fof(t41_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ( k3_partit1(A,k3_pua2mss1(A),B) = B
            & k2_partit1(A,k3_pua2mss1(A),B) = k3_pua2mss1(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_partit1,t38_partit1,d3_eqrel_1,t38_partit1,t36_partit1,t24_partit1,d2_xboole_0,d3_tarski,t7_xboole_1,d10_xboole_0,t24_partit1,t36_partit1,t33_partit1,t5_partit1,t28_partit1,t38_partit1,t17_eqrel_1,t38_partit1,t29_partit1]),
    [file(partit1,t41_partit1),interesting(1.00)]).

fof(t31_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => k2_partit1(A,B,k3_partit1(A,B,C)) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t28_partit1,t27_partit1,t24_eqrel_1,t29_partit1]),
    [file(partit1,t31_partit1),interesting(0.96)]).

fof(t32_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => k3_partit1(A,B,k2_partit1(A,B,C)) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_partit1,t28_partit1,t25_eqrel_1,t29_partit1]),
    [file(partit1,t32_partit1),interesting(0.96)]).

fof(t20_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => k3_partit1(A,B,B) = B ) ) ),
    inference(mizar_proof,[status(thm)],[t19_partit1,d5_partit1,d2_partit1,d1_partit1,d6_eqrel_1,d6_eqrel_1,d4_tarski,d4_tarski,d3_xboole_0,d7_xboole_0,d6_eqrel_1,t92_zfmisc_1,t8_partit1,d2_partit1,d2_setfam_1,t5_partit1]),
    [file(partit1,t20_partit1),interesting(0.89)]).

fof(t40_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ( k3_partit1(A,k6_partit1(A),B) = k6_partit1(A)
            & k2_partit1(A,k6_partit1(A),B) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_partit1,t37_partit1,t4_msualg_5,d3_eqrel_1,t24_partit1,t36_partit1,t5_partit1,t28_partit1,t37_partit1,t18_eqrel_1,t29_partit1]),
    [file(partit1,t40_partit1),interesting(0.89)]).

fof(t9_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => r1_partit1(A,B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t80_zfmisc_1,t31_zfmisc_1,d1_tarski,d1_tarski,d1_tarski,d6_eqrel_1,d6_eqrel_1,d3_tarski,d4_tarski,d1_tarski,d2_setfam_1,d1_tarski,t8_partit1]),
    [file(partit1,t9_partit1),interesting(0.88)]).

fof(t30_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( m1_eqrel_1(D,A)
                 => k3_partit1(A,k3_partit1(A,B,C),D) = k3_partit1(A,B,k3_partit1(A,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_partit1,t27_partit1,t1_msualg_5,t27_partit1,t27_partit1,t29_partit1]),
    [file(partit1,t30_partit1),interesting(0.88)]).

fof(t13_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( m1_subset_1(D,A)
                 => ? [E] : 
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                      & r2_hidden(D,E)
                      & r2_partit1(A,B,C,E) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,t9_partit1,t9_partit1,s7_domain_1,d1_setfam_1,d10_setfam_1,t10_partit1,t10_partit1,d1_partit1,t95_zfmisc_1,d3_tarski,d10_setfam_1,d1_setfam_1,d10_xboole_0,t95_zfmisc_1,d5_subset_1,d4_xboole_0,t44_subset_1,d7_xboole_0,t12_partit1,t12_partit1,d3_tarski,d10_setfam_1,d1_setfam_1,t26_xboole_1,t3_xboole_1,t26_xboole_1,t3_xboole_1,d1_xboole_0,t92_zfmisc_1,d1_xboole_0,d2_partit1,d10_setfam_1]),
    [file(partit1,t13_partit1),interesting(0.83)]).

fof(t33_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( m1_eqrel_1(D,A)
                 => ( ( r1_setfam_1(B,D)
                      & r1_setfam_1(C,D) )
                   => r1_setfam_1(k3_partit1(A,B,C),D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t24_partit1,t24_partit1,t27_partit1,t8_xboole_1,d3_eqrel_1,t24_partit1]),
    [file(partit1,t33_partit1),interesting(0.83)]).

fof(t27_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => k4_partit1(A,k3_partit1(A,B,C)) = k5_eqrel_1(A,k4_partit1(A,B),k4_partit1(A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t19_partit1,t19_partit1,d6_eqrel_1,d6_eqrel_1,d3_relat_1,d6_partit1,d5_partit1,d2_partit1,d1_partit1,d4_tarski,d8_finseq_1,t57_finseq_1,t57_finseq_1,t57_finseq_1,d4_tarski,d3_xboole_0,d3_tarski,d1_partit1,d3_tarski,d4_tarski,d4_tarski,d3_xboole_0,d4_tarski,d3_tarski,d4_tarski,d4_tarski,t35_finseq_1,t57_finseq_1,t8_xreal_1,t2_xreal_1,t59_finseq_1,t59_finseq_1,d7_xboole_0,t3_xboole_0,d3_xboole_0,d3_finseq_1,t3_finseq_1,d7_finseq_1,t38_nat_1,t38_nat_1,t1_xreal_1,d4_tarski,d10_xboole_0,d3_tarski,d1_partit1,t37_xboole_1,d1_xboole_0,d4_xboole_0,d4_tarski,t35_finseq_1,t57_finseq_1,t8_xreal_1,t2_xreal_1,t59_finseq_1,t59_finseq_1,d7_xboole_0,t3_xboole_0,d3_xboole_0,d3_finseq_1,t3_finseq_1,d7_finseq_1,t38_nat_1,t38_nat_1,t1_xreal_1,d3_xboole_0,t25_partit1,d2_partit1,d4_tarski,d4_tarski,t35_finseq_1,t57_finseq_1,t8_xreal_1,t59_finseq_1,t59_finseq_1,d7_xboole_0,t3_xboole_0,d3_xboole_0,d3_finseq_1,t3_finseq_1,d7_finseq_1,t38_nat_1,t38_nat_1,t1_xreal_1,t2_xreal_1,t57_finseq_1,d3_xboole_0,d6_partit1,d6_partit1,t7_xboole_1,t7_xboole_1,t26_partit1,d2_xboole_0,d6_partit1,d2_setfam_1,d6_partit1,d2_setfam_1,d6_partit1,d3_relat_1,d3_eqrel_1,d10_xboole_0]),
    [file(partit1,t27_partit1),interesting(0.82)]).

fof(t15_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => k2_partit1(A,B,B) = B ) ) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d5_setfam_1,t17_xboole_1,d2_setfam_1,d6_eqrel_1,d1_tarski,d5_setfam_1,d4_xboole_0,d2_setfam_1,t5_partit1]),
    [file(partit1,t15_partit1),interesting(0.80)]).

fof(t19_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => r1_setfam_1(B,k3_partit1(A,B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,d3_tarski,d6_eqrel_1,d4_tarski,d5_partit1,d2_partit1,d1_partit1,d4_tarski,d3_xboole_0,d7_xboole_0,d6_eqrel_1,t92_zfmisc_1,d2_setfam_1]),
    [file(partit1,t19_partit1),interesting(0.79)]).

fof(t5_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( ( r1_setfam_1(C,B)
                  & r1_setfam_1(B,C) )
               => B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_partit1,t4_partit1,d10_xboole_0]),
    [file(partit1,t5_partit1),interesting(0.77)]).

fof(t8_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( r1_setfam_1(C,B)
               => ! [D] : 
                    ( r2_hidden(D,B)
                   => r1_partit1(A,C,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,t7_partit1,d3_setfam_1,d3_tarski,d4_tarski,d2_setfam_1,d6_eqrel_1,t3_xboole_0,d4_tarski,d3_tarski,d4_tarski,d3_tarski,d10_xboole_0,d1_partit1]),
    [file(partit1,t8_partit1),interesting(0.76)]).

fof(t7_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( r1_setfam_1(C,B)
               => r2_setfam_1(C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,d6_eqrel_1,d4_tarski,d2_setfam_1,d6_eqrel_1,t3_xboole_0,d3_setfam_1]),
    [file(partit1,t7_partit1),interesting(0.75)]).

fof(t4_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( ( r1_setfam_1(C,B)
                  & r1_setfam_1(B,C) )
               => r1_tarski(C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_setfam_1,d2_setfam_1,t1_xboole_1,t1_partit1,d10_xboole_0]),
    [file(partit1,t4_partit1),interesting(0.75)]).

fof(t38_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => k4_partit1(A,k3_pua2mss1(A)) = k6_relat_1(A) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,d3_relat_1,d6_partit1,t13_pua2mss1,d1_tarski,d1_tarski,d10_relat_1,d3_relat_1,d10_relat_1,d4_tarski,d6_partit1,d10_xboole_0]),
    [file(partit1,t38_partit1),interesting(0.74)]).

fof(t1_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C,D] : 
              ( ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & r1_tarski(C,D) )
             => C = D ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,d6_eqrel_1,t3_xboole_0]),
    [file(partit1,t1_partit1),interesting(0.73)]).

fof(t29_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( k4_partit1(A,B) = k4_partit1(A,C)
               => B = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t24_partit1,t5_partit1]),
    [file(partit1,t29_partit1),interesting(0.73)]).

fof(t39_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => r1_setfam_1(k3_pua2mss1(A),k6_partit1(A)) ) ),
    inference(mizar_proof,[status(thm)],[t37_partit1,d1_eqrel_1,t24_partit1]),
    [file(partit1,t39_partit1),interesting(0.72)]).

fof(t17_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => r1_setfam_1(k2_partit1(A,B,C),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d5_setfam_1,t17_xboole_1,d2_setfam_1]),
    [file(partit1,t17_partit1),interesting(0.71)]).

fof(t12_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( r1_partit1(A,B,C)
               => ( C = A
                  | r1_partit1(A,B,k3_subset_1(A,C)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_partit1,d1_partit1,d6_eqrel_1,d5_subset_1,t37_xboole_1,d10_xboole_0,t3_t_1topsp,t36_xboole_1]),
    [file(partit1,t12_partit1),interesting(0.70)]).

fof(t36_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ( r1_setfam_1(B,k6_partit1(A))
            & r1_setfam_1(k3_pua2mss1(A),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,d3_tarski,d6_eqrel_1,d4_tarski,d1_tarski,t13_pua2mss1,d6_eqrel_1,d1_tarski,d3_tarski,d6_eqrel_1,d4_tarski,d3_tarski,d1_tarski,d2_setfam_1]),
    [file(partit1,t36_partit1),interesting(0.70)]).

fof(t24_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( r1_setfam_1(B,C)
              <=> r1_tarski(k4_partit1(A,B),k4_partit1(A,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_partit1,d2_setfam_1,d6_partit1,d3_relat_1,d6_eqrel_1,d3_tarski,d6_partit1,d6_partit1,d6_partit1,d3_tarski,d6_partit1,t3_xboole_0,d6_eqrel_1,d3_tarski,d6_partit1,d10_xboole_0,d2_setfam_1]),
    [file(partit1,t24_partit1),interesting(0.69)]).

fof(t37_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => k4_partit1(A,k6_partit1(A)) = k1_eqrel_1(A) ) ),
    inference(mizar_proof,[status(thm)],[d3_relat_1,d1_eqrel_1,d3_relat_1,d1_tarski,t106_zfmisc_1,d6_partit1,d10_xboole_0]),
    [file(partit1,t37_partit1),interesting(0.69)]).

fof(t10_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ( ! [D] : 
                    ( r2_hidden(D,C)
                   => r1_partit1(A,B,D) )
               => ( k8_setfam_1(A,C) = k1_xboole_0
                  | r1_partit1(A,B,k8_setfam_1(A,C)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d10_setfam_1,t9_partit1,d1_partit1,s1_zfrefle1,d3_tarski,d5_funct_1,t1_xboole_1,d1_funct_2,t11_relset_1,s8_domain_1,d1_partit1,d3_tarski,d1_xboole_0,t58_setfam_1,d1_xboole_0,d1_xboole_0,d10_setfam_1,d1_setfam_1,d4_tarski,d1_t_1topsp,d1_setfam_1,d10_setfam_1,d3_tarski,d10_setfam_1,d1_xboole_0,d1_t_1topsp,d1_setfam_1,d4_tarski,d1_t_1topsp,d1_setfam_1,d10_setfam_1,d4_tarski,d3_tarski,d1_xboole_0,d4_tarski,d10_setfam_1,d1_setfam_1,d4_tarski,d1_setfam_1,d10_setfam_1,d10_xboole_0]),
    [file(partit1,t10_partit1),interesting(0.66)]).

fof(t21_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B,C,D,E] : 
          ( m1_eqrel_1(E,A)
         => ! [F] : 
              ( m1_eqrel_1(F,A)
             => ( ( r1_setfam_1(E,F)
                  & r2_hidden(B,F)
                  & r2_hidden(C,E)
                  & r2_hidden(D,B)
                  & r2_hidden(D,C) )
               => r1_tarski(C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_setfam_1,d6_eqrel_1,t3_xboole_0]),
    [file(partit1,t21_partit1),interesting(0.65)]).

fof(t28_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => k4_partit1(A,k2_partit1(A,B,C)) = k4_eqrel_1(A,k4_partit1(A,B),k4_partit1(A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t17_partit1,d6_partit1,d2_setfam_1,d2_setfam_1,d6_partit1,d3_xboole_0,d3_xboole_0,d6_partit1,d6_partit1,d3_xboole_0,d5_setfam_1,d1_tarski,d4_xboole_0,d3_xboole_0,d3_xboole_0,d6_partit1,d2_relat_1]),
    [file(partit1,t28_partit1),interesting(0.63)]).

fof(t34_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( m1_eqrel_1(D,A)
                 => ( ( r1_setfam_1(D,B)
                      & r1_setfam_1(D,C) )
                   => r1_setfam_1(D,k2_partit1(A,B,C)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t24_partit1,t24_partit1,t28_partit1,t19_xboole_1,t24_partit1]),
    [file(partit1,t34_partit1),interesting(0.63)]).

fof(t3_partit1,theorem,(
    ! [A] : k3_tarski(k4_xboole_0(A,k1_tarski(k1_xboole_0))) = k3_tarski(A) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_tarski,d4_xboole_0,d4_tarski,d3_tarski,d4_tarski,d1_tarski,d4_xboole_0,d4_tarski,d10_xboole_0]),
    [file(partit1,t3_partit1),interesting(0.62)]).

fof(t16_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( m1_eqrel_1(D,A)
                 => k2_partit1(A,k2_partit1(A,B,C),D) = k2_partit1(A,B,k2_partit1(A,C,D)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_xboole_0,d5_setfam_1,d4_xboole_0,d5_setfam_1,t16_xboole_1,d5_setfam_1,d4_xboole_0,t16_xboole_1,d1_tarski,d1_tarski,d4_xboole_0,d5_setfam_1,d4_xboole_0,d2_setfam_1,d4_xboole_0,d5_setfam_1,d4_xboole_0,d5_setfam_1,t16_xboole_1,d5_setfam_1,d4_xboole_0,d1_tarski,d1_tarski,d4_xboole_0,d5_setfam_1,d4_xboole_0,t16_xboole_1,d2_setfam_1,t5_partit1]),
    [file(partit1,t16_partit1),interesting(0.61)]).

fof(t22_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B,C,D,E] : 
          ( m1_eqrel_1(E,A)
         => ! [F] : 
              ( m1_eqrel_1(F,A)
             => ( ( r2_hidden(B,k3_partit1(A,E,F))
                  & r2_hidden(C,E)
                  & r2_hidden(D,B)
                  & r2_hidden(D,C) )
               => r1_tarski(C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t19_partit1,t21_partit1]),
    [file(partit1,t22_partit1),interesting(0.61)]).

fof(t35_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => k3_pua2mss1(A) = a_1_0_partit1(A) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t13_pua2mss1,t37_zfmisc_1,d3_tarski,t13_pua2mss1,d10_xboole_0]),
    [file(partit1,t35_partit1),interesting(0.60)]).

fof(t11_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( ( r1_partit1(A,B,C)
                      & r1_partit1(A,B,D) )
                   => ( r1_xboole_0(C,D)
                      | r1_partit1(A,B,k5_subset_1(A,C,D)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_partit1,d1_partit1,d3_tarski,d3_xboole_0,d4_tarski,d4_tarski,t3_xboole_0,d6_eqrel_1,d3_xboole_0,d4_tarski,d3_tarski,d4_tarski,d3_xboole_0,d4_tarski,d3_xboole_0,d10_xboole_0,t8_xboole_1,t17_xboole_1,t7_xboole_1,t1_xboole_1,t1_xboole_1,t3_xboole_0,d3_xboole_0,t2_zfmisc_1,d1_partit1]),
    [file(partit1,t11_partit1),interesting(0.54)]).

fof(t14_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,A)
             => ? [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                  & r2_hidden(C,D)
                  & r2_hidden(D,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_eqrel_1,t28_eqrel_1,d5_eqrel_1]),
    [file(partit1,t14_partit1),interesting(0.54)]).

fof(t23_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ? [C] : 
              ( v3_relat_2(C)
              & v8_relat_2(C)
              & v1_partfun1(C,A,A)
              & m2_relset_1(C,A,A)
              & ! [D,E] : 
                  ( r2_hidden(k4_tarski(D,E),C)
                <=> ? [F] : 
                      ( m1_subset_1(F,k1_zfmisc_1(A))
                      & r2_hidden(F,B)
                      & r2_hidden(D,F)
                      & r2_hidden(E,F) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_eqrel_1,d4_tarski,d6_eqrel_1,t3_xboole_0,s1_eqrel_1]),
    [file(partit1,t23_partit1),interesting(0.42)]).

fof(d1_partit1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( r1_partit1(A,B,C)
            <=> ? [D] : 
                  ( r1_tarski(D,B)
                  & D != k1_xboole_0
                  & C = k3_tarski(D) ) ) ) ) ),
    file(partit1,d1_partit1),
    [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(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(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(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(d6_eqrel_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( A != k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> ( k5_setfam_1(A,B) = A
              & ! [C] : 
                  ( m1_subset_1(C,k1_zfmisc_1(A))
                 => ( r2_hidden(C,B)
                   => ( C != k1_xboole_0
                      & ! [D] : 
                          ( m1_subset_1(D,k1_zfmisc_1(A))
                         => ~ ( r2_hidden(D,B)
                              & C != D
                              & ~ r1_xboole_0(C,D) ) ) ) ) ) ) ) )
        & ( A = k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> B = k1_xboole_0 ) ) ) ) ),
    file(eqrel_1,d6_eqrel_1),
    [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(t8_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    file(xboole_1,t8_xboole_1),
    [interesting(0.00)]).

fof(t17_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ),
    file(xboole_1,t17_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(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(t2_zfmisc_1,theorem,(
    k3_tarski(k1_xboole_0) = k1_xboole_0 ),
    file(zfmisc_1,t2_zfmisc_1),
    [interesting(0.00)]).

fof(t80_zfmisc_1,theorem,(
    ! [A] : r1_tarski(k1_tarski(A),k1_zfmisc_1(A)) ),
    file(zfmisc_1,t80_zfmisc_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(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(d3_setfam_1,definition,(
    ! [A,B] : 
      ( r2_setfam_1(A,B)
    <=> ! [C] : ~ ( r2_hidden(C,B)
          & ! [D] : ~ ( r2_hidden(D,A)
              & r1_tarski(D,C) ) ) ) ),
    file(setfam_1,d3_setfam_1),
    [interesting(0.00)]).

fof(s7_domain_1,theorem,(
    m1_subset_1(a_0_0_domain_1,k1_zfmisc_1(f1_s7_domain_1)) ),
    file(domain_1,s7_domain_1),
    [interesting(0.00)]).

fof(d1_setfam_1,definition,(
    ! [A,B] : 
      ( ( A != k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ! [D] : 
                  ( r2_hidden(D,A)
                 => r2_hidden(C,D) ) ) ) )
      & ( A = k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> B = k1_xboole_0 ) ) ) ),
    file(setfam_1,d1_setfam_1),
    [interesting(0.00)]).

fof(d10_setfam_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( B != k1_xboole_0
         => k8_setfam_1(A,B) = k6_setfam_1(A,B) )
        & ( B = k1_xboole_0
         => k8_setfam_1(A,B) = A ) ) ) ),
    file(setfam_1,d10_setfam_1),
    [interesting(0.00)]).

fof(s1_zfrefle1,theorem,
    ( ! [A] : ~ ( r2_hidden(A,f1_s1_zfrefle1)
        & ! [B] : ~ p1_s1_zfrefle1(A,B) )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s1_zfrefle1
        & ! [B] : 
            ( r2_hidden(B,f1_s1_zfrefle1)
           => p1_s1_zfrefle1(B,k1_funct_1(A,B)) ) ) ),
    file(zfrefle1,s1_zfrefle1),
    [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(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(t11_relset_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => ( ( r1_tarski(k1_relat_1(C),A)
          & r1_tarski(k2_relat_1(C),B) )
       => m2_relset_1(C,A,B) ) ) ),
    file(relset_1,t11_relset_1),
    [interesting(0.00)]).

fof(s8_domain_1,theorem,(
    m1_subset_1(a_0_1_domain_1,k1_zfmisc_1(f2_s8_domain_1)) ),
    file(domain_1,s8_domain_1),
    [interesting(0.00)]).

fof(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.00)]).

fof(t58_setfam_1,theorem,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( r2_hidden(B,A)
       => ( r2_hidden(B,k8_setfam_1(A,C))
        <=> ! [D] : 
              ( r2_hidden(D,C)
             => r2_hidden(B,D) ) ) ) ) ),
    file(setfam_1,t58_setfam_1),
    [interesting(0.00)]).

fof(d1_t_1topsp,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( D = k1_t_1topsp(A,B,C)
                  <=> ( r2_hidden(B,D)
                      & r2_hidden(D,C) ) ) ) ) ) ) ),
    file(t_1topsp,d1_t_1topsp),
    [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(d5_subset_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k3_subset_1(A,B) = k4_xboole_0(A,B) ) ),
    file(subset_1,d5_subset_1),
    [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(t44_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,k3_subset_1(A,C))
          <=> r1_tarski(B,C) ) ) ) ),
    file(subset_1,t44_subset_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(t37_xboole_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,B) = k1_xboole_0
    <=> r1_tarski(A,B) ) ),
    file(xboole_1,t37_xboole_1),
    [interesting(0.00)]).

fof(t3_t_1topsp,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(B))
             => k4_xboole_0(k5_setfam_1(A,B),k3_tarski(C)) = k3_tarski(k4_xboole_0(B,C)) ) ) ) ),
    file(t_1topsp,t3_t_1topsp),
    [interesting(0.00)]).

fof(t36_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ),
    file(xboole_1,t36_xboole_1),
    [interesting(0.00)]).

fof(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(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(d2_partit1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( r2_partit1(A,B,C,D)
                <=> ( r1_partit1(A,B,D)
                    & r1_partit1(A,C,D)
                    & ! [E] : 
                        ( ( r1_tarski(E,D)
                          & r1_partit1(A,B,E)
                          & r1_partit1(A,C,E) )
                       => E = D ) ) ) ) ) ) ),
    file(partit1,d2_partit1),
    [interesting(0.00)]).

fof(t43_eqrel_1,theorem,(
    ! [A,B] : 
      ( m1_eqrel_1(B,A)
     => ? [C] : 
          ( v3_relat_2(C)
          & v8_relat_2(C)
          & v1_partfun1(C,A,A)
          & m2_relset_1(C,A,A)
          & B = k7_eqrel_1(A,C) ) ) ),
    file(eqrel_1,t43_eqrel_1),
    [interesting(0.00)]).

fof(t28_eqrel_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_2(B)
        & v3_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,k6_eqrel_1(A,B,C)) ) ) ),
    file(eqrel_1,t28_eqrel_1),
    [interesting(0.00)]).

fof(d5_eqrel_1,definition,(
    ! [A,B] : 
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = k7_eqrel_1(A,B)
          <=> ! [D] : 
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> ? [E] : 
                      ( r2_hidden(E,A)
                      & D = k6_eqrel_1(A,B,E) ) ) ) ) ) ) ),
    file(eqrel_1,d5_eqrel_1),
    [interesting(0.00)]).

fof(d5_setfam_1,definition,(
    ! [A,B,C] : 
      ( C = k3_setfam_1(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ? [E,F] : 
              ( r2_hidden(E,A)
              & r2_hidden(F,B)
              & D = k3_xboole_0(E,F) ) ) ) ),
    file(setfam_1,d5_setfam_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(t18_partit1,theorem,(
    $true ),
    file(partit1,t18_partit1),
    [interesting(0.00)]).

fof(d5_partit1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D] : 
                  ( m1_eqrel_1(D,A)
                 => ( D = k3_partit1(A,B,C)
                  <=> ! [E] : 
                        ( r2_hidden(E,D)
                      <=> r2_partit1(A,B,C,E) ) ) ) ) ) ) ),
    file(partit1,d5_partit1),
    [interesting(0.00)]).

fof(s1_eqrel_1,theorem,
    ( ( ! [A] : 
          ( r2_hidden(A,f1_s1_eqrel_1)
         => p1_s1_eqrel_1(A,A) )
      & ! [A,B] : 
          ( p1_s1_eqrel_1(A,B)
         => p1_s1_eqrel_1(B,A) )
      & ! [A,B,C] : 
          ( ( p1_s1_eqrel_1(A,B)
            & p1_s1_eqrel_1(B,C) )
         => p1_s1_eqrel_1(A,C) ) )
   => ? [A] : 
        ( v3_relat_2(A)
        & v8_relat_2(A)
        & v1_partfun1(A,f1_s1_eqrel_1,f1_s1_eqrel_1)
        & m2_relset_1(A,f1_s1_eqrel_1,f1_s1_eqrel_1)
        & ! [B,C] : 
            ( r2_hidden(k4_tarski(B,C),A)
          <=> ( r2_hidden(B,f1_s1_eqrel_1)
              & r2_hidden(C,f1_s1_eqrel_1)
              & p1_s1_eqrel_1(B,C) ) ) ) ),
    file(eqrel_1,s1_eqrel_1),
    [interesting(0.00)]).

fof(t2_partit1,theorem,(
    $true ),
    file(partit1,t2_partit1),
    [interesting(0.00)]).

fof(d3_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(A,B)
          <=> ! [C,D] : 
                ( r2_hidden(k4_tarski(C,D),A)
               => r2_hidden(k4_tarski(C,D),B) ) ) ) ) ),
    file(relat_1,d3_relat_1),
    [interesting(0.00)]).

fof(d6_partit1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => ( C = k4_partit1(A,B)
              <=> ! [D,E] : 
                    ( r2_hidden(k4_tarski(D,E),C)
                  <=> ? [F] : 
                        ( m1_subset_1(F,k1_zfmisc_1(A))
                        & r2_hidden(F,B)
                        & r2_hidden(D,F)
                        & r2_hidden(E,F) ) ) ) ) ) ) ),
    file(partit1,d6_partit1),
    [interesting(0.00)]).

fof(d8_finseq_1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( B = k9_finseq_1(A)
      <=> ( k1_relat_1(B) = k2_finseq_1(1)
          & k1_funct_1(B,1) = A ) ) ) ),
    file(finseq_1,d8_finseq_1),
    [interesting(0.00)]).

fof(t57_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( B = k9_finseq_1(A)
      <=> ( k3_finseq_1(B) = 1
          & k1_funct_1(B,1) = A ) ) ) ),
    file(finseq_1,t57_finseq_1),
    [interesting(0.00)]).

fof(t35_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ),
    file(finseq_1,t35_finseq_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(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(t59_finseq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k1_funct_1(k7_finseq_1(B,k9_finseq_1(A)),k1_nat_1(k3_finseq_1(B),1)) = A ) ),
    file(finseq_1,t59_finseq_1),
    [interesting(0.00)]).

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

fof(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(d7_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( C = k7_finseq_1(A,B)
              <=> ( k4_finseq_1(C) = k2_finseq_1(k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)))
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(A))
                       => k1_funct_1(C,D) = k1_funct_1(A,D) ) )
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(B))
                       => k1_funct_1(C,k1_nat_1(k3_finseq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ),
    file(finseq_1,d7_finseq_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(t1_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(A,B)
              & r1_xreal_0(B,A) )
           => A = B ) ) ) ),
    file(xreal_1,t1_xreal_1),
    [interesting(0.00)]).

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

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

fof(t25_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ! [D,E,F,G] : 
                  ( m2_finseq_1(G,A)
                 => ( ( r1_tarski(D,A)
                      & r2_hidden(E,D)
                      & k1_funct_1(G,1) = E
                      & k1_funct_1(G,k3_finseq_1(G)) = F
                      & r1_xreal_0(1,k3_finseq_1(G))
                      & ! [H] : 
                          ( m2_subset_1(H,k1_numbers,k5_numbers)
                         => ~ ( r1_xreal_0(1,H)
                              & ~ r1_xreal_0(k3_finseq_1(G),H)
                              & ! [I,J,K] : ~ ( r2_hidden(I,B)
                                  & r2_hidden(J,C)
                                  & r2_hidden(k1_funct_1(G,H),I)
                                  & r2_hidden(K,I)
                                  & r2_hidden(K,J)
                                  & r2_hidden(k1_funct_1(G,k1_nat_1(H,1)),J) ) ) )
                      & r1_partit1(A,B,D)
                      & r1_partit1(A,C,D) )
                   => r2_hidden(F,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_partit1,d1_partit1,d6_eqrel_1,d6_eqrel_1,t38_nat_1,t26_nat_1,t38_nat_1,d4_tarski,d6_eqrel_1,t38_nat_1,d4_tarski,t3_xboole_0,t92_zfmisc_1,d6_eqrel_1,d4_tarski,t3_xboole_0,t3_xboole_0,t92_zfmisc_1,t3_xboole_0,s1_nat_1]),
    [file(partit1,t25_partit1),interesting(0.00)]).

fof(d3_eqrel_1,definition,(
    ! [A,B] : 
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => ! [D] : 
              ( ( v3_relat_2(D)
                & v8_relat_2(D)
                & v1_partfun1(D,A,A)
                & m2_relset_1(D,A,A) )
             => ( D = k5_eqrel_1(A,B,C)
              <=> ( r1_tarski(k3_eqrel_1(A,B,C),D)
                  & ! [E] : 
                      ( ( v3_relat_2(E)
                        & v8_relat_2(E)
                        & v1_partfun1(E,A,A)
                        & m2_relset_1(E,A,A) )
                     => ( r1_tarski(k3_eqrel_1(A,B,C),E)
                       => r1_tarski(D,E) ) ) ) ) ) ) ) ),
    file(eqrel_1,d3_eqrel_1),
    [interesting(0.00)]).

fof(t13_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] : 
          ( m2_finseq_1(C,B)
         => ( r2_hidden(A,k4_finseq_1(C))
           => r2_hidden(k1_funct_1(C,A),B) ) ) ) ),
    file(finseq_2,t13_finseq_2),
    [interesting(0.00)]).

fof(t62_finseq_1,theorem,(
    ! [A,B,C,D] : 
      ( ( v1_relat_1(D)
        & v1_funct_1(D)
        & v1_finseq_1(D) )
     => ( D = k11_finseq_1(A,B,C)
      <=> ( k3_finseq_1(D) = 3
          & k1_funct_1(D,1) = A
          & k1_funct_1(D,2) = B
          & k1_funct_1(D,3) = C ) ) ) ),
    file(finseq_1,t62_finseq_1),
    [interesting(0.00)]).

fof(t3_cqc_the1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ~ ( r1_xreal_0(A,2)
          & A != 0
          & A != 1
          & A != 2 ) ) ),
    file(cqc_the1,t3_cqc_the1),
    [interesting(0.00)]).

fof(t36_eqrel_1,theorem,(
    ! [A,B,C,D] : 
      ( ( v3_relat_2(D)
        & v8_relat_2(D)
        & v1_partfun1(D,B,B)
        & m2_relset_1(D,B,B) )
     => ! [E] : 
          ( ( v3_relat_2(E)
            & v8_relat_2(E)
            & v1_partfun1(E,B,B)
            & m2_relset_1(E,B,B) )
         => ( r2_hidden(A,B)
           => ( r2_hidden(k4_tarski(A,C),k5_eqrel_1(B,D,E))
            <=> ? [F] : 
                  ( v1_relat_1(F)
                  & v1_funct_1(F)
                  & v1_finseq_1(F)
                  & r1_xreal_0(1,k3_finseq_1(F))
                  & A = k1_funct_1(F,1)
                  & C = k1_funct_1(F,k3_finseq_1(F))
                  & ! [G] : 
                      ( m2_subset_1(G,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(1,G)
                       => ( r1_xreal_0(k3_finseq_1(F),G)
                          | r2_hidden(k4_tarski(k1_funct_1(F,G),k1_funct_1(F,k1_nat_1(G,1))),k3_eqrel_1(B,D,E)) ) ) ) ) ) ) ) ) ),
    file(eqrel_1,t36_eqrel_1),
    [interesting(0.00)]).

fof(t13_eqrel_1,theorem,(
    ! [A,B,C,D,E] : 
      ( ( v8_relat_2(E)
        & v1_partfun1(E,A,A)
        & m2_relset_1(E,A,A) )
     => ( ( r2_hidden(k4_tarski(B,C),E)
          & r2_hidden(k4_tarski(C,D),E) )
       => r2_hidden(k4_tarski(B,D),E) ) ) ),
    file(eqrel_1,t13_eqrel_1),
    [interesting(0.00)]).

fof(t11_eqrel_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_2(C)
        & v1_partfun1(C,A,A)
        & m2_relset_1(C,A,A) )
     => ( r2_hidden(B,A)
       => r2_hidden(k4_tarski(B,B),C) ) ) ),
    file(eqrel_1,t11_eqrel_1),
    [interesting(0.00)]).

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

fof(t26_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] : 
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => ! [D] : 
                  ( m2_finseq_1(D,A)
                 => ! [E,F] : 
                      ( ( r2_hidden(E,A)
                        & r2_hidden(F,A)
                        & k1_funct_1(D,1) = E
                        & k1_funct_1(D,k3_finseq_1(D)) = F
                        & r1_xreal_0(1,k3_finseq_1(D))
                        & ! [G] : 
                            ( m2_subset_1(G,k1_numbers,k5_numbers)
                           => ~ ( r1_xreal_0(1,G)
                                & ~ r1_xreal_0(k3_finseq_1(D),G)
                                & ! [H] : ~ ( r2_hidden(H,A)
                                    & r2_hidden(k4_tarski(k1_funct_1(D,G),H),k3_eqrel_1(A,B,C))
                                    & r2_hidden(k4_tarski(H,k1_funct_1(D,k1_nat_1(G,1))),k3_eqrel_1(A,B,C)) ) ) ) )
                     => r2_hidden(k4_tarski(E,F),k5_eqrel_1(A,B,C)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t38_nat_1,t26_nat_1,d3_eqrel_1,d3_finseq_1,t3_finseq_1,t13_finseq_2,t3_finseq_1,t13_finseq_2,t62_finseq_1,t62_finseq_1,t38_nat_1,t3_cqc_the1,t36_eqrel_1,t13_eqrel_1,t38_nat_1,t11_eqrel_1,d2_xboole_0,s1_nat_1]),
    [file(partit1,t26_partit1),interesting(0.00)]).

fof(t1_msualg_5,theorem,(
    ! [A,B] : 
      ( ( v1_partfun1(B,A,A)
        & v3_relat_2(B)
        & v8_relat_2(B)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( ( v1_partfun1(C,A,A)
            & v3_relat_2(C)
            & v8_relat_2(C)
            & m2_relset_1(C,A,A) )
         => ! [D] : 
              ( ( v1_partfun1(D,A,A)
                & v3_relat_2(D)
                & v8_relat_2(D)
                & m2_relset_1(D,A,A) )
             => k5_eqrel_1(A,k5_eqrel_1(A,B,C),D) = k5_eqrel_1(A,B,k5_eqrel_1(A,C,D)) ) ) ) ),
    file(msualg_5,t1_msualg_5),
    [interesting(0.00)]).

fof(d2_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( A = B
          <=> ! [C,D] : 
                ( r2_hidden(k4_tarski(C,D),A)
              <=> r2_hidden(k4_tarski(C,D),B) ) ) ) ) ),
    file(relat_1,d2_relat_1),
    [interesting(0.00)]).

fof(t24_eqrel_1,theorem,(
    ! [A,B] : 
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => k4_eqrel_1(A,B,k5_eqrel_1(A,B,C)) = B ) ) ),
    file(eqrel_1,t24_eqrel_1),
    [interesting(0.00)]).

fof(t25_eqrel_1,theorem,(
    ! [A,B] : 
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => ! [C] : 
          ( ( v3_relat_2(C)
            & v8_relat_2(C)
            & v1_partfun1(C,A,A)
            & m2_relset_1(C,A,A) )
         => k5_eqrel_1(A,B,k4_eqrel_1(A,B,C)) = B ) ) ),
    file(eqrel_1,t25_eqrel_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(t13_pua2mss1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => k3_pua2mss1(A) = a_1_0_pua2mss1(A) ) ),
    file(pua2mss1,t13_pua2mss1),
    [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(d1_eqrel_1,definition,(
    ! [A] : k1_eqrel_1(A) = k2_zfmisc_1(A,A) ),
    file(eqrel_1,d1_eqrel_1),
    [interesting(0.00)]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.00)]).

fof(t4_msualg_5,theorem,(
    ! [A,B] : 
      ( m2_relset_1(B,A,A)
     => k3_eqrel_1(A,k1_eqrel_1(A),B) = k1_eqrel_1(A) ) ),
    file(msualg_5,t4_msualg_5),
    [interesting(0.00)]).

fof(t18_eqrel_1,theorem,(
    ! [A,B] : 
      ( m2_relset_1(B,A,A)
     => k2_eqrel_1(A,k1_eqrel_1(A),B) = B ) ),
    file(eqrel_1,t18_eqrel_1),
    [interesting(0.00)]).

fof(d10_relat_1,definition,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( B = k6_relat_1(A)
      <=> ! [C,D] : 
            ( r2_hidden(k4_tarski(C,D),B)
          <=> ( r2_hidden(C,A)
              & C = D ) ) ) ) ),
    file(relat_1,d10_relat_1),
    [interesting(0.00)]).

fof(t17_eqrel_1,theorem,(
    ! [A,B] : 
      ( ( v3_relat_2(B)
        & v8_relat_2(B)
        & v1_partfun1(B,A,A)
        & m2_relset_1(B,A,A) )
     => k4_eqrel_1(A,k6_partfun1(A),B) = k6_partfun1(A) ) ),
    file(eqrel_1,t17_eqrel_1),
    [interesting(0.00)]).

fof(t6_partit1,theorem,(
    $true ),
    file(partit1,t6_partit1),
    [interesting(0.00)]).