t26_tops_2,theorem,(
    ! [A] : 
      ( ( v2_pre_topc(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)
           => v3_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ),
    file(tops_2,t26_tops_2),
    [interesting(0.00)]).

fof(t43_subset_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r1_xboole_0(B,C)
          <=> r1_tarski(B,k3_subset_1(A,C)) ) ) ) ),
    file(subset_1,t43_subset_1),
    [interesting(0.00)]).

fof(t13_pre_topc,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => r2_hidden(B,k2_pre_topc(A)) ) ) ),
    file(pre_topc,t13_pre_topc),
    [interesting(0.00)]).

fof(t5_pcomps_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(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)
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ? [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & r2_hidden(C,D)
                    & r2_hidden(D,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_pre_topc,t13_pre_topc,d4_tarski]),
    [file(pcomps_1,t5_pcomps_1),interesting(0.00)]).

fof(d5_compts_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v4_compts_1(A)
      <=> ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ~ ( C != k1_xboole_0
                    & v4_pre_topc(C,A)
                    & r2_hidden(B,k3_subset_1(u1_struct_0(A),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)
                                & r1_tarski(C,E)
                                & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ),
    file(compts_1,d5_compts_1),
    [interesting(0.00)]).

fof(d6_compts_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v5_compts_1(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)))
               => ~ ( B != k1_xboole_0
                    & C != k1_xboole_0
                    & v4_pre_topc(B,A)
                    & v4_pre_topc(C,A)
                    & r1_xboole_0(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)
                                & r1_tarski(B,D)
                                & r1_tarski(C,E)
                                & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ),
    file(compts_1,d6_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(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(t22_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,k2_xcmplx_0(B,C))
              <=> r1_xreal_0(k6_xcmplx_0(A,B),C) ) ) ) ) ),
    file(xreal_1,t22_xreal_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(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(t52_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(B,A)
              & r1_xreal_0(k6_xcmplx_0(B,A),0) ) ) ) ),
    file(xreal_1,t52_xreal_1),
    [interesting(0.00)]).

fof(t35_square_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => r1_xreal_0(k1_square_1(A,B),A) ) ) ),
    file(square_1,t35_square_1),
    [interesting(0.00)]).

fof(t38_square_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( k1_square_1(A,B) = A
            | k1_square_1(A,B) = B ) ) ) ),
    file(square_1,t38_square_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(t32_pcomps_1,theorem,(
    $true ),
    file(pcomps_1,t32_pcomps_1),
    [interesting(0.00)]).

fof(t29_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ? [C] : 
              ( m1_subset_1(C,k1_numbers)
              & ~ r1_xreal_0(C,0)
              & r1_tarski(k9_metric_1(A,B,C),u1_struct_0(A)) ) ) ) ),
    file(pcomps_1,t29_pcomps_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(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(t19_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C) )
     => r1_tarski(A,k3_xboole_0(B,C)) ) ),
    file(xboole_1,t19_xboole_1),
    [interesting(0.00)]).

fof(t35_pcomps_1,theorem,(
    ! [A] : 
      ( l1_metric_1(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)))
             => ( ( r2_hidden(B,k4_pcomps_1(A))
                  & r2_hidden(C,k4_pcomps_1(A)) )
               => r2_hidden(k5_subset_1(u1_struct_0(A),B,C),k4_pcomps_1(A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_xboole_0,d5_pcomps_1,d5_pcomps_1,t35_square_1,t38_square_1,t38_square_1,t1_pcomps_1,t1_xboole_1,t35_square_1,t38_square_1,t1_pcomps_1,t1_xboole_1,t19_xboole_1,d5_pcomps_1]),
    [file(pcomps_1,t35_pcomps_1),interesting(0.00)]).

fof(d1_pre_topc,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v2_pre_topc(A)
      <=> ( r2_hidden(u1_struct_0(A),u1_pre_topc(A))
          & ! [B] : 
              ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( r1_tarski(B,u1_pre_topc(A))
               => r2_hidden(k5_setfam_1(u1_struct_0(A),B),u1_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)))
                 => ( ( r2_hidden(B,u1_pre_topc(A))
                      & r2_hidden(C,u1_pre_topc(A)) )
                   => r2_hidden(k5_subset_1(u1_struct_0(A),B,C),u1_pre_topc(A)) ) ) ) ) ) ) ),
    file(pre_topc,d1_pre_topc),
    [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(t5_metric_1,theorem,(
    ! [A] : 
      ( ( v6_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] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(0,k4_metric_1(A,B,C)) ) ) ) ),
    file(metric_1,t5_metric_1),
    [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(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(t10_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ~ ( ~ r1_xreal_0(B,A)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ),
    file(xreal_1,t10_xreal_1),
    [interesting(0.00)]).

fof(d5_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(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(d7_pcomps_1,definition,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( r1_pcomps_1(A,B)
      <=> ! [C] : 
            ( m1_subset_1(C,A)
           => ! [D] : 
                ( m1_subset_1(D,A)
               => ! [E] : 
                    ( m1_subset_1(E,A)
                   => ( ( k1_metric_1(A,A,B,C,D) = 0
                       => C = D )
                      & ( C = D
                       => k1_metric_1(A,A,B,C,D) = 0 )
                      & k1_metric_1(A,A,B,C,D) = k1_metric_1(A,A,B,D,C)
                      & r1_xreal_0(k1_metric_1(A,A,B,C,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,D,E))) ) ) ) ) ) ) ),
    file(pcomps_1,d7_pcomps_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(t3_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) = k2_metric_1(A,C,B) ) )
      <=> v8_metric_1(A) ) ) ),
    file(metric_1,t3_metric_1),
    [interesting(0.00)]).

fof(t39_pcomps_1,theorem,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
        & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
     => ( r1_pcomps_1(A,B)
      <=> ( v6_metric_1(g1_metric_1(A,B))
          & v7_metric_1(g1_metric_1(A,B))
          & v8_metric_1(g1_metric_1(A,B))
          & v9_metric_1(g1_metric_1(A,B))
          & l1_metric_1(g1_metric_1(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_metric_1,d7_pcomps_1,d1_metric_1,d7_pcomps_1,d1_metric_1,d1_metric_1,d7_pcomps_1,t6_metric_1,d1_metric_1,t1_metric_1,t2_metric_1,d1_metric_1,t3_metric_1,d1_metric_1,d1_metric_1,t4_metric_1,d7_pcomps_1]),
    [file(pcomps_1,t39_pcomps_1),interesting(0.00)]).

fof(t3_pcomps_1,theorem,(
    $true ),
    file(pcomps_1,t3_pcomps_1),
    [interesting(0.00)]).

fof(t4_pcomps_1,theorem,(
    $true ),
    file(pcomps_1,t4_pcomps_1),
    [interesting(0.00)]).

fof(t6_pcomps_1,theorem,(
    $true ),
    file(pcomps_1,t6_pcomps_1),
    [interesting(0.00)]).

fof(t7_pcomps_1,theorem,(
    ! [A] : u1_pre_topc(k2_pcomps_1(A)) = k1_pcomps_1(A) ),
    file(pcomps_1,t7_pcomps_1),
    [interesting(0.00)]).

fof(t8_pcomps_1,theorem,(
    ! [A] : u1_struct_0(k2_pcomps_1(A)) = A ),
    file(pcomps_1,t8_pcomps_1),
    [interesting(0.00)]).

fof(t24_finset_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
    <=> v1_finset_1(k1_zfmisc_1(A)) ) ),
    file(finset_1,t24_finset_1),
    [interesting(0.00)]).