fof(t81_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ( r2_hidden(a_1_0_arytm_3(A),k6_arytm_3)
      <=> A = k12_arytm_3 ) ) ),
    inference(mizar_proof,[status(thm)],[t80_arytm_3,t23_ordinal1,t23_ordinal1,t35_arytm_3,t37_arytm_3,t71_arytm_3,t73_arytm_3,d2_tarski,d3_tarski,t71_arytm_3,t3_xboole_1]),
    [file(arytm_3,t81_arytm_3),interesting(1.00)]).

fof(t80_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ~ ( A != k12_arytm_3
          & ! [B] : 
              ( m1_subset_1(B,k6_arytm_3)
             => ~ ( ~ r3_arytm_3(A,B)
                  & ~ r2_hidden(B,k5_ordinal2) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t48_ordinal2,t14_ordinal1,t66_arytm_3,d12_arytm_3,t64_arytm_3,t73_arytm_3,t74_arytm_3,d8_arytm_3,l1_arytm_3,t56_ordinal2,t46_arytm_3,t27_ordinal3,t19_ordinal3,t14_ordinal1,t44_ordinal2,t66_arytm_3,d12_arytm_3,t56_arytm_3,t69_arytm_3,t73_arytm_3,t74_arytm_3,t56_arytm_3,t71_arytm_3,t56_arytm_3,t73_arytm_3,t79_arytm_3]),
    [file(arytm_3,t80_arytm_3),interesting(0.89)]).

fof(t66_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ? [B] : 
          ( m1_subset_1(B,k6_arytm_3)
          & A = k10_arytm_3(B,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t64_arytm_3,t29_ordinal3,t60_arytm_3,t59_arytm_3,t58_arytm_3,t63_arytm_3,t59_arytm_3,t59_arytm_3]),
    [file(arytm_3,t66_arytm_3),interesting(0.85)]).

fof(t102_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ~ ! [B] : 
            ( m1_subset_1(B,k6_arytm_3)
           => r3_arytm_3(B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t56_arytm_3,d12_arytm_3,t69_arytm_3,t73_arytm_3]),
    [file(arytm_3,t102_arytm_3),interesting(0.83)]).

fof(t71_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => r3_arytm_3(k12_arytm_3,A) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,t56_arytm_3]),
    [file(arytm_3,t71_arytm_3),interesting(0.82)]).

fof(t59_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => k11_arytm_3(A,k13_arytm_3) = A ) ),
    inference(mizar_proof,[status(thm)],[d7_arytm_3,d8_arytm_3,l1_arytm_3,t56_ordinal2,t45_arytm_3]),
    [file(arytm_3,t59_arytm_3),interesting(0.81)]).

fof(t73_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ( ( r3_arytm_3(A,B)
              & r3_arytm_3(B,A) )
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,d12_arytm_3,t56_arytm_3,t57_arytm_3,t69_arytm_3,t70_arytm_3,t56_arytm_3]),
    [file(arytm_3,t73_arytm_3),interesting(0.81)]).

fof(t56_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => k10_arytm_3(A,k12_arytm_3) = A ) ),
    inference(mizar_proof,[status(thm)],[d7_arytm_3,d8_arytm_3,t19_ordinal2,t44_ordinal2,t52_ordinal2,t56_ordinal2,t45_arytm_3]),
    [file(arytm_3,t56_arytm_3),interesting(0.79)]).

fof(t101_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ~ ( ~ r3_arytm_3(B,A)
              & ! [C] : 
                  ( m1_subset_1(C,k6_arytm_3)
                 => ~ ( ~ r3_arytm_3(C,A)
                      & ~ r3_arytm_3(B,C) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,t66_arytm_3,t57_arytm_3,t56_arytm_3,d12_arytm_3,t73_arytm_3,d12_arytm_3,t69_arytm_3,t73_arytm_3]),
    [file(arytm_3,t101_arytm_3),interesting(0.77)]).

fof(t54_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => k11_arytm_3(A,k12_arytm_3) = k12_arytm_3 ) ),
    inference(mizar_proof,[status(thm)],[d7_arytm_3,d8_arytm_3,t19_ordinal2,t52_ordinal2,t56_ordinal2,t46_arytm_3]),
    [file(arytm_3,t54_arytm_3),interesting(0.75)]).

fof(t68_arytm_3,theorem,(
    ! [A] : ~ r2_hidden(k4_tarski(k12_arytm_3,A),k6_arytm_3) ),
    inference(mizar_proof,[status(thm)],[t38_arytm_3,t35_arytm_3,t33_zfmisc_1,t7_arytm_3]),
    [file(arytm_3,t68_arytm_3),interesting(0.74)]).

fof(t72_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ( r3_arytm_3(A,k12_arytm_3)
       => A = k12_arytm_3 ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,t70_arytm_3]),
    [file(arytm_3,t72_arytm_3),interesting(0.74)]).

fof(t74_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ( ( r3_arytm_3(A,B)
                  & r3_arytm_3(B,C) )
               => r3_arytm_3(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,d12_arytm_3,d12_arytm_3,t57_arytm_3]),
    [file(arytm_3,t74_arytm_3),interesting(0.74)]).

fof(t75_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ( ~ r3_arytm_3(B,A)
          <=> ( r3_arytm_3(A,B)
              & A != B ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t73_arytm_3]),
    [file(arytm_3,t75_arytm_3),interesting(0.73)]).

fof(t83_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ( r3_arytm_3(k10_arytm_3(A,B),k10_arytm_3(C,B))
              <=> r3_arytm_3(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,d12_arytm_3,t57_arytm_3,t69_arytm_3,d12_arytm_3,d12_arytm_3,t57_arytm_3]),
    [file(arytm_3,t83_arytm_3),interesting(0.73)]).

fof(t45_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => k9_arytm_3(k7_arytm_3(A),k8_arytm_3(A)) = A ) ),
    inference(mizar_proof,[status(thm)],[t41_arytm_3,t40_arytm_3,t28_arytm_3,t42_arytm_3,d7_arytm_3,d9_arytm_3,d8_arytm_3,t42_arytm_3,d9_arytm_3]),
    [file(arytm_3,t45_arytm_3),interesting(0.72)]).

fof(t70_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ( k10_arytm_3(A,B) = k12_arytm_3
           => A = k12_arytm_3 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_arytm_3,t34_ordinal3,d7_arytm_3,d8_arytm_3,t19_ordinal2,t45_arytm_3,t51_arytm_3,t52_ordinal2,t34_ordinal3,t29_ordinal3,t34_ordinal3,t43_arytm_3]),
    [file(arytm_3,t70_arytm_3),interesting(0.72)]).

fof(t90_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ( r3_arytm_3(A,B)
               => r3_arytm_3(k11_arytm_3(A,C),k11_arytm_3(B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,d12_arytm_3,t63_arytm_3]),
    [file(arytm_3,t90_arytm_3),interesting(0.71)]).

fof(t61_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ~ ( A != k12_arytm_3
              & ! [C] : 
                  ( m1_subset_1(C,k6_arytm_3)
                 => B != k11_arytm_3(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_arytm_3,t59_arytm_3,t58_arytm_3]),
    [file(arytm_3,t61_arytm_3),interesting(0.70)]).

fof(t28_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r1_arytm_3(A,B)
           => k5_arytm_3(A,B) = A ) ) ) ),
    inference(mizar_proof,[status(thm)],[t25_arytm_3,t84_ordinal3]),
    [file(arytm_3,t28_arytm_3),interesting(0.69)]).

fof(t69_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ( k10_arytm_3(A,B) = k10_arytm_3(C,B)
               => A = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_arytm_3,t34_ordinal3,t51_arytm_3,t58_ordinal3,t58_ordinal3,t36_ordinal3,t54_ordinal3,t54_ordinal3,t58_ordinal3,t24_ordinal3,t58_ordinal3,t58_ordinal3,t36_ordinal3,t51_arytm_3,t45_arytm_3,t45_arytm_3]),
    [file(arytm_3,t69_arytm_3),interesting(0.69)]).

fof(t60_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ~ ( A != k12_arytm_3
          & ! [B] : 
              ( m1_subset_1(B,k6_arytm_3)
             => k11_arytm_3(A,B) != k13_arytm_3 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_arytm_3,t43_arytm_3,t40_arytm_3,t49_arytm_3,t34_ordinal3,t47_arytm_3]),
    [file(arytm_3,t60_arytm_3),interesting(0.68)]).

fof(t58_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => k11_arytm_3(k11_arytm_3(A,B),C) = k11_arytm_3(A,k11_arytm_3(B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t45_arytm_3,t55_arytm_3,t58_ordinal3,t58_ordinal3,t55_arytm_3]),
    [file(arytm_3,t58_arytm_3),interesting(0.68)]).

fof(t57_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => k10_arytm_3(k10_arytm_3(A,B),C) = k10_arytm_3(A,k10_arytm_3(B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_arytm_3,t34_ordinal3,t45_arytm_3,t52_arytm_3,t54_ordinal3,t58_ordinal3,t58_ordinal3,t58_ordinal3,t33_ordinal3,t54_ordinal3,t58_ordinal3,t52_arytm_3,t45_arytm_3]),
    [file(arytm_3,t57_arytm_3),interesting(0.68)]).

fof(t87_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( r3_arytm_3(A,k11_arytm_3(B,C))
                  & ! [D] : 
                      ( m1_subset_1(D,k6_arytm_3)
                     => ~ ( A = k11_arytm_3(B,D)
                          & r3_arytm_3(D,C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,t54_arytm_3,t70_arytm_3,t61_arytm_3,t61_arytm_3,d12_arytm_3,t63_arytm_3,t62_arytm_3]),
    [file(arytm_3,t87_arytm_3),interesting(0.68)]).

fof(t64_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & m1_subset_1(A,k6_arytm_3) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & m1_subset_1(B,k6_arytm_3) )
         => k10_arytm_3(A,B) = k1_arytm_3(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2,d8_arytm_3,t56_ordinal2,t56_ordinal2,t56_ordinal2,t46_arytm_3,d7_arytm_3,d7_arytm_3]),
    [file(arytm_3,t64_arytm_3),interesting(0.66)]).

fof(t86_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ~ ( k11_arytm_3(A,B) = k12_arytm_3
              & A != k12_arytm_3
              & B != k12_arytm_3 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_arytm_3,t34_ordinal3,d7_arytm_3,d8_arytm_3,t19_ordinal2,t45_arytm_3,t51_arytm_3,t52_ordinal2,t34_ordinal3,t34_ordinal3,t43_arytm_3]),
    [file(arytm_3,t86_arytm_3),interesting(0.66)]).

fof(t29_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ( k5_arytm_3(A,k4_ordinal2) = A
        & k5_arytm_3(k4_ordinal2,A) = k4_ordinal2 ) ) ),
    inference(mizar_proof,[status(thm)],[t6_arytm_3,t25_arytm_3,t84_ordinal3]),
    [file(arytm_3,t29_arytm_3),interesting(0.65)]).

fof(t63_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => k11_arytm_3(A,k10_arytm_3(B,C)) = k10_arytm_3(k11_arytm_3(A,B),k11_arytm_3(A,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t45_arytm_3,t41_arytm_3,t34_ordinal3,t55_arytm_3,t54_ordinal3,t58_ordinal3,t58_ordinal3,t58_ordinal3,t50_arytm_3,t54_ordinal3,t58_ordinal3,t58_ordinal3,t58_ordinal3,t52_arytm_3]),
    [file(arytm_3,t63_arytm_3),interesting(0.65)]).

fof(t62_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ( k11_arytm_3(A,B) = k11_arytm_3(A,C)
               => ( A = k12_arytm_3
                  | B = C ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_arytm_3,t58_arytm_3,t59_arytm_3]),
    [file(arytm_3,t62_arytm_3),interesting(0.65)]).

fof(t92_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ( A = k12_arytm_3
          <=> k10_arytm_3(A,B) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t56_arytm_3,t69_arytm_3]),
    [file(arytm_3,t92_arytm_3),interesting(0.65)]).

fof(t77_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( ~ r3_arytm_3(B,A)
                  & ~ r3_arytm_3(C,B)
                  & r3_arytm_3(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t74_arytm_3]),
    [file(arytm_3,t77_arytm_3),interesting(0.64)]).

fof(t78_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ( ( r2_hidden(A,k5_ordinal2)
              & r2_hidden(k10_arytm_3(A,B),k5_ordinal2) )
           => r2_hidden(B,k5_ordinal2) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t45_arytm_3,d8_arytm_3,t41_arytm_3,t34_ordinal3,t56_ordinal2,t51_arytm_3,t56_ordinal2,t56_ordinal2,t65_ordinal3,t76_ordinal3,d2_arytm_3,d4_arytm_3,t40_arytm_3,t25_arytm_3,t46_arytm_3]),
    [file(arytm_3,t78_arytm_3),interesting(0.64)]).

fof(t97_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( r3_arytm_3(A,B)
                  & r3_arytm_3(B,C)
                  & B != C
                  & A = C ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t73_arytm_3]),
    [file(arytm_3,t97_arytm_3),interesting(0.63)]).

fof(t22_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => k4_arytm_3(k2_arytm_3(A,B),k2_arytm_3(C,B)) = k2_arytm_3(k4_arytm_3(A,C),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t52_ordinal2,t21_arytm_3,d21_ordinal2,d4_arytm_3,t11_arytm_3,t58_ordinal3,d2_arytm_3,d4_arytm_3,t11_arytm_3,t58_ordinal3,d4_arytm_3,t11_arytm_3,t58_ordinal3,t36_ordinal3,d2_arytm_3,d4_arytm_3,t11_arytm_3,t58_ordinal3,t56_ordinal2,t36_ordinal3,t20_arytm_3,t41_ordinal3,t56_ordinal2,t19_arytm_3,t52_ordinal2,t19_arytm_3]),
    [file(arytm_3,t22_arytm_3),interesting(0.60)]).

fof(t76_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( ( ( ~ r3_arytm_3(B,A)
                      & r3_arytm_3(B,C) )
                    | ( r3_arytm_3(A,B)
                      & ~ r3_arytm_3(C,B) ) )
                  & r3_arytm_3(C,A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t74_arytm_3]),
    [file(arytm_3,t76_arytm_3),interesting(0.60)]).

fof(t46_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( k9_arytm_3(k1_xboole_0,A) = k1_xboole_0
            & k9_arytm_3(B,k4_ordinal2) = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t31_arytm_3,d9_arytm_3,t29_arytm_3,d9_arytm_3]),
    [file(arytm_3,t46_arytm_3),interesting(0.59)]).

fof(t79_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & m1_subset_1(B,k6_arytm_3) )
         => ~ ( ~ r3_arytm_3(A,B)
              & ~ r3_arytm_3(k10_arytm_3(B,k13_arytm_3),A)
              & r2_hidden(A,k5_ordinal2) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t37_arytm_3,d12_arytm_3,d12_arytm_3,t64_arytm_3,t37_arytm_3,t78_arytm_3,t57_arytm_3,t69_arytm_3,t64_arytm_3,t27_ordinal3,t19_ordinal3,t56_arytm_3]),
    [file(arytm_3,t79_arytm_3),interesting(0.59)]).

fof(t88_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ( r3_arytm_3(k11_arytm_3(B,A),k11_arytm_3(C,A))
               => ( A = k12_arytm_3
                  | r3_arytm_3(B,C) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t87_arytm_3,t62_arytm_3]),
    [file(arytm_3,t88_arytm_3),interesting(0.59)]).

fof(t91_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ! [D] : 
                  ( m1_subset_1(D,k6_arytm_3)
                 => ~ ( k11_arytm_3(A,B) = k11_arytm_3(C,D)
                      & ~ r3_arytm_3(A,C)
                      & ~ r3_arytm_3(B,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t71_arytm_3,t86_arytm_3,t54_arytm_3,t62_arytm_3,t90_arytm_3,t73_arytm_3,t90_arytm_3]),
    [file(arytm_3,t91_arytm_3),interesting(0.59)]).

fof(t65_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & m1_subset_1(A,k6_arytm_3) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & m1_subset_1(B,k6_arytm_3) )
         => k11_arytm_3(A,B) = k2_arytm_3(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2,d8_arytm_3,t56_ordinal2,t46_arytm_3,d7_arytm_3,d7_arytm_3]),
    [file(arytm_3,t65_arytm_3),interesting(0.55)]).

fof(t50_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ( A != k1_xboole_0
               => k9_arytm_3(k2_arytm_3(B,A),k2_arytm_3(C,A)) = k9_arytm_3(B,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_arytm_3,t52_ordinal2,d9_arytm_3,t34_ordinal3,t48_arytm_3,t33_arytm_3,t48_arytm_3,t48_arytm_3,t33_arytm_3,t48_arytm_3,t45_arytm_3,t45_arytm_3]),
    [file(arytm_3,t50_arytm_3),interesting(0.54)]).

fof(t89_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ! [D] : 
                  ( m1_subset_1(D,k6_arytm_3)
                 => ~ ( k10_arytm_3(A,B) = k10_arytm_3(C,D)
                      & ~ r3_arytm_3(A,C)
                      & ~ r3_arytm_3(B,D) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t83_arytm_3]),
    [file(arytm_3,t89_arytm_3),interesting(0.53)]).

fof(t53_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ( A != k1_xboole_0
               => k10_arytm_3(k9_arytm_3(B,A),k9_arytm_3(C,A)) = k9_arytm_3(k1_arytm_3(B,C),A) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t52_arytm_3,t54_ordinal3,t50_arytm_3]),
    [file(arytm_3,t53_arytm_3),interesting(0.52)]).

fof(t93_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ! [D] : 
                  ( m1_subset_1(D,k6_arytm_3)
                 => ( ( k10_arytm_3(A,B) = k10_arytm_3(C,D)
                      & r3_arytm_3(A,C) )
                   => r3_arytm_3(D,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,d12_arytm_3,t57_arytm_3,t69_arytm_3]),
    [file(arytm_3,t93_arytm_3),interesting(0.51)]).

fof(t94_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( r3_arytm_3(A,B)
                  & r3_arytm_3(B,k10_arytm_3(A,C))
                  & ! [D] : 
                      ( m1_subset_1(D,k6_arytm_3)
                     => ~ ( B = k10_arytm_3(A,D)
                          & r3_arytm_3(D,C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3,t83_arytm_3]),
    [file(arytm_3,t94_arytm_3),interesting(0.51)]).

fof(t33_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ( A != k1_xboole_0
               => k5_arytm_3(k2_arytm_3(B,A),k2_arytm_3(C,A)) = k5_arytm_3(B,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_arytm_3,t23_arytm_3,t34_ordinal3,t52_ordinal2,t83_ordinal3,t22_arytm_3,t11_arytm_3,t58_ordinal3,t81_ordinal3]),
    [file(arytm_3,t33_arytm_3),interesting(0.50)]).

fof(t24_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( ~ ( A = k1_xboole_0
                & B = k1_xboole_0 )
           => r1_arytm_3(k6_ordinal3(A,k4_arytm_3(A,B)),k6_ordinal3(B,k4_arytm_3(A,B))) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t56_ordinal2,t19_arytm_3,t81_ordinal3,t6_arytm_3,d4_arytm_3,t11_arytm_3,d1_arytm_3,t52_ordinal2,t2_arytm_3,t58_ordinal3,d2_arytm_3,d4_arytm_3,t11_arytm_3,t58_ordinal3,t52_ordinal2,t56_ordinal2,t36_ordinal3,t41_ordinal3]),
    [file(arytm_3,t24_arytm_3),interesting(0.50)]).

fof(t27_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( ~ ( A = k1_xboole_0
                & B = k1_xboole_0 )
           => r1_arytm_3(k5_arytm_3(A,B),k5_arytm_3(B,A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t24_arytm_3]),
    [file(arytm_3,t27_arytm_3),interesting(0.46)]).

fof(t99_arytm_3,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_arytm_3)) )
     => ~ ( r2_hidden(A,k6_arytm_3)
          & ! [B] : 
              ( m1_subset_1(B,k6_arytm_3)
             => ~ ( r2_hidden(B,A)
                  & ! [C] : 
                      ( m1_subset_1(C,k6_arytm_3)
                     => ( r2_hidden(C,A)
                       => r3_arytm_3(C,B) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_tarski,t23_ordinal1,t10_ordinal3,d1_tarski,t7_arytm_3,t35_arytm_3,d2_tarski,d1_tarski,d2_tarski,t56_ordinal2,d1_tarski,d1_arytm_3,t35_arytm_3,t10_ordinal3,d5_ordinal2,t7_ordinal1,t42_ordinal1,t10_ordinal1,t10_ordinal1,t23_ordinal1,d21_ordinal2,t34_arytm_3,t34_ordinal1,d6_ordinal3,t64_arytm_3,d12_arytm_3]),
    [file(arytm_3,t99_arytm_3),interesting(0.46)]).

fof(t49_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_ordinal2)
     => ! [B] : 
          ( m1_subset_1(B,k5_ordinal2)
         => ( r1_arytm_3(A,B)
           => ( B = k1_xboole_0
              | ( k7_arytm_3(k9_arytm_3(A,B)) = A
                & k8_arytm_3(k9_arytm_3(A,B)) = B ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t28_arytm_3,t48_arytm_3]),
    [file(arytm_3,t49_arytm_3),interesting(0.43)]).

fof(t103_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ~ ( A != k12_arytm_3
              & ! [C] : 
                  ( m1_subset_1(C,k6_arytm_3)
                 => ~ ( r2_hidden(C,k5_ordinal2)
                      & r3_arytm_3(B,k11_arytm_3(C,A)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t43_arytm_3,d21_ordinal2,t34_arytm_3,d21_ordinal2,d12_arytm_3,t41_arytm_3,t34_ordinal3,t10_ordinal3,t33_ordinal1,d6_ordinal3,t58_ordinal3,t54_ordinal3,t58_ordinal3,t51_arytm_3,t53_arytm_3,t56_ordinal2,t45_arytm_3,t56_ordinal2,t55_arytm_3,t46_arytm_3,t45_arytm_3]),
    [file(arytm_3,t103_arytm_3),interesting(0.42)]).

fof(l44_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r2_hidden(k4_tarski(A,B),c3)
           => ( r1_arytm_3(A,B)
              & B != k1_xboole_0 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t33_zfmisc_1]),
    [file(arytm_3,l44_arytm_3),interesting(0.41)]).

fof(t44_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ( r2_hidden(A,k5_ordinal2)
      <=> k8_arytm_3(A) = k4_ordinal2 ) ) ),
    inference(mizar_proof,[status(thm)],[d8_arytm_3,t42_arytm_3]),
    [file(arytm_3,t44_arytm_3),interesting(0.41)]).

fof(t52_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ! [D] : 
                  ( ( v3_ordinal1(D)
                    & v4_ordinal2(D) )
                 => ~ ( B != k1_xboole_0
                      & A != k1_xboole_0
                      & k10_arytm_3(k9_arytm_3(C,B),k9_arytm_3(D,A)) != k9_arytm_3(k1_arytm_3(k2_arytm_3(C,A),k2_arytm_3(B,D)),k2_arytm_3(B,A)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_arytm_3,t48_arytm_3,t50_arytm_3,t58_ordinal3,t26_arytm_3,t54_ordinal3,t58_ordinal3,t26_arytm_3,t58_ordinal3,t26_arytm_3,t50_arytm_3,t58_ordinal3,t26_arytm_3,t54_ordinal3,t58_ordinal3,t26_arytm_3,t58_ordinal3,t26_arytm_3]),
    [file(arytm_3,t52_arytm_3),interesting(0.36)]).

fof(t42_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ( ~ r2_hidden(A,k5_ordinal2)
       => ( A = k4_tarski(k7_arytm_3(A),k8_arytm_3(A))
          & k8_arytm_3(A) != k4_ordinal2 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_arytm_3,d7_arytm_3,d8_arytm_3]),
    [file(arytm_3,t42_arytm_3),interesting(0.35)]).

fof(t18_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( A != k1_xboole_0
           => r2_arytm_3(k6_ordinal3(k2_arytm_3(B,A),k3_arytm_3(B,A)),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_arytm_3,d3_arytm_3,t17_arytm_3,t11_arytm_3,t58_ordinal3,t36_ordinal3]),
    [file(arytm_3,t18_arytm_3),interesting(0.35)]).

fof(t48_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( A != k1_xboole_0
           => ( k7_arytm_3(k9_arytm_3(B,A)) = k5_arytm_3(B,A)
              & k8_arytm_3(k9_arytm_3(B,A)) = k5_arytm_3(A,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_arytm_3,d7_arytm_3,d8_arytm_3,d9_arytm_3,t38_arytm_3,d7_arytm_3,d8_arytm_3]),
    [file(arytm_3,t48_arytm_3),interesting(0.33)]).

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 ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t66_arytm_3,d12_arytm_3,t71_arytm_3,t56_arytm_3,t56_arytm_3,t69_arytm_3]),
    [file(arytm_3,t82_arytm_3),interesting(0.20)]).

fof(t39_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_ordinal2)
     => ! [B] : 
          ( m1_subset_1(B,k5_ordinal2)
         => ( r2_hidden(k4_tarski(A,B),k6_arytm_3)
          <=> ( r1_arytm_3(A,B)
              & B != k1_xboole_0
              & B != k4_ordinal2 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t38_arytm_3,d2_xboole_0,d4_xboole_0,l44_arytm_3,t33_zfmisc_1,d4_xboole_0,d2_xboole_0]),
    [file(arytm_3,t39_arytm_3),interesting(0.14)]).

fof(d21_ordinal2,definition,(
    ! [A] : 
      ( v4_ordinal2(A)
    <=> r2_hidden(A,k5_ordinal2) ) ),
    file(ordinal2,d21_ordinal2),
    [interesting(0.00)]).

fof(l40_arytm_3,theorem,(
    m1_subset_1(k4_ordinal2,k5_ordinal2) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2]),
    [file(arytm_3,l40_arytm_3),interesting(0.00)]).

fof(d1_arytm_3,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_arytm_3(A,B)
          <=> ! [C] : 
                ( v3_ordinal1(C)
               => ! [D] : 
                    ( v3_ordinal1(D)
                   => ! [E] : 
                        ( v3_ordinal1(E)
                       => ( ( A = k15_ordinal2(C,D)
                            & B = k15_ordinal2(C,E) )
                         => C = k4_ordinal2 ) ) ) ) ) ) ) ),
    file(arytm_3,d1_arytm_3),
    [interesting(0.00)]).

fof(t41_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( k15_ordinal2(A,B) = k4_ordinal2
           => ( A = k4_ordinal2
              & B = k4_ordinal2 ) ) ) ) ),
    file(ordinal3,t41_ordinal3),
    [interesting(0.00)]).

fof(t6_arytm_3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => r1_arytm_3(k4_ordinal2,A) ) ),
    inference(mizar_proof,[status(thm)],[d1_arytm_3,t41_ordinal3]),
    [file(arytm_3,t6_arytm_3),interesting(0.00)]).

fof(l41_arytm_3,theorem,
    ( c2 != k1_xboole_0
    & r1_arytm_3(c2,c2) ),
    inference(mizar_proof,[status(thm)],[t6_arytm_3]),
    [file(arytm_3,l41_arytm_3),interesting(0.00)]).

fof(l42_arytm_3,theorem,(
    r2_hidden(k4_tarski(c2,c2),a_0_0_arytm_3) ),
    inference(mizar_proof,[status(thm)],[l41_arytm_3]),
    [file(arytm_3,l42_arytm_3),interesting(0.00)]).

fof(l43_arytm_3,theorem,(
    ~ v1_xboole_0(a_0_0_arytm_3) ),
    inference(mizar_proof,[status(thm)],[l42_arytm_3]),
    [file(arytm_3,l43_arytm_3),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(s1_ordinal1,theorem,
    ( ? [A] : 
        ( v3_ordinal1(A)
        & p1_s1_ordinal1(A) )
   => ? [A] : 
        ( v3_ordinal1(A)
        & p1_s1_ordinal1(A)
        & ! [B] : 
            ( v3_ordinal1(B)
           => ( p1_s1_ordinal1(B)
             => r1_ordinal1(A,B) ) ) ) ),
    file(ordinal1,s1_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(d5_ordinal2,definition,(
    ! [A] : 
      ( A = k5_ordinal2
    <=> ( r2_hidden(k1_xboole_0,A)
        & v4_ordinal1(A)
        & v3_ordinal1(A)
        & ! [B] : 
            ( v3_ordinal1(B)
           => ( ( r2_hidden(k1_xboole_0,B)
                & v4_ordinal1(B) )
             => r1_tarski(A,B) ) ) ) ) ),
    file(ordinal2,d5_ordinal2),
    [interesting(0.00)]).

fof(t42_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( ~ ( ~ v4_ordinal1(A)
            & ! [B] : 
                ( v3_ordinal1(B)
               => A != k1_ordinal1(B) ) )
        & ~ ( ? [B] : 
                ( v3_ordinal1(B)
                & A = k1_ordinal1(B) )
            & v4_ordinal1(A) ) ) ) ),
    file(ordinal1,t42_ordinal1),
    [interesting(0.00)]).

fof(t13_ordinal1,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,k1_ordinal1(B))
    <=> ( r2_hidden(A,B)
        | A = B ) ) ),
    file(ordinal1,t13_ordinal1),
    [interesting(0.00)]).

fof(t7_ordinal1,theorem,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & r1_tarski(B,A) ) ),
    file(ordinal1,t7_ordinal1),
    [interesting(0.00)]).

fof(t19_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v1_ordinal1(C)
             => ( ( r2_hidden(C,A)
                  & r2_hidden(A,B) )
               => r2_hidden(C,B) ) ) ) ) ),
    file(ordinal1,t19_ordinal1),
    [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(t34_arytm_3,theorem,(
    r1_tarski(k5_ordinal2,k6_arytm_3) ),
    inference(mizar_proof,[status(thm)],[t7_xboole_1]),
    [file(arytm_3,t34_arytm_3),interesting(0.00)]).

fof(t48_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k14_ordinal2(A,k4_ordinal2) = k1_ordinal1(A) ) ),
    file(ordinal2,t48_ordinal2),
    [interesting(0.00)]).

fof(d8_arytm_3,definition,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k5_ordinal2)
         => ( ( r2_hidden(A,k5_ordinal2)
             => ( B = k8_arytm_3(A)
              <=> B = k4_ordinal2 ) )
            & ( ~ r2_hidden(A,k5_ordinal2)
             => ( B = k8_arytm_3(A)
              <=> ? [C] : 
                    ( v3_ordinal1(C)
                    & v4_ordinal2(C)
                    & A = k4_tarski(C,B) ) ) ) ) ) ) ),
    file(arytm_3,d8_arytm_3),
    [interesting(0.00)]).

fof(t56_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( k15_ordinal2(k4_ordinal2,A) = A
        & k15_ordinal2(A,k4_ordinal2) = A ) ) ),
    file(ordinal2,t56_ordinal2),
    [interesting(0.00)]).

fof(t83_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( k6_ordinal3(k1_xboole_0,A) = k1_xboole_0
        & k7_ordinal3(k1_xboole_0,A) = k1_xboole_0
        & k7_ordinal3(A,k1_xboole_0) = A ) ) ),
    file(ordinal3,t83_ordinal3),
    [interesting(0.00)]).

fof(t52_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k15_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) ),
    file(ordinal2,t52_ordinal2),
    [interesting(0.00)]).

fof(d2_arytm_3,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_arytm_3(A,B)
          <=> ? [C] : 
                ( v3_ordinal1(C)
                & B = k15_ordinal2(A,C) ) ) ) ) ),
    file(arytm_3,d2_arytm_3),
    [interesting(0.00)]).

fof(t14_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ( r2_arytm_3(A,k1_xboole_0)
        & r2_arytm_3(k4_ordinal2,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t52_ordinal2,d2_arytm_3,t56_ordinal2,d2_arytm_3]),
    [file(arytm_3,t14_arytm_3),interesting(0.00)]).

fof(d3_arytm_3,definition,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( m1_subset_1(C,k5_ordinal2)
             => ( C = k3_arytm_3(A,B)
              <=> ( r2_arytm_3(A,C)
                  & r2_arytm_3(B,C)
                  & ! [D] : 
                      ( ( v3_ordinal1(D)
                        & v4_ordinal2(D) )
                     => ( ( r2_arytm_3(A,D)
                          & r2_arytm_3(B,D) )
                       => r2_arytm_3(C,D) ) ) ) ) ) ) ) ),
    file(arytm_3,d3_arytm_3),
    [interesting(0.00)]).

fof(d4_arytm_3,definition,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( m1_subset_1(C,k5_ordinal2)
             => ( C = k4_arytm_3(A,B)
              <=> ( r2_arytm_3(C,A)
                  & r2_arytm_3(C,B)
                  & ! [D] : 
                      ( ( v3_ordinal1(D)
                        & v4_ordinal2(D) )
                     => ( ( r2_arytm_3(D,A)
                          & r2_arytm_3(D,B) )
                       => r2_arytm_3(D,C) ) ) ) ) ) ) ) ),
    file(arytm_3,d4_arytm_3),
    [interesting(0.00)]).

fof(t19_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ( k4_arytm_3(A,k1_xboole_0) = A
        & k3_arytm_3(A,k1_xboole_0) = k1_xboole_0 ) ) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2,t14_arytm_3,t14_arytm_3,d3_arytm_3,d4_arytm_3]),
    [file(arytm_3,t19_arytm_3),interesting(0.00)]).

fof(t81_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( A != k1_xboole_0
           => k6_ordinal3(k15_ordinal2(B,A),A) = B ) ) ) ),
    file(ordinal3,t81_ordinal3),
    [interesting(0.00)]).

fof(t31_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ( k5_arytm_3(k1_xboole_0,A) = k1_xboole_0
        & ( A != k1_xboole_0
         => k5_arytm_3(A,k1_xboole_0) = k4_ordinal2 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t83_ordinal3,t19_arytm_3,t56_ordinal2,t81_ordinal3]),
    [file(arytm_3,t31_arytm_3),interesting(0.00)]).

fof(d9_arytm_3,definition,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( ( B = k1_xboole_0
             => k9_arytm_3(A,B) = k1_xboole_0 )
            & ( k5_arytm_3(B,A) = k4_ordinal2
             => k9_arytm_3(A,B) = k5_arytm_3(A,B) )
            & ~ ( B != k1_xboole_0
                & k5_arytm_3(B,A) != k4_ordinal2
                & k9_arytm_3(A,B) != k4_tarski(k5_arytm_3(A,B),k5_arytm_3(B,A)) ) ) ) ) ),
    file(arytm_3,d9_arytm_3),
    [interesting(0.00)]).

fof(t55_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k15_ordinal2(A,k1_xboole_0) = k1_xboole_0 ) ),
    file(ordinal2,t55_ordinal2),
    [interesting(0.00)]).

fof(t39_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( A != k1_xboole_0
           => ( r1_ordinal1(B,k15_ordinal2(B,A))
              & r1_ordinal1(B,k15_ordinal2(A,B)) ) ) ) ) ),
    file(ordinal3,t39_ordinal3),
    [interesting(0.00)]).

fof(t22_ordinal1,theorem,(
    ! [A] : 
      ( v1_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( ( r1_tarski(A,B)
                  & r2_hidden(B,C) )
               => r2_hidden(A,C) ) ) ) ) ),
    file(ordinal1,t22_ordinal1),
    [interesting(0.00)]).

fof(t2_arytm_3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( v4_ordinal2(k15_ordinal2(A,B))
           => ( v1_xboole_0(k15_ordinal2(A,B))
              | ( r2_hidden(A,k5_ordinal2)
                & r2_hidden(B,k5_ordinal2) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2,t52_ordinal2,t55_ordinal2,t39_ordinal3,t22_ordinal1]),
    [file(arytm_3,t2_arytm_3),interesting(0.00)]).

fof(t9_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r2_arytm_3(A,B)
          <=> ? [C] : 
                ( v3_ordinal1(C)
                & v4_ordinal2(C)
                & B = k2_arytm_3(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_arytm_3,t55_ordinal2,t2_arytm_3,d2_arytm_3]),
    [file(arytm_3,t9_arytm_3),interesting(0.00)]).

fof(t44_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k14_ordinal2(A,k1_xboole_0) = A ) ),
    file(ordinal2,t44_ordinal2),
    [interesting(0.00)]).

fof(d7_ordinal3,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( ( B != k1_xboole_0
                 => ( C = k6_ordinal3(A,B)
                  <=> ? [D] : 
                        ( v3_ordinal1(D)
                        & A = k14_ordinal2(k15_ordinal2(C,B),D)
                        & r2_hidden(D,B) ) ) )
                & ( B = k1_xboole_0
                 => ( C = k6_ordinal3(A,B)
                  <=> C = k1_xboole_0 ) ) ) ) ) ) ),
    file(ordinal3,d7_ordinal3),
    [interesting(0.00)]).

fof(t11_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r2_arytm_3(B,A)
          <=> A = k2_arytm_3(B,k6_ordinal3(A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_arytm_3,t52_ordinal2,t44_ordinal2,t10_ordinal3,d2_arytm_3,d7_ordinal3]),
    [file(arytm_3,t11_arytm_3),interesting(0.00)]).

fof(t25_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r1_arytm_3(A,B)
          <=> k4_arytm_3(A,B) = k4_ordinal2 ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_arytm_3,t11_arytm_3,d1_arytm_3,d1_arytm_3,t19_arytm_3,t2_arytm_3,d2_arytm_3,d4_arytm_3,t9_arytm_3,t41_ordinal3]),
    [file(arytm_3,t25_arytm_3),interesting(0.00)]).

fof(t84_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( k6_ordinal3(A,k4_ordinal2) = A
        & k7_ordinal3(A,k4_ordinal2) = k1_xboole_0 ) ) ),
    file(ordinal3,t84_ordinal3),
    [interesting(0.00)]).

fof(d7_arytm_3,definition,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k5_ordinal2)
         => ( ( r2_hidden(A,k5_ordinal2)
             => ( B = k7_arytm_3(A)
              <=> B = A ) )
            & ( ~ r2_hidden(A,k5_ordinal2)
             => ( B = k7_arytm_3(A)
              <=> ? [C] : 
                    ( v3_ordinal1(C)
                    & v4_ordinal2(C)
                    & A = k4_tarski(B,C) ) ) ) ) ) ) ),
    file(arytm_3,d7_arytm_3),
    [interesting(0.00)]).

fof(s2_arytm_3,theorem,
    ( ( f2_s2_arytm_3 = k13_arytm_3
      & f1_s2_arytm_3 = k12_arytm_3
      & r2_hidden(f3_s2_arytm_3,k5_ordinal2)
      & p1_s2_arytm_3(f1_s2_arytm_3)
      & ~ p1_s2_arytm_3(f3_s2_arytm_3) )
   => ? [A] : 
        ( m1_subset_1(A,k6_arytm_3)
        & r2_hidden(A,k5_ordinal2)
        & p1_s2_arytm_3(A)
        & ~ p1_s2_arytm_3(k10_arytm_3(A,f2_s2_arytm_3)) ) ),
    inference(mizar_proof,[status(thm)],[t23_ordinal1,s1_ordinal1,t10_ordinal3,d5_ordinal2,t42_ordinal1,t13_ordinal1,t7_ordinal1,t19_ordinal1,t34_arytm_3,t48_ordinal2,t64_arytm_3]),
    [file(arytm_3,s2_arytm_3),interesting(0.00)]).

fof(d12_arytm_3,definition,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ( r3_arytm_3(A,B)
          <=> ? [C] : 
                ( m1_subset_1(C,k6_arytm_3)
                & B = k10_arytm_3(A,C) ) ) ) ) ),
    file(arytm_3,d12_arytm_3),
    [interesting(0.00)]).

fof(t100_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ~ ! [C] : 
                ( m1_subset_1(C,k6_arytm_3)
               => ( k10_arytm_3(A,C) != B
                  & k10_arytm_3(B,C) != A ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3]),
    [file(arytm_3,t100_arytm_3),interesting(0.00)]).

fof(t29_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( k14_ordinal2(A,B) = k1_xboole_0
           => ( A = k1_xboole_0
              & B = k1_xboole_0 ) ) ) ) ),
    file(ordinal3,t29_ordinal3),
    [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(t35_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ~ ( ~ r2_hidden(A,k5_ordinal2)
          & ! [B] : 
              ( m1_subset_1(B,k5_ordinal2)
             => ! [C] : 
                  ( m1_subset_1(C,k5_ordinal2)
                 => ~ ( A = k4_tarski(B,C)
                      & r1_arytm_3(B,C)
                      & C != k1_xboole_0
                      & C != k4_ordinal2 ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_xboole_0,d4_xboole_0]),
    [file(arytm_3,t35_arytm_3),interesting(0.00)]).

fof(t41_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => k8_arytm_3(A) != k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d8_arytm_3,t35_arytm_3,d8_arytm_3]),
    [file(arytm_3,t41_arytm_3),interesting(0.00)]).

fof(t24_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ~ ( ~ r2_hidden(A,B)
              & A != B
              & ~ r2_hidden(B,A) ) ) ) ),
    file(ordinal1,t24_ordinal1),
    [interesting(0.00)]).

fof(t5_arytm_3,theorem,(
    ~ r1_arytm_3(k1_xboole_0,k1_xboole_0) ),
    inference(mizar_proof,[status(thm)],[d1_arytm_3,t52_ordinal2]),
    [file(arytm_3,t5_arytm_3),interesting(0.00)]).

fof(t17_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( r2_hidden(A,k4_ordinal2)
       => A = k1_xboole_0 ) ) ),
    file(ordinal3,t17_ordinal3),
    [interesting(0.00)]).

fof(t7_arytm_3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( r1_arytm_3(k1_xboole_0,A)
       => A = k4_ordinal2 ) ) ),
    inference(mizar_proof,[status(thm)],[t24_ordinal1,t5_arytm_3,t55_ordinal2,t56_ordinal2,t17_ordinal3,d1_arytm_3]),
    [file(arytm_3,t7_arytm_3),interesting(0.00)]).

fof(t43_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ( ~ ( A != k1_xboole_0
            & k7_arytm_3(A) = k1_xboole_0 )
        & ~ ( k7_arytm_3(A) != k1_xboole_0
            & A = k1_xboole_0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_arytm_3,t35_arytm_3,d7_arytm_3,t7_arytm_3,d5_ordinal2,d7_arytm_3]),
    [file(arytm_3,t43_arytm_3),interesting(0.00)]).

fof(t40_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => r1_arytm_3(k7_arytm_3(A),k8_arytm_3(A)) ) ),
    inference(mizar_proof,[status(thm)],[d8_arytm_3,t6_arytm_3,t35_arytm_3,d7_arytm_3,d8_arytm_3]),
    [file(arytm_3,t40_arytm_3),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(t38_arytm_3,theorem,(
    ! [A,B] : ~ r2_hidden(k4_tarski(A,B),k5_ordinal2) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal3,d2_tarski]),
    [file(arytm_3,t38_arytm_3),interesting(0.00)]).

fof(t34_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ~ ( k15_ordinal2(A,B) = k1_xboole_0
              & A != k1_xboole_0
              & B != k1_xboole_0 ) ) ) ),
    file(ordinal3,t34_ordinal3),
    [interesting(0.00)]).

fof(t21_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ( k4_arytm_3(A,A) = A
        & k3_arytm_3(A,A) = A ) ) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2,d4_arytm_3,d3_arytm_3]),
    [file(arytm_3,t21_arytm_3),interesting(0.00)]).

fof(t32_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ( A != k1_xboole_0
       => k5_arytm_3(A,A) = k4_ordinal2 ) ) ),
    inference(mizar_proof,[status(thm)],[t21_arytm_3,t56_ordinal2,t81_ordinal3]),
    [file(arytm_3,t32_arytm_3),interesting(0.00)]).

fof(t47_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ( A != k1_xboole_0
       => k9_arytm_3(A,A) = k4_ordinal2 ) ) ),
    inference(mizar_proof,[status(thm)],[t32_arytm_3,d9_arytm_3]),
    [file(arytm_3,t47_arytm_3),interesting(0.00)]).

fof(t41_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v4_ordinal1(A)
      <=> ! [B] : 
            ( v3_ordinal1(B)
           => ( r2_hidden(B,A)
             => r2_hidden(k1_ordinal1(B),A) ) ) ) ) ),
    file(ordinal1,t41_ordinal1),
    [interesting(0.00)]).

fof(t19_ordinal2,theorem,
    ( r2_hidden(k1_xboole_0,k5_ordinal2)
    & v4_ordinal1(k5_ordinal2)
    & ! [A] : 
        ( v3_ordinal1(A)
       => ( ( r2_hidden(k1_xboole_0,A)
            & v4_ordinal1(A) )
         => r1_ordinal1(k5_ordinal2,A) ) ) ),
    file(ordinal2,t19_ordinal2),
    [interesting(0.00)]).

fof(l1_arytm_3,theorem,(
    r2_hidden(k4_ordinal2,k5_ordinal2) ),
    inference(mizar_proof,[status(thm)],[t41_ordinal1,t19_ordinal2]),
    [file(arytm_3,l1_arytm_3),interesting(0.00)]).

fof(t20_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( k4_arytm_3(A,B) = k1_xboole_0
           => A = k1_xboole_0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_arytm_3,t11_arytm_3,t52_ordinal2]),
    [file(arytm_3,t20_arytm_3),interesting(0.00)]).

fof(t23_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( A != k1_xboole_0
           => ( k4_arytm_3(B,A) != k1_xboole_0
              & k6_ordinal3(A,k4_arytm_3(B,A)) != k1_xboole_0 ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_arytm_3,t11_arytm_3,t52_ordinal2]),
    [file(arytm_3,t23_arytm_3),interesting(0.00)]).

fof(t58_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => k15_ordinal2(k15_ordinal2(A,B),C) = k15_ordinal2(A,k15_ordinal2(B,C)) ) ) ) ),
    file(ordinal3,t58_ordinal3),
    [interesting(0.00)]).

fof(t36_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( k15_ordinal2(A,B) = k15_ordinal2(C,B)
               => ( B = k1_xboole_0
                  | A = C ) ) ) ) ) ),
    file(ordinal3,t36_ordinal3),
    [interesting(0.00)]).

fof(t26_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => k2_arytm_3(k5_arytm_3(A,B),k4_arytm_3(A,B)) = A ) ) ),
    inference(mizar_proof,[status(thm)],[d4_arytm_3,t11_arytm_3]),
    [file(arytm_3,t26_arytm_3),interesting(0.00)]).

fof(t55_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ! [D] : 
                  ( ( v3_ordinal1(D)
                    & v4_ordinal2(D) )
                 => k11_arytm_3(k9_arytm_3(B,C),k9_arytm_3(D,A)) = k9_arytm_3(k2_arytm_3(B,D),k2_arytm_3(C,A)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t20_arytm_3,t48_arytm_3,t50_arytm_3,t58_ordinal3,t26_arytm_3,t58_ordinal3,t26_arytm_3,t50_arytm_3,t58_ordinal3,t26_arytm_3,t58_ordinal3,t26_arytm_3,d9_arytm_3,t52_ordinal2,d9_arytm_3,t54_arytm_3]),
    [file(arytm_3,t55_arytm_3),interesting(0.00)]).

fof(t54_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => k15_ordinal2(k14_ordinal2(A,B),C) = k14_ordinal2(k15_ordinal2(A,C),k15_ordinal2(B,C)) ) ) ) ),
    file(ordinal3,t54_ordinal3),
    [interesting(0.00)]).

fof(t33_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => k14_ordinal2(k14_ordinal2(A,B),C) = k14_ordinal2(A,k14_ordinal2(B,C)) ) ) ) ),
    file(ordinal3,t33_ordinal3),
    [interesting(0.00)]).

fof(t51_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ! [D] : 
                  ( ( v3_ordinal1(D)
                    & v4_ordinal2(D) )
                 => ~ ( B != k1_xboole_0
                      & A != k1_xboole_0
                      & ~ ( k9_arytm_3(C,B) = k9_arytm_3(D,A)
                        <=> k2_arytm_3(C,A) = k2_arytm_3(B,D) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t48_arytm_3,t48_arytm_3,t26_arytm_3,t58_ordinal3,t58_ordinal3,t26_arytm_3,t58_ordinal3,t26_arytm_3,t33_arytm_3,t33_arytm_3,t45_arytm_3,t45_arytm_3]),
    [file(arytm_3,t51_arytm_3),interesting(0.00)]).

fof(t24_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( k14_ordinal2(A,B) = k14_ordinal2(A,C)
               => B = C ) ) ) ) ),
    file(ordinal3,t24_ordinal3),
    [interesting(0.00)]).

fof(t33_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r1_ordinal1(k1_ordinal1(A),B) ) ) ) ),
    file(ordinal1,t33_ordinal1),
    [interesting(0.00)]).

fof(d6_ordinal3,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( ( r1_ordinal1(B,A)
                 => ( C = k5_ordinal3(A,B)
                  <=> A = k14_ordinal2(B,C) ) )
                & ( ~ r1_ordinal1(B,A)
                 => ( C = k5_ordinal3(A,B)
                  <=> C = k1_xboole_0 ) ) ) ) ) ) ),
    file(ordinal3,d6_ordinal3),
    [interesting(0.00)]).

fof(d8_ordinal3,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k7_ordinal3(A,B) = k5_ordinal3(A,k15_ordinal2(k6_ordinal3(A,B),B)) ) ) ),
    file(ordinal3,d8_ordinal3),
    [interesting(0.00)]).

fof(t65_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k5_ordinal3(k14_ordinal2(A,B),A) = B ) ) ),
    file(ordinal3,t65_ordinal3),
    [interesting(0.00)]).

fof(t10_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( r2_hidden(k1_xboole_0,A)
           => r2_hidden(k7_ordinal3(B,A),A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_ordinal3,d8_ordinal3,t65_ordinal3]),
    [file(arytm_3,t10_arytm_3),interesting(0.00)]).

fof(t12_arytm_3,theorem,(
    $true ),
    file(arytm_3,t12_arytm_3),
    [interesting(0.00)]).

fof(t13_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( ( r2_arytm_3(A,B)
              & r2_arytm_3(B,A) )
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t11_arytm_3,t56_ordinal2,t58_ordinal3,t52_ordinal2,t41_ordinal3,t56_ordinal2,t36_ordinal3]),
    [file(arytm_3,t13_arytm_3),interesting(0.00)]).

fof(t15_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ( ( r2_hidden(k1_xboole_0,B)
              & r2_arytm_3(A,B) )
           => r1_ordinal1(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_arytm_3,t55_ordinal2,t39_ordinal3]),
    [file(arytm_3,t15_arytm_3),interesting(0.00)]).

fof(t76_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => k5_ordinal3(k15_ordinal2(A,B),k15_ordinal2(C,B)) = k15_ordinal2(k5_ordinal3(A,C),B) ) ) ) ),
    file(ordinal3,t76_ordinal3),
    [interesting(0.00)]).

fof(t16_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ! [C] : 
              ( ( v3_ordinal1(C)
                & v4_ordinal2(C) )
             => ( ( r2_arytm_3(A,B)
                  & r2_arytm_3(A,k1_arytm_3(B,C)) )
               => r2_arytm_3(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_arytm_3,t9_arytm_3,t65_ordinal3,t76_ordinal3,d2_arytm_3]),
    [file(arytm_3,t16_arytm_3),interesting(0.00)]).

fof(t17_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => r2_arytm_3(k3_arytm_3(A,B),k2_arytm_3(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_arytm_3,d3_arytm_3]),
    [file(arytm_3,t17_arytm_3),interesting(0.00)]).

fof(t27_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,k14_ordinal2(A,B))
            & r1_ordinal1(B,k14_ordinal2(A,B)) ) ) ) ),
    file(ordinal3,t27_ordinal3),
    [interesting(0.00)]).

fof(t1_arytm_3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( v4_ordinal2(k14_ordinal2(A,B))
           => ( r2_hidden(A,k5_ordinal2)
              & r2_hidden(B,k5_ordinal2) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2,t27_ordinal3,t22_ordinal1]),
    [file(arytm_3,t1_arytm_3),interesting(0.00)]).

fof(t30_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ~ ( A != k1_xboole_0
              & k5_arytm_3(A,B) = k1_xboole_0 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t23_arytm_3]),
    [file(arytm_3,t30_arytm_3),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(t53_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k15_ordinal2(k1_ordinal1(A),B) = k14_ordinal2(k15_ordinal2(A,B),B) ) ) ),
    file(ordinal2,t53_ordinal2),
    [interesting(0.00)]).

fof(t45_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k14_ordinal2(A,k1_ordinal1(B)) = k1_ordinal1(k14_ordinal2(A,B)) ) ) ),
    file(ordinal2,t45_ordinal2),
    [interesting(0.00)]).

fof(t47_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k14_ordinal2(k1_xboole_0,A) = A ) ),
    file(ordinal2,t47_ordinal2),
    [interesting(0.00)]).

fof(t10_ordinal1,theorem,(
    ! [A] : r2_hidden(A,k1_ordinal1(A)) ),
    file(ordinal1,t10_ordinal1),
    [interesting(0.00)]).

fof(t2_xboole_1,theorem,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    file(xboole_1,t2_xboole_1),
    [interesting(0.00)]).

fof(d8_xboole_0,definition,(
    ! [A,B] : 
      ( r2_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        & A != B ) ) ),
    file(xboole_0,d8_xboole_0),
    [interesting(0.00)]).

fof(t21_ordinal1,theorem,(
    ! [A] : 
      ( v1_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_xboole_0(A,B)
           => r2_hidden(A,B) ) ) ) ),
    file(ordinal1,t21_ordinal1),
    [interesting(0.00)]).

fof(s1_ordinal2,theorem,
    ( ( p1_s1_ordinal2(k1_xboole_0)
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( p1_s1_ordinal2(A)
           => p1_s1_ordinal2(k1_ordinal1(A)) ) )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( ( v4_ordinal1(A)
              & ! [B] : 
                  ( v3_ordinal1(B)
                 => ( r2_hidden(B,A)
                   => p1_s1_ordinal2(B) ) ) )
           => ( A = k1_xboole_0
              | p1_s1_ordinal2(A) ) ) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => p1_s1_ordinal2(A) ) ),
    file(ordinal2,s1_ordinal2),
    [interesting(0.00)]).

fof(s1_arytm_3,theorem,
    ( ( p1_s1_arytm_3(k1_xboole_0)
      & ! [A] : 
          ( ( v3_ordinal1(A)
            & v4_ordinal2(A) )
         => ( p1_s1_arytm_3(A)
           => p1_s1_arytm_3(k1_ordinal1(A)) ) ) )
   => ! [A] : 
        ( ( v3_ordinal1(A)
          & v4_ordinal2(A) )
       => p1_s1_arytm_3(A) ) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal1,d21_ordinal2,t19_ordinal1,t2_xboole_1,d8_xboole_0,t21_ordinal1,d5_ordinal2,d21_ordinal2,s1_ordinal2]),
    [file(arytm_3,s1_arytm_3),interesting(0.00)]).

fof(t3_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => k14_ordinal2(A,B) = k14_ordinal2(B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t44_ordinal2,t47_ordinal2,t44_ordinal2,t44_ordinal2,t45_ordinal2,t45_ordinal2,s1_arytm_3,t45_ordinal2,s1_arytm_3]),
    [file(arytm_3,t3_arytm_3),interesting(0.00)]).

fof(t4_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => k15_ordinal2(A,B) = k15_ordinal2(B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t55_ordinal2,t52_ordinal2,t52_ordinal2,t44_ordinal2,t52_ordinal2,t53_ordinal2,t33_ordinal3,t45_ordinal2,t45_ordinal2,t3_arytm_3,t33_ordinal3,t53_ordinal2,t45_ordinal2,s1_arytm_3,t53_ordinal2,s1_arytm_3]),
    [file(arytm_3,t4_arytm_3),interesting(0.00)]).

fof(t67_arytm_3,theorem,(
    $true ),
    file(arytm_3,t67_arytm_3),
    [interesting(0.00)]).

fof(t14_ordinal1,theorem,(
    ! [A] : A != k1_ordinal1(A) ),
    file(ordinal1,t14_ordinal1),
    [interesting(0.00)]).

fof(t19_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ~ ( r1_ordinal1(A,k4_ordinal2)
          & A != k1_xboole_0
          & A != k4_ordinal2 ) ) ),
    file(ordinal3,t19_ordinal3),
    [interesting(0.00)]).

fof(t36_arytm_3,theorem,(
    ! [A,B] : ~ v3_ordinal1(k4_tarski(A,B)) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal3,d2_tarski]),
    [file(arytm_3,t36_arytm_3),interesting(0.00)]).

fof(t37_arytm_3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( r2_hidden(A,k6_arytm_3)
       => r2_hidden(A,k5_ordinal2) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_arytm_3,t36_arytm_3]),
    [file(arytm_3,t37_arytm_3),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(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(t84_arytm_3,theorem,(
    $true ),
    file(arytm_3,t84_arytm_3),
    [interesting(0.00)]).

fof(t85_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => r3_arytm_3(A,k10_arytm_3(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d12_arytm_3]),
    [file(arytm_3,t85_arytm_3),interesting(0.00)]).

fof(t38_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( r1_ordinal1(k15_ordinal2(A,B),k15_ordinal2(C,B))
               => ( B = k1_xboole_0
                  | r1_ordinal1(A,C) ) ) ) ) ) ),
    file(ordinal3,t38_ordinal3),
    [interesting(0.00)]).

fof(t22_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ! [D] : 
                  ( v3_ordinal1(D)
                 => ( r2_hidden(A,B)
                   => ( ( ~ ( r1_ordinal1(C,D)
                            & D != k1_xboole_0 )
                        & ~ r2_hidden(C,D) )
                      | r2_hidden(k15_ordinal2(A,C),k15_ordinal2(B,D)) ) ) ) ) ) ) ),
    file(ordinal3,t22_ordinal3),
    [interesting(0.00)]).

fof(t8_arytm_3,theorem,(
    ! [A] : 
      ( ( v3_ordinal1(A)
        & v4_ordinal2(A) )
     => ! [B] : 
          ( ( v3_ordinal1(B)
            & v4_ordinal2(B) )
         => ~ ( ~ ( A = k1_xboole_0
                  & B = k1_xboole_0 )
              & ! [C] : 
                  ( ( v3_ordinal1(C)
                    & v4_ordinal2(C) )
                 => ! [D] : 
                      ( ( v3_ordinal1(D)
                        & v4_ordinal2(D) )
                     => ! [E] : 
                          ( ( v3_ordinal1(E)
                            & v4_ordinal2(E) )
                         => ~ ( r1_arytm_3(D,E)
                              & A = k2_arytm_3(C,D)
                              & B = k2_arytm_3(C,E) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d21_ordinal2,t3_xboole_1,d21_ordinal2,t3_xboole_1,s1_ordinal1,t22_ordinal1,t56_ordinal2,d1_arytm_3,t52_ordinal2,t55_ordinal2,t39_ordinal3,t22_ordinal1,t38_ordinal3,t58_ordinal3,t24_ordinal1,t17_ordinal3,t22_ordinal3,t56_ordinal2,t7_ordinal1]),
    [file(arytm_3,t8_arytm_3),interesting(0.00)]).

fof(t95_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( r3_arytm_3(A,k10_arytm_3(B,C))
                  & ! [D] : 
                      ( m1_subset_1(D,k6_arytm_3)
                     => ! [E] : 
                          ( m1_subset_1(E,k6_arytm_3)
                         => ~ ( A = k10_arytm_3(D,E)
                              & r3_arytm_3(D,B)
                              & r3_arytm_3(E,C) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t94_arytm_3,t56_arytm_3,t71_arytm_3]),
    [file(arytm_3,t95_arytm_3),interesting(0.00)]).

fof(t96_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( ~ r3_arytm_3(B,A)
                  & ~ r3_arytm_3(C,A)
                  & ! [D] : 
                      ( m1_subset_1(D,k6_arytm_3)
                     => ~ ( r3_arytm_3(D,B)
                          & r3_arytm_3(D,C)
                          & ~ r3_arytm_3(D,A) ) ) ) ) ) ) ),
    file(arytm_3,t96_arytm_3),
    [interesting(0.00)]).

fof(t98_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ! [B] : 
          ( m1_subset_1(B,k6_arytm_3)
         => ! [C] : 
              ( m1_subset_1(C,k6_arytm_3)
             => ~ ( ~ r3_arytm_3(k10_arytm_3(B,C),A)
                  & C != k12_arytm_3
                  & ! [D] : 
                      ( m1_subset_1(D,k6_arytm_3)
                     => ! [E] : 
                          ( m1_subset_1(E,k6_arytm_3)
                         => ~ ( A = k10_arytm_3(D,E)
                              & r3_arytm_3(D,B)
                              & r3_arytm_3(E,C)
                              & E != C ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t94_arytm_3,t56_arytm_3,t71_arytm_3]),
    [file(arytm_3,t98_arytm_3),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(t34_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,k1_ordinal1(B))
          <=> r1_ordinal1(A,B) ) ) ) ),
    file(ordinal1,t34_ordinal1),
    [interesting(0.00)]).