fof(t1_arytm_0,theorem,(
    r1_tarski(k2_arytm_2,k1_numbers) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1,t3_arytm_2,d1_numbers,t40_zfmisc_1]),
    [file(arytm_0,t1_arytm_0),interesting(1.00)]).

fof(t18_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => r2_hidden(k2_arytm_0(A,A),k2_arytm_2) ) ),
    inference(mizar_proof,[status(thm)],[d1_numbers,d4_xboole_0,d3_arytm_0,d3_arytm_0,d2_xboole_0]),
    [file(arytm_0,t18_arytm_0),interesting(0.95)]).

fof(l1_arytm_0,theorem,(
    r2_hidden(k12_arytm_3,k1_tarski(k12_arytm_3)) ),
    inference(mizar_proof,[status(thm)],[d1_tarski]),
    [file(arytm_0,l1_arytm_0),interesting(0.93)]).

fof(l16_arytm_0,theorem,(
    ! [A,B,C] : 
      ( k4_tarski(A,B) = k1_tarski(C)
     => ( C = k1_tarski(A)
        & A = B ) ) ),
    inference(mizar_proof,[status(thm)],[d5_tarski,d2_tarski,d1_tarski,d1_tarski,t9_zfmisc_1]),
    [file(arytm_0,l16_arytm_0),interesting(0.90)]).

fof(l23_arytm_0,theorem,(
    m1_subset_1(0,k1_numbers) ),
    inference(mizar_proof,[status(thm)],[t1_arytm_0,t21_arytm_2]),
    [file(arytm_0,l23_arytm_0),interesting(0.89)]).

fof(l33_arytm_0,theorem,(
    m1_subset_1(k13_arytm_3,k1_numbers) ),
    inference(mizar_proof,[status(thm)],[t1_arytm_0,t21_arytm_2]),
    [file(arytm_0,l33_arytm_0),interesting(0.87)]).

fof(t7_arytm_0,theorem,(
    ! [A,B] : k13_arytm_3 != k4_tarski(A,B) ),
    inference(mizar_proof,[status(thm)],[t18_ordinal3,d5_tarski,d2_tarski,d1_tarski]),
    [file(arytm_0,t7_arytm_0),interesting(0.87)]).

fof(t24_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => k4_arytm_0(k2_arytm_0(A,B)) = k2_arytm_0(k4_arytm_0(A),k4_arytm_0(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t14_arytm_0,d5_arytm_0,d5_arytm_0,t14_arytm_0,d5_arytm_0,d5_arytm_0,t23_arytm_0,t15_arytm_0,t15_arytm_0,t21_arytm_0,d5_arytm_0]),
    [file(arytm_0,t24_arytm_0),interesting(0.83)]).

fof(l2_arytm_0,theorem,(
    ! [A] : k13_arytm_3 != k4_tarski(0,A) ),
    inference(mizar_proof,[status(thm)],[d1_ordinal1,d4_ordinal2,d5_tarski,d2_tarski]),
    [file(arytm_0,l2_arytm_0),interesting(0.81)]).

fof(t21_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B = k13_arytm_3
           => k2_arytm_0(A,B) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[t14_arytm_0,d5_arytm_0,t14_arytm_0,d5_arytm_0,d3_arytm_0,t21_arytm_2,t15_arytm_0,d5_arytm_0,t16_arytm_2,t20_arytm_0]),
    [file(arytm_0,t21_arytm_0),interesting(0.81)]).

fof(t25_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => k1_arytm_0(A,k1_arytm_0(B,C)) = k1_arytm_0(k1_arytm_0(A,B),C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_numbers,d4_xboole_0,d2_arytm_0,d2_arytm_0,d2_arytm_0,d2_arytm_0,t7_arytm_2,d2_arytm_0,d2_arytm_0,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d2_arytm_0,t20_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,t33_zfmisc_1,t21_arytm_1,d2_arytm_0,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d2_arytm_0,d2_arytm_1,d2_arytm_0,t12_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,d2_arytm_1,d2_arytm_0,t33_zfmisc_1,t22_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,d2_arytm_1,d2_arytm_0,t22_arytm_1,t33_zfmisc_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,t33_zfmisc_1,t23_arytm_1,t33_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d2_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t24_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t20_arytm_2,t3_arytm_1,d2_arytm_1,t13_arytm_1,t33_zfmisc_1,d2_arytm_0,d2_arytm_0,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d2_arytm_0,t20_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,t33_zfmisc_1,t21_arytm_1,d2_arytm_0,d2_arytm_0,d2_arytm_1,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d2_arytm_0,t33_zfmisc_1,t25_arytm_1,d2_arytm_1,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t33_zfmisc_1,t11_arytm_1,t3_arytm_1,d2_arytm_1,t14_arytm_1,t33_zfmisc_1,d2_arytm_1,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t33_zfmisc_1,t11_arytm_1,t3_arytm_1,t33_zfmisc_1,t14_arytm_1,d2_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t15_arytm_1,t33_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d2_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t24_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t20_arytm_2,t3_arytm_1,d2_arytm_1,t13_arytm_1,t33_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t5_arytm_0,t3_xboole_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t7_arytm_2,t33_zfmisc_1,d2_xboole_0]),
    [file(arytm_0,t25_arytm_0),interesting(0.79)]).

fof(t15_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => k2_arytm_0(A,k2_arytm_0(B,C)) = k2_arytm_0(k2_arytm_0(A,B),C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_arytm_0,d3_arytm_0,d3_arytm_0,d3_arytm_0,t13_arytm_2,d4_xboole_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t13_arytm_2,t33_zfmisc_1,d4_xboole_0,d3_arytm_0,t2_arytm_1,d3_arytm_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t13_arytm_2,t33_zfmisc_1,d4_xboole_0,d4_xboole_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,d3_arytm_0,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t13_arytm_2,d4_xboole_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,d3_arytm_0,t2_arytm_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t13_arytm_2,d4_xboole_0,d4_xboole_0,d3_arytm_0,d3_arytm_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t13_arytm_2,t33_zfmisc_1,d4_xboole_0,d4_xboole_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t13_arytm_2,t33_zfmisc_1,d4_xboole_0,d4_xboole_0,d4_xboole_0,d3_arytm_0,d4_xboole_0,d1_tarski,t2_arytm_1,d3_arytm_0,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,d4_xboole_0,d1_tarski,t2_arytm_1,d3_arytm_0,t33_zfmisc_1,t13_arytm_2,t14_arytm_0,t14_arytm_0,t14_arytm_0,t14_arytm_0,t14_arytm_0,t14_arytm_0,t14_arytm_0,t14_arytm_0,t14_arytm_0,t14_arytm_0,t3_arytm_2,d1_numbers,t42_xboole_1,t65_zfmisc_1,d2_xboole_0]),
    [file(arytm_0,t15_arytm_0),interesting(0.74)]).

fof(t23_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ~ ( k2_arytm_0(A,B) = 0
              & A != 0
              & B != 0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_arytm_0,t21_arytm_0,t15_arytm_0,t14_arytm_0]),
    [file(arytm_0,t23_arytm_0),interesting(0.72)]).

fof(t16_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => k2_arytm_0(A,k1_arytm_0(B,C)) = k1_arytm_0(k2_arytm_0(A,B),k2_arytm_0(A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t14_arytm_0,t13_arytm_0,t14_arytm_0,t14_arytm_0,d3_arytm_0,d3_arytm_0,d2_arytm_0,d2_arytm_0,d3_arytm_0,t14_arytm_2,d4_xboole_0,d3_arytm_0,d3_arytm_0,d2_arytm_0,t33_zfmisc_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,d2_arytm_1,d3_arytm_0,t26_arytm_1,t33_zfmisc_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t27_arytm_1,t33_zfmisc_1,d4_xboole_0,d3_arytm_0,d3_arytm_0,d2_arytm_0,t33_zfmisc_1,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,d2_arytm_1,d3_arytm_0,t26_arytm_1,t33_zfmisc_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t27_arytm_1,t33_zfmisc_1,d4_xboole_0,d4_xboole_0,d3_arytm_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t5_arytm_0,t3_xboole_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t14_arytm_2,t33_zfmisc_1,d4_xboole_0,d2_arytm_0,d3_arytm_0,d3_arytm_0,l1_arytm_0,t106_zfmisc_1,t5_arytm_0,t3_xboole_0,d2_arytm_0,t6_arytm_2,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t14_arytm_2,t33_zfmisc_1,t14_arytm_0,t13_arytm_0,t14_arytm_0,t14_arytm_0,t13_arytm_0,t13_arytm_0,t14_arytm_0,t13_arytm_0,t13_arytm_0,t14_arytm_0,d4_xboole_0,d4_xboole_0,d3_arytm_0,d3_arytm_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,d1_numbers,d4_xboole_0,d1_tarski,t28_arytm_1,t33_zfmisc_1,t10_arytm_1,t33_zfmisc_1,t33_zfmisc_1,t14_arytm_0,t18_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t26_arytm_1,t33_zfmisc_1,t13_arytm_0,t13_arytm_0,t14_arytm_0,d4_xboole_0,d4_xboole_0,d3_arytm_0,d3_arytm_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,d2_arytm_0,t33_zfmisc_1,d2_arytm_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,d1_numbers,d4_xboole_0,d1_tarski,t28_arytm_1,t33_zfmisc_1,t10_arytm_1,t33_zfmisc_1,t33_zfmisc_1,t14_arytm_0,t18_arytm_1,d2_arytm_1,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t26_arytm_1,t33_zfmisc_1,t13_arytm_0,t13_arytm_0,t14_arytm_0,d4_xboole_0,d4_xboole_0,d4_xboole_0,d3_arytm_0,d3_arytm_0,d2_arytm_0,t5_arytm_0,t3_xboole_0,d2_arytm_0,l1_arytm_0,t106_zfmisc_1,d3_arytm_0,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t33_zfmisc_1,t14_arytm_2,t3_arytm_2,d1_numbers,t42_xboole_1,t65_zfmisc_1,d2_xboole_0]),
    [file(arytm_0,t16_arytm_0),interesting(0.67)]).

fof(t17_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => k2_arytm_0(k3_arytm_0(A),B) = k3_arytm_0(k2_arytm_0(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t16_arytm_0,d4_arytm_0,t14_arytm_0,d4_arytm_0]),
    [file(arytm_0,t17_arytm_0),interesting(0.66)]).

fof(t14_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B = 0
           => k2_arytm_0(A,B) = 0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t21_arytm_2,t4_arytm_2,d3_arytm_0,t5_arytm_0,t21_arytm_2,t3_xboole_0,d3_arytm_0]),
    [file(arytm_0,t14_arytm_0),interesting(0.64)]).

fof(t5_arytm_0,theorem,(
    r1_subset_1(k2_arytm_2,k2_zfmisc_1(k1_tarski(k12_arytm_3),k2_arytm_2)) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0,t103_zfmisc_1,d1_tarski,t3_arytm_2]),
    [file(arytm_0,t5_arytm_0),interesting(0.63)]).

fof(t13_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B = 0
           => k1_arytm_0(A,B) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[t21_arytm_2,d8_arytm_2,d2_arytm_0,d1_numbers,d4_xboole_0,d2_xboole_0,t103_zfmisc_1,d1_tarski,t3_arytm_0,t19_arytm_1,d2_arytm_0]),
    [file(arytm_0,t13_arytm_0),interesting(0.63)]).

fof(t27_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => k3_arytm_0(k1_arytm_0(A,B)) = k1_arytm_0(k3_arytm_0(A),k3_arytm_0(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_arytm_0,t25_arytm_0,d4_arytm_0,t13_arytm_0,d4_arytm_0,d4_arytm_0]),
    [file(arytm_0,t27_arytm_0),interesting(0.55)]).

fof(t3_arytm_0,theorem,(
    ! [A] : ~ ( r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers)
      & A = k12_arytm_3 ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,d1_numbers,d4_xboole_0]),
    [file(arytm_0,t3_arytm_0),interesting(0.54)]).

fof(t6_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( k2_arytm_1(A,B) = k12_arytm_3
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_arytm_1,d2_arytm_1,t10_arytm_1]),
    [file(arytm_0,t6_arytm_0),interesting(0.52)]).

fof(t8_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( k8_arytm_2(A,B) = k8_arytm_2(A,C)
               => ( A = k12_arytm_3
                  | B = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t26_arytm_1,t18_arytm_1,t2_arytm_1,d2_arytm_1,t6_arytm_0,t26_arytm_1,t18_arytm_1,t2_arytm_1,d2_arytm_1,t6_arytm_0]),
    [file(arytm_0,t8_arytm_0),interesting(0.41)]).

fof(t22_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B != 0
           => k2_arytm_0(k2_arytm_0(A,B),k4_arytm_0(B)) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[t15_arytm_0,d5_arytm_0,t21_arytm_0]),
    [file(arytm_0,t22_arytm_0),interesting(0.38)]).

fof(t4_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => r2_hidden(k2_arytm_1(A,B),k1_numbers) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_arytm_1,t1_arytm_0,d2_arytm_1,t9_arytm_1,t2_arytm_0]),
    [file(arytm_0,t4_arytm_0),interesting(0.33)]).

fof(t10_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ~ r2_hidden(k5_funct_4(k1_numbers,0,k13_arytm_3,A,B),k1_numbers) ) ) ),
    inference(mizar_proof,[status(thm)],[t71_funct_4,d5_tarski,d2_tarski,d2_tarski,d5_tarski,d5_tarski,t69_enumset1,t69_enumset1,d2_tarski,d1_tarski,t33_zfmisc_1,t9_zfmisc_1,l16_arytm_0,d2_tarski,d2_tarski,d1_tarski,d2_tarski,d1_tarski,l16_arytm_0,d2_tarski,d2_tarski,d1_tarski,d2_tarski,d1_tarski,t9_zfmisc_1,d2_tarski,d4_xboole_0,t23_ordinal1,t10_ordinal3,d2_tarski,d6_arytm_3,d2_xboole_0,d2_tarski,t82_arytm_3,d2_tarski,d1_arytm_2,d4_xboole_0,d2_xboole_0,d2_arytm_2,d4_xboole_0,t103_zfmisc_1,d1_tarski,d5_tarski,t71_funct_4,d2_tarski,d2_tarski,d1_tarski,d2_tarski,d2_tarski,d2_tarski,d5_tarski,d2_xboole_0,d1_numbers,d4_xboole_0]),
    [file(arytm_0,t10_arytm_0),interesting(0.25)]).

fof(t11_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_numbers)
     => ? [B] : 
          ( m1_subset_1(B,k1_numbers)
          & ? [C] : 
              ( m1_subset_1(C,k1_numbers)
              & A = k5_arytm_0(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t1_arytm_0,t21_arytm_2,d7_arytm_0,d2_numbers,d2_xboole_0,d4_xboole_0,d2_funct_2,d2_tarski,t12_funct_1,t69_funct_4,t66_funct_4,d4_xboole_0,d7_arytm_0]),
    [file(arytm_0,t11_arytm_0),interesting(0.24)]).

fof(t12_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ! [D] : 
                  ( m1_subset_1(D,k1_numbers)
                 => ( k5_arytm_0(A,B) = k5_arytm_0(C,D)
                   => ( A = C
                      & B = D ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_arytm_0,d7_arytm_0,t10_arytm_0,d7_arytm_0,d7_arytm_0,d7_arytm_0,t10_arytm_0,d7_arytm_0,t72_funct_4]),
    [file(arytm_0,t12_arytm_0),interesting(0.24)]).

fof(t20_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ( ( k2_arytm_0(A,B) = k13_arytm_3
                  & k2_arytm_0(A,C) = k13_arytm_3 )
               => ( A = 0
                  | B = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t5_arytm_0,t21_arytm_2,t3_xboole_0,d1_numbers,d4_xboole_0,d3_arytm_0,t21_arytm_2,t4_arytm_2,t5_arytm_0,t3_xboole_0,d3_arytm_0,d2_xboole_0,d3_arytm_0,t8_arytm_0,d3_arytm_0,d3_arytm_0,t33_zfmisc_1,t3_arytm_0,t8_arytm_0,d3_arytm_0,l2_arytm_0,d3_arytm_0,l2_arytm_0,d3_arytm_0,l2_arytm_0,d3_arytm_0,l2_arytm_0,d3_arytm_0,d3_arytm_0]),
    [file(arytm_0,t20_arytm_0),interesting(0.19)]).

fof(t19_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( k1_arytm_0(k2_arytm_0(A,A),k2_arytm_0(B,B)) = 0
           => A = 0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_arytm_0,d2_arytm_0,t6_arytm_2,d1_numbers,d4_xboole_0,d3_arytm_0,t2_arytm_1,d3_arytm_0,t33_zfmisc_1,t2_arytm_1,d1_tarski,d1_numbers,d4_xboole_0,d2_xboole_0]),
    [file(arytm_0,t19_arytm_0),interesting(0.17)]).

fof(t2_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ( A != k12_arytm_3
       => r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,t33_zfmisc_1,d1_tarski,t106_zfmisc_1,d2_xboole_0,d1_numbers,d4_xboole_0]),
    [file(arytm_0,t2_arytm_0),interesting(0.11)]).

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(t3_arytm_2,theorem,(
    ! [A] : ~ r2_hidden(k4_tarski(k12_arytm_3,A),k2_arytm_2) ),
    file(arytm_2,t3_arytm_2),
    [interesting(0.00)]).

fof(d1_numbers,definition,(
    k1_numbers = k4_xboole_0(k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)),k1_tarski(k4_tarski(0,0))) ),
    file(numbers,d1_numbers),
    [interesting(0.00)]).

fof(t40_zfmisc_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => ( r2_hidden(C,A)
        | r1_tarski(A,k4_xboole_0(B,k1_tarski(C))) ) ) ),
    file(zfmisc_1,t40_zfmisc_1),
    [interesting(0.00)]).

fof(t21_arytm_2,theorem,
    ( r2_hidden(k12_arytm_3,k2_arytm_2)
    & r2_hidden(k13_arytm_3,k2_arytm_2) ),
    file(arytm_2,t21_arytm_2),
    [interesting(0.00)]).

fof(d7_arytm_0,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( ( B = 0
             => k5_arytm_0(A,B) = A )
            & ( B != 0
             => k5_arytm_0(A,B) = k5_funct_4(k1_numbers,0,k13_arytm_3,A,B) ) ) ) ) ),
    file(arytm_0,d7_arytm_0),
    [interesting(0.00)]).

fof(d2_numbers,definition,(
    k2_numbers = k2_xboole_0(k4_xboole_0(k1_funct_2(k2_tarski(0,k13_arytm_3),k1_numbers),a_0_0_numbers),k1_numbers) ),
    file(numbers,d2_numbers),
    [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(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(d2_funct_2,definition,(
    ! [A,B,C] : 
      ( C = k1_funct_2(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ? [E] : 
              ( v1_relat_1(E)
              & v1_funct_1(E)
              & D = E
              & k1_relat_1(E) = A
              & r1_tarski(k2_relat_1(E),B) ) ) ) ),
    file(funct_2,d2_funct_2),
    [interesting(0.00)]).

fof(d2_tarski,definition,(
    ! [A,B,C] : 
      ( C = k2_tarski(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( D = A
            | D = B ) ) ) ),
    file(tarski,d2_tarski),
    [interesting(0.00)]).

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

fof(t69_funct_4,theorem,(
    ! [A,B,C,D,E] : 
      ( ( v1_relat_1(E)
        & v1_funct_1(E) )
     => ( ( k1_relat_1(E) = k2_tarski(A,B)
          & k1_funct_1(E,A) = C
          & k1_funct_1(E,B) = D )
       => E = k4_funct_4(A,B,C,D) ) ) ),
    file(funct_4,t69_funct_4),
    [interesting(0.00)]).

fof(t66_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( A != B
     => ( k1_funct_1(k4_funct_4(A,B,C,D),A) = C
        & k1_funct_1(k4_funct_4(A,B,C,D),B) = D ) ) ),
    file(funct_4,t66_funct_4),
    [interesting(0.00)]).

fof(t71_funct_4,theorem,(
    ! [A,B,C,D] : 
      ( A != C
     => k4_funct_4(A,C,B,D) = k2_tarski(k4_tarski(A,B),k4_tarski(C,D)) ) ),
    file(funct_4,t71_funct_4),
    [interesting(0.00)]).

fof(d5_tarski,definition,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    [interesting(0.00)]).

fof(t69_enumset1,theorem,(
    ! [A] : k2_tarski(A,A) = k1_tarski(A) ),
    file(enumset1,t69_enumset1),
    [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(t33_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( k4_tarski(A,B) = k4_tarski(C,D)
     => ( A = C
        & B = D ) ) ),
    file(zfmisc_1,t33_zfmisc_1),
    [interesting(0.00)]).

fof(t9_zfmisc_1,theorem,(
    ! [A,B,C] : 
      ( k1_tarski(A) = k2_tarski(B,C)
     => B = C ) ),
    file(zfmisc_1,t9_zfmisc_1),
    [interesting(0.00)]).

fof(t23_ordinal1,theorem,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r2_hidden(A,B)
       => v3_ordinal1(A) ) ) ),
    file(ordinal1,t23_ordinal1),
    [interesting(0.00)]).

fof(t10_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( A != k1_xboole_0
       => r2_hidden(k1_xboole_0,A) ) ) ),
    file(ordinal3,t10_ordinal3),
    [interesting(0.00)]).

fof(d6_arytm_3,definition,(
    k6_arytm_3 = k2_xboole_0(k4_xboole_0(a_0_0_arytm_3,a_0_1_arytm_3),k5_ordinal2) ),
    file(arytm_3,d6_arytm_3),
    [interesting(0.00)]).

fof(t82_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k6_arytm_3))
     => ~ ( ? [B] : 
              ( m1_subset_1(B,k6_arytm_3)
              & r2_hidden(B,A)
              & B != k12_arytm_3 )
          & ! [B] : 
              ( m1_subset_1(B,k6_arytm_3)
             => ! [C] : 
                  ( m1_subset_1(C,k6_arytm_3)
                 => ( ( r2_hidden(B,A)
                      & r3_arytm_3(C,B) )
                   => r2_hidden(C,A) ) ) )
          & ! [B] : 
              ( m1_subset_1(B,k6_arytm_3)
             => ! [C] : 
                  ( m1_subset_1(C,k6_arytm_3)
                 => ! [D] : 
                      ( m1_subset_1(D,k6_arytm_3)
                     => ~ ( r2_hidden(B,A)
                          & r2_hidden(C,A)
                          & r2_hidden(D,A)
                          & B != C
                          & C != D
                          & D != B ) ) ) ) ) ) ),
    file(arytm_3,t82_arytm_3),
    [interesting(0.00)]).

fof(d1_arytm_2,definition,(
    k1_arytm_2 = k4_xboole_0(a_0_0_arytm_2,k1_tarski(k6_arytm_3)) ),
    file(arytm_2,d1_arytm_2),
    [interesting(0.00)]).

fof(d2_arytm_2,definition,(
    k2_arytm_2 = k4_xboole_0(k2_xboole_0(k6_arytm_3,k1_arytm_2),a_0_1_arytm_2) ),
    file(arytm_2,d2_arytm_2),
    [interesting(0.00)]).

fof(t103_zfmisc_1,theorem,(
    ! [A,B,C,D] : ~ ( r1_tarski(A,k2_zfmisc_1(B,C))
      & r2_hidden(D,A)
      & ! [E,F] : ~ ( r2_hidden(E,B)
          & r2_hidden(F,C)
          & D = k4_tarski(E,F) ) ) ),
    file(zfmisc_1,t103_zfmisc_1),
    [interesting(0.00)]).

fof(t72_funct_4,theorem,(
    ! [A,B,C,D,E,F] : 
      ( k4_funct_4(A,B,C,D) = k4_funct_4(A,B,E,F)
     => ( A = B
        | ( C = E
          & D = F ) ) ) ),
    file(funct_4,t72_funct_4),
    [interesting(0.00)]).

fof(t4_arytm_2,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k12_arytm_3
           => k8_arytm_2(A,B) = k12_arytm_3 ) ) ) ),
    file(arytm_2,t4_arytm_2),
    [interesting(0.00)]).

fof(d3_arytm_0,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_arytm_2) )
                 => ( C = k2_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = E
                            & C = k8_arytm_2(D,E) ) ) ) )
                & ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                 => ( A = 0
                    | ( C = k2_arytm_0(A,B)
                    <=> ? [D] : 
                          ( m1_subset_1(D,k2_arytm_2)
                          & ? [E] : 
                              ( m1_subset_1(E,k2_arytm_2)
                              & A = D
                              & B = k4_tarski(0,E)
                              & C = k4_tarski(0,k8_arytm_2(D,E)) ) ) ) ) )
                & ( ( r2_hidden(B,k2_arytm_2)
                    & r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                 => ( B = 0
                    | ( C = k2_arytm_0(A,B)
                    <=> ? [D] : 
                          ( m1_subset_1(D,k2_arytm_2)
                          & ? [E] : 
                              ( m1_subset_1(E,k2_arytm_2)
                              & A = k4_tarski(0,D)
                              & B = E
                              & C = k4_tarski(0,k8_arytm_2(E,D)) ) ) ) ) )
                & ( ( r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                 => ( C = k2_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = k4_tarski(0,D)
                            & B = k4_tarski(0,E)
                            & C = k8_arytm_2(E,D) ) ) ) )
                & ~ ( ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_arytm_2) )
                    & ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
                        & A != 0 )
                    & ~ ( r2_hidden(B,k2_arytm_2)
                        & r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
                        & B != 0 )
                    & ~ ( r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
                        & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                    & ~ ( C = k2_arytm_0(A,B)
                      <=> C = 0 ) ) ) ) ) ) ),
    file(arytm_0,d3_arytm_0),
    [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(d8_arytm_2,definition,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( B = k12_arytm_3
             => k7_arytm_2(A,B) = A )
            & ( A = k12_arytm_3
             => k7_arytm_2(A,B) = B )
            & ~ ( B != k12_arytm_3
                & A != k12_arytm_3
                & k7_arytm_2(A,B) != k4_arytm_2(k5_arytm_2(k3_arytm_2(A),k3_arytm_2(B))) ) ) ) ) ),
    file(arytm_2,d8_arytm_2),
    [interesting(0.00)]).

fof(d2_arytm_0,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_arytm_2) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = E
                            & C = k7_arytm_2(D,E) ) ) ) )
                & ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = k4_tarski(0,E)
                            & C = k2_arytm_1(D,E) ) ) ) )
                & ( ( r2_hidden(B,k2_arytm_2)
                    & r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = k4_tarski(0,D)
                            & B = E
                            & C = k2_arytm_1(E,D) ) ) ) )
                & ~ ( ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_arytm_2) )
                    & ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                    & ~ ( r2_hidden(B,k2_arytm_2)
                        & r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                    & ~ ( C = k1_arytm_0(A,B)
                      <=> ? [D] : 
                            ( m1_subset_1(D,k2_arytm_2)
                            & ? [E] : 
                                ( m1_subset_1(E,k2_arytm_2)
                                & A = k4_tarski(0,D)
                                & B = k4_tarski(0,E)
                                & C = k4_tarski(0,k7_arytm_2(D,E)) ) ) ) ) ) ) ) ) ),
    file(arytm_0,d2_arytm_0),
    [interesting(0.00)]).

fof(t19_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k1_xboole_0
           => ( B = k1_xboole_0
              | k2_arytm_1(A,B) = k4_tarski(k1_xboole_0,B) ) ) ) ) ),
    file(arytm_1,t19_arytm_1),
    [interesting(0.00)]).

fof(t14_arytm_2,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => k8_arytm_2(A,k7_arytm_2(B,C)) = k7_arytm_2(k8_arytm_2(A,B),k8_arytm_2(A,C)) ) ) ) ),
    file(arytm_2,t14_arytm_2),
    [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(d2_arytm_1,definition,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( r1_arytm_2(B,A)
             => k2_arytm_1(A,B) = k1_arytm_1(A,B) )
            & ( ~ r1_arytm_2(B,A)
             => k2_arytm_1(A,B) = k4_tarski(k1_xboole_0,k1_arytm_1(B,A)) ) ) ) ) ),
    file(arytm_1,d2_arytm_1),
    [interesting(0.00)]).

fof(t26_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( r1_arytm_2(A,B)
               => k8_arytm_2(C,k1_arytm_1(B,A)) = k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) ) ) ) ) ),
    file(arytm_1,t26_arytm_1),
    [interesting(0.00)]).

fof(t27_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ~ ( ~ r1_arytm_2(A,B)
                  & C != k1_xboole_0
                  & k4_tarski(k1_xboole_0,k8_arytm_2(C,k1_arytm_1(A,B))) != k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) ) ) ) ) ),
    file(arytm_1,t27_arytm_1),
    [interesting(0.00)]).

fof(t6_arytm_2,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( k7_arytm_2(A,B) = k12_arytm_3
           => A = k12_arytm_3 ) ) ) ),
    file(arytm_2,t6_arytm_2),
    [interesting(0.00)]).

fof(t28_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( r1_arytm_2(B,A)
               => ( k1_arytm_1(A,B) = k1_xboole_0
                  | C = k1_xboole_0
                  | k2_arytm_1(k8_arytm_2(C,B),k8_arytm_2(C,A)) = k4_tarski(k1_xboole_0,k8_arytm_2(C,k1_arytm_1(A,B))) ) ) ) ) ) ),
    file(arytm_1,t28_arytm_1),
    [interesting(0.00)]).

fof(t10_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( r1_arytm_2(A,B)
              & k1_arytm_1(B,A) = k1_xboole_0 )
           => A = B ) ) ) ),
    file(arytm_1,t10_arytm_1),
    [interesting(0.00)]).

fof(t18_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => k2_arytm_1(A,A) = k1_xboole_0 ) ),
    file(arytm_1,t18_arytm_1),
    [interesting(0.00)]).

fof(t42_xboole_1,theorem,(
    ! [A,B,C] : k4_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,C),k4_xboole_0(B,C)) ),
    file(xboole_1,t42_xboole_1),
    [interesting(0.00)]).

fof(t65_zfmisc_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,k1_tarski(B)) = A
    <=> ~ r2_hidden(B,A) ) ),
    file(zfmisc_1,t65_zfmisc_1),
    [interesting(0.00)]).

fof(d4_arytm_0,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B = k3_arytm_0(A)
          <=> k1_arytm_0(A,B) = 0 ) ) ) ),
    file(arytm_0,d4_arytm_0),
    [interesting(0.00)]).

fof(t2_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ~ ( k8_arytm_2(A,B) = k1_xboole_0
              & A != k1_xboole_0
              & B != k1_xboole_0 ) ) ) ),
    file(arytm_1,t2_arytm_1),
    [interesting(0.00)]).

fof(t13_arytm_2,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => k8_arytm_2(A,k8_arytm_2(B,C)) = k8_arytm_2(k8_arytm_2(A,B),C) ) ) ) ),
    file(arytm_2,t13_arytm_2),
    [interesting(0.00)]).

fof(d5_arytm_0,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( ( A != 0
             => ( B = k4_arytm_0(A)
              <=> k2_arytm_0(A,B) = k13_arytm_3 ) )
            & ( A = 0
             => ( B = k4_arytm_0(A)
              <=> B = 0 ) ) ) ) ) ),
    file(arytm_0,d5_arytm_0),
    [interesting(0.00)]).

fof(t16_arytm_2,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k13_arytm_3
           => k8_arytm_2(A,B) = B ) ) ) ),
    file(arytm_2,t16_arytm_2),
    [interesting(0.00)]).

fof(d1_ordinal1,definition,(
    ! [A] : k1_ordinal1(A) = k2_xboole_0(A,k1_tarski(A)) ),
    file(ordinal1,d1_ordinal1),
    [interesting(0.00)]).

fof(d4_ordinal2,definition,(
    k4_ordinal2 = k1_ordinal1(k1_xboole_0) ),
    file(ordinal2,d4_ordinal2),
    [interesting(0.00)]).

fof(t26_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( r2_hidden(k5_arytm_0(A,B),k1_numbers)
           => B = 0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_arytm_0,t10_arytm_0]),
    [file(arytm_0,t26_arytm_0),interesting(0.00)]).

fof(t7_arytm_2,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => k7_arytm_2(A,k7_arytm_2(B,C)) = k7_arytm_2(k7_arytm_2(A,B),C) ) ) ) ),
    file(arytm_2,t7_arytm_2),
    [interesting(0.00)]).

fof(t20_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( r1_arytm_2(A,B)
               => k7_arytm_2(C,k1_arytm_1(B,A)) = k2_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) ),
    file(arytm_1,t20_arytm_1),
    [interesting(0.00)]).

fof(t21_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( ~ r1_arytm_2(A,B)
               => k2_arytm_1(C,k1_arytm_1(A,B)) = k2_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) ),
    file(arytm_1,t21_arytm_1),
    [interesting(0.00)]).

fof(t12_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( ( r1_arytm_2(A,B)
                  & r1_arytm_2(A,C) )
               => k7_arytm_2(B,k1_arytm_1(C,A)) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) ),
    file(arytm_1,t12_arytm_1),
    [interesting(0.00)]).

fof(t22_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( r1_arytm_2(A,B)
               => ( r1_arytm_2(A,C)
                  | k2_arytm_1(B,k1_arytm_1(A,C)) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) ) ),
    file(arytm_1,t22_arytm_1),
    [interesting(0.00)]).

fof(t23_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ~ ( ~ r1_arytm_2(A,B)
                  & ~ r1_arytm_2(A,C)
                  & k2_arytm_1(B,k1_arytm_1(A,C)) != k2_arytm_1(C,k1_arytm_1(A,B)) ) ) ) ) ),
    file(arytm_1,t23_arytm_1),
    [interesting(0.00)]).

fof(t24_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( r1_arytm_2(A,B)
               => k2_arytm_1(B,k7_arytm_2(A,C)) = k2_arytm_1(k1_arytm_1(B,A),C) ) ) ) ) ),
    file(arytm_1,t24_arytm_1),
    [interesting(0.00)]).

fof(t20_arytm_2,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( A = k7_arytm_2(B,C)
               => r1_arytm_2(C,A) ) ) ) ) ),
    file(arytm_2,t20_arytm_2),
    [interesting(0.00)]).

fof(t3_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( ( r1_arytm_2(A,B)
                  & r1_arytm_2(B,C) )
               => r1_arytm_2(A,C) ) ) ) ) ),
    file(arytm_1,t3_arytm_1),
    [interesting(0.00)]).

fof(t13_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( r1_arytm_2(A,B)
               => k7_arytm_2(C,k1_arytm_1(B,A)) = k1_arytm_1(k7_arytm_2(C,B),A) ) ) ) ) ),
    file(arytm_1,t13_arytm_1),
    [interesting(0.00)]).

fof(t25_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( ( r1_arytm_2(A,B)
                  & r1_arytm_2(C,B) )
               => k2_arytm_1(k1_arytm_1(B,C),A) = k2_arytm_1(k1_arytm_1(B,A),C) ) ) ) ) ),
    file(arytm_1,t25_arytm_1),
    [interesting(0.00)]).

fof(t11_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => r1_arytm_2(k1_arytm_1(A,B),A) ) ) ),
    file(arytm_1,t11_arytm_1),
    [interesting(0.00)]).

fof(t14_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( ( r1_arytm_2(A,B)
                  & r1_arytm_2(C,A) )
               => k7_arytm_2(k1_arytm_1(B,A),C) = k1_arytm_1(B,k1_arytm_1(A,C)) ) ) ) ) ),
    file(arytm_1,t14_arytm_1),
    [interesting(0.00)]).

fof(t15_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( ( r1_arytm_2(A,B)
                  & r1_arytm_2(A,C) )
               => k7_arytm_2(k1_arytm_1(C,A),B) = k7_arytm_2(k1_arytm_1(B,A),C) ) ) ) ) ),
    file(arytm_1,t15_arytm_1),
    [interesting(0.00)]).

fof(t9_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ~ ( ~ r1_arytm_2(A,B)
              & k1_arytm_1(A,B) = k1_xboole_0 ) ) ) ),
    file(arytm_1,t9_arytm_1),
    [interesting(0.00)]).

fof(t18_ordinal3,theorem,(
    k4_ordinal2 = k1_tarski(k1_xboole_0) ),
    file(ordinal3,t18_ordinal3),
    [interesting(0.00)]).

fof(t9_arytm_0,theorem,(
    $true ),
    file(arytm_0,t9_arytm_0),
    [interesting(0.00)]).