fof(t1_binop_2,theorem,(
    k3_binop_1(k2_numbers,k27_binop_2) = 0 ),
    inference(mizar_proof,[status(thm)],[l27_binop_2,d8_binop_1]),
    [file(binop_2,t1_binop_2),interesting(1.00)]).

fof(t2_binop_2,theorem,(
    k3_binop_1(k1_numbers,k33_binop_2) = 0 ),
    inference(mizar_proof,[status(thm)],[l28_binop_2,d8_binop_1]),
    [file(binop_2,t2_binop_2),interesting(1.00)]).

fof(t3_binop_2,theorem,(
    k3_binop_1(k3_numbers,k39_binop_2) = 0 ),
    inference(mizar_proof,[status(thm)],[l30_binop_2,d8_binop_1]),
    [file(binop_2,t3_binop_2),interesting(1.00)]).

fof(t4_binop_2,theorem,(
    k3_binop_1(k4_numbers,k44_binop_2) = 0 ),
    inference(mizar_proof,[status(thm)],[l32_binop_2,d8_binop_1]),
    [file(binop_2,t4_binop_2),interesting(1.00)]).

fof(t6_binop_2,theorem,(
    k3_binop_1(k2_numbers,k29_binop_2) = 1 ),
    inference(mizar_proof,[status(thm)],[l36_binop_2,d8_binop_1]),
    [file(binop_2,t6_binop_2),interesting(1.00)]).

fof(t7_binop_2,theorem,(
    k3_binop_1(k1_numbers,k35_binop_2) = 1 ),
    inference(mizar_proof,[status(thm)],[l37_binop_2,d8_binop_1]),
    [file(binop_2,t7_binop_2),interesting(1.00)]).

fof(t8_binop_2,theorem,(
    k3_binop_1(k3_numbers,k41_binop_2) = 1 ),
    inference(mizar_proof,[status(thm)],[l39_binop_2,d8_binop_1]),
    [file(binop_2,t8_binop_2),interesting(1.00)]).

fof(t9_binop_2,theorem,(
    k3_binop_1(k4_numbers,k46_binop_2) = 1 ),
    inference(mizar_proof,[status(thm)],[l41_binop_2,d8_binop_1]),
    [file(binop_2,t9_binop_2),interesting(1.00)]).

fof(t10_binop_2,theorem,(
    k3_binop_1(k5_numbers,k48_binop_2) = 1 ),
    inference(mizar_proof,[status(thm)],[l42_binop_2,d8_binop_1]),
    [file(binop_2,t10_binop_2),interesting(0.97)]).

fof(t5_binop_2,theorem,(
    k3_binop_1(k5_numbers,k47_binop_2) = 0 ),
    inference(mizar_proof,[status(thm)],[l33_binop_2,d8_binop_1]),
    [file(binop_2,t5_binop_2),interesting(0.97)]).

fof(l30_binop_2,theorem,(
    r3_binop_1(k3_numbers,c2,k39_binop_2) ),
    inference(mizar_proof,[status(thm)],[d15_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d15_binop_2]),
    [file(binop_2,l30_binop_2),interesting(0.90)]).

fof(l32_binop_2,theorem,(
    r3_binop_1(k4_numbers,c3,k44_binop_2) ),
    inference(mizar_proof,[status(thm)],[d20_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d20_binop_2]),
    [file(binop_2,l32_binop_2),interesting(0.90)]).

fof(l39_binop_2,theorem,(
    r3_binop_1(k3_numbers,c5,k41_binop_2) ),
    inference(mizar_proof,[status(thm)],[d17_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d17_binop_2]),
    [file(binop_2,l39_binop_2),interesting(0.90)]).

fof(l41_binop_2,theorem,(
    r3_binop_1(k4_numbers,c6,k46_binop_2) ),
    inference(mizar_proof,[status(thm)],[d22_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d22_binop_2]),
    [file(binop_2,l41_binop_2),interesting(0.90)]).

fof(l27_binop_2,theorem,(
    r3_binop_1(k2_numbers,c1,k27_binop_2) ),
    inference(mizar_proof,[status(thm)],[d3_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d3_binop_2]),
    [file(binop_2,l27_binop_2),interesting(0.89)]).

fof(l36_binop_2,theorem,(
    r3_binop_1(k2_numbers,c4,k29_binop_2) ),
    inference(mizar_proof,[status(thm)],[d5_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d5_binop_2]),
    [file(binop_2,l36_binop_2),interesting(0.89)]).

fof(l28_binop_2,theorem,(
    r3_binop_1(k1_numbers,0,k33_binop_2) ),
    inference(mizar_proof,[status(thm)],[d9_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d9_binop_2]),
    [file(binop_2,l28_binop_2),interesting(0.88)]).

fof(l37_binop_2,theorem,(
    r3_binop_1(k1_numbers,1,k35_binop_2) ),
    inference(mizar_proof,[status(thm)],[d11_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d11_binop_2]),
    [file(binop_2,l37_binop_2),interesting(0.88)]).

fof(l42_binop_2,theorem,(
    r3_binop_1(k5_numbers,1,k48_binop_2) ),
    inference(mizar_proof,[status(thm)],[d24_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d24_binop_2]),
    [file(binop_2,l42_binop_2),interesting(0.86)]).

fof(l33_binop_2,theorem,(
    r3_binop_1(k5_numbers,0,k47_binop_2) ),
    inference(mizar_proof,[status(thm)],[d23_binop_2,d7_binop_1,d16_binop_1,d17_binop_1,d23_binop_2]),
    [file(binop_2,l33_binop_2),interesting(0.86)]).

fof(l26_binop_2,theorem,(
    m1_subset_1(0,k2_numbers) ),
    inference(mizar_proof,[status(thm)],[l25_binop_2,t20_numbers]),
    [file(binop_2,l26_binop_2),interesting(0.72)]).

fof(l29_binop_2,theorem,(
    m1_subset_1(0,k3_numbers) ),
    inference(mizar_proof,[status(thm)],[l25_binop_2,t18_numbers]),
    [file(binop_2,l29_binop_2),interesting(0.72)]).

fof(l31_binop_2,theorem,(
    m1_subset_1(0,k4_numbers) ),
    inference(mizar_proof,[status(thm)],[l25_binop_2,t17_numbers]),
    [file(binop_2,l31_binop_2),interesting(0.72)]).

fof(l35_binop_2,theorem,(
    m1_subset_1(1,k2_numbers) ),
    inference(mizar_proof,[status(thm)],[l34_binop_2,t20_numbers]),
    [file(binop_2,l35_binop_2),interesting(0.72)]).

fof(l38_binop_2,theorem,(
    m1_subset_1(1,k3_numbers) ),
    inference(mizar_proof,[status(thm)],[l34_binop_2,t18_numbers]),
    [file(binop_2,l38_binop_2),interesting(0.72)]).

fof(l40_binop_2,theorem,(
    m1_subset_1(1,k4_numbers) ),
    inference(mizar_proof,[status(thm)],[l34_binop_2,t17_numbers]),
    [file(binop_2,l40_binop_2),interesting(0.72)]).

fof(s2_binop_2,theorem,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(f1_s2_binop_2,f1_s2_binop_2),f1_s2_binop_2)
        & m2_relset_1(A,k2_zfmisc_1(f1_s2_binop_2,f1_s2_binop_2),f1_s2_binop_2) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(f1_s2_binop_2,f1_s2_binop_2),f1_s2_binop_2)
            & m2_relset_1(B,k2_zfmisc_1(f1_s2_binop_2,f1_s2_binop_2),f1_s2_binop_2) )
         => ( ( ! [C] : 
                  ( m1_subset_1(C,f1_s2_binop_2)
                 => ! [D] : 
                      ( m1_subset_1(D,f1_s2_binop_2)
                     => k2_binop_1(f1_s2_binop_2,f1_s2_binop_2,f1_s2_binop_2,A,C,D) = f2_s2_binop_2(C,D) ) )
              & ! [C] : 
                  ( m1_subset_1(C,f1_s2_binop_2)
                 => ! [D] : 
                      ( m1_subset_1(D,f1_s2_binop_2)
                     => k2_binop_1(f1_s2_binop_2,f1_s2_binop_2,f1_s2_binop_2,B,C,D) = f2_s2_binop_2(C,D) ) ) )
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_binop_1]),
    [file(binop_2,s2_binop_2),interesting(0.03)]).

fof(l25_binop_2,theorem,(
    r2_hidden(0,k5_numbers) ),
    file(binop_2,l25_binop_2),
    [interesting(0.00)]).

fof(t20_numbers,theorem,(
    r1_tarski(k5_numbers,k2_numbers) ),
    file(numbers,t20_numbers),
    [interesting(0.00)]).

fof(t18_numbers,theorem,(
    r1_tarski(k5_numbers,k3_numbers) ),
    file(numbers,t18_numbers),
    [interesting(0.00)]).

fof(t17_numbers,theorem,(
    r1_tarski(k5_numbers,k4_numbers) ),
    file(numbers,t17_numbers),
    [interesting(0.00)]).

fof(l34_binop_2,theorem,(
    r2_hidden(1,k5_numbers) ),
    file(binop_2,l34_binop_2),
    [interesting(0.00)]).

fof(t113_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) )
         => ( ! [E] : 
                ( m1_subset_1(E,A)
               => k1_funct_1(C,E) = k1_funct_1(D,E) )
           => C = D ) ) ) ),
    file(funct_2,t113_funct_2),
    [interesting(0.00)]).

fof(s1_binop_2,theorem,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,f1_s1_binop_2,f2_s1_binop_2)
        & m2_relset_1(A,f1_s1_binop_2,f2_s1_binop_2) )
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,f1_s1_binop_2,f2_s1_binop_2)
            & m2_relset_1(B,f1_s1_binop_2,f2_s1_binop_2) )
         => ( ( ! [C] : 
                  ( m1_subset_1(C,f1_s1_binop_2)
                 => k8_funct_2(f1_s1_binop_2,f2_s1_binop_2,A,C) = f3_s1_binop_2(C) )
              & ! [C] : 
                  ( m1_subset_1(C,f1_s1_binop_2)
                 => k8_funct_2(f1_s1_binop_2,f2_s1_binop_2,B,C) = f3_s1_binop_2(C) ) )
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t113_funct_2]),
    [file(binop_2,s1_binop_2),interesting(0.00)]).

fof(t2_binop_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ~ v1_xboole_0(C)
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(A,B),C)
                    & m2_relset_1(D,k2_zfmisc_1(A,B),C) )
                 => ! [E] : 
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k2_zfmisc_1(A,B),C)
                        & m2_relset_1(E,k2_zfmisc_1(A,B),C) )
                     => ( ! [F] : 
                            ( m1_subset_1(F,A)
                           => ! [G] : 
                                ( m1_subset_1(G,B)
                               => k2_binop_1(A,B,C,D,F,G) = k2_binop_1(A,B,C,E,F,G) ) )
                       => D = E ) ) ) ) ) ) ),
    file(binop_1,t2_binop_1),
    [interesting(0.00)]).

fof(d24_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )
     => ( A = k48_binop_2
      <=> ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(k5_numbers,k5_numbers,k5_numbers,A,B,C) = k24_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d24_binop_2),
    [interesting(0.00)]).

fof(d7_binop_1,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,A)
     => ! [C] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r3_binop_1(A,B,C)
          <=> ( r1_binop_1(A,B,C)
              & r2_binop_1(A,B,C) ) ) ) ) ),
    file(binop_1,d7_binop_1),
    [interesting(0.00)]).

fof(d16_binop_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( r1_binop_1(A,B,C)
              <=> ! [D] : 
                    ( m1_subset_1(D,A)
                   => k2_binop_1(A,A,A,C,B,D) = D ) ) ) ) ) ),
    file(binop_1,d16_binop_1),
    [interesting(0.00)]).

fof(d17_binop_1,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( r2_binop_1(A,B,C)
              <=> ! [D] : 
                    ( m1_subset_1(D,A)
                   => k2_binop_1(A,A,A,C,D,B) = D ) ) ) ) ) ),
    file(binop_1,d17_binop_1),
    [interesting(0.00)]).

fof(d8_binop_1,definition,(
    ! [A,B] : 
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k2_zfmisc_1(A,A),A)
        & m2_relset_1(B,k2_zfmisc_1(A,A),A) )
     => ( ? [C] : 
            ( m1_subset_1(C,A)
            & r3_binop_1(A,C,B) )
       => ! [C] : 
            ( m1_subset_1(C,A)
           => ( C = k3_binop_1(A,B)
            <=> r3_binop_1(A,C,B) ) ) ) ) ),
    file(binop_1,d8_binop_1),
    [interesting(0.00)]).

fof(d3_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k2_numbers,k2_numbers),k2_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k2_numbers,k2_numbers),k2_numbers) )
     => ( A = k27_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k2_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k2_numbers)
               => k2_binop_1(k2_numbers,k2_numbers,k2_numbers,A,B,C) = k3_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d3_binop_2),
    [interesting(0.00)]).

fof(d9_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )
     => ( A = k33_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k1_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => k2_binop_1(k1_numbers,k1_numbers,k1_numbers,A,B,C) = k9_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d9_binop_2),
    [interesting(0.00)]).

fof(d15_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k3_numbers,k3_numbers),k3_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k3_numbers,k3_numbers),k3_numbers) )
     => ( A = k39_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k3_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k3_numbers)
               => k2_binop_1(k3_numbers,k3_numbers,k3_numbers,A,B,C) = k15_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d15_binop_2),
    [interesting(0.00)]).

fof(d20_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers) )
     => ( A = k44_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k4_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k4_numbers)
               => k2_binop_1(k4_numbers,k4_numbers,k4_numbers,A,B,C) = k20_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d20_binop_2),
    [interesting(0.00)]).

fof(d23_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) )
     => ( A = k47_binop_2
      <=> ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_binop_1(k5_numbers,k5_numbers,k5_numbers,A,B,C) = k23_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d23_binop_2),
    [interesting(0.00)]).

fof(d5_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k2_numbers,k2_numbers),k2_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k2_numbers,k2_numbers),k2_numbers) )
     => ( A = k29_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k2_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k2_numbers)
               => k2_binop_1(k2_numbers,k2_numbers,k2_numbers,A,B,C) = k5_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d5_binop_2),
    [interesting(0.00)]).

fof(d11_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) )
     => ( A = k35_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k1_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k1_numbers)
               => k2_binop_1(k1_numbers,k1_numbers,k1_numbers,A,B,C) = k11_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d11_binop_2),
    [interesting(0.00)]).

fof(d17_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k3_numbers,k3_numbers),k3_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k3_numbers,k3_numbers),k3_numbers) )
     => ( A = k41_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k3_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k3_numbers)
               => k2_binop_1(k3_numbers,k3_numbers,k3_numbers,A,B,C) = k17_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d17_binop_2),
    [interesting(0.00)]).

fof(d22_binop_2,definition,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers)
        & m2_relset_1(A,k2_zfmisc_1(k4_numbers,k4_numbers),k4_numbers) )
     => ( A = k46_binop_2
      <=> ! [B] : 
            ( m1_subset_1(B,k4_numbers)
           => ! [C] : 
                ( m1_subset_1(C,k4_numbers)
               => k2_binop_1(k4_numbers,k4_numbers,k4_numbers,A,B,C) = k22_binop_2(B,C) ) ) ) ) ),
    file(binop_2,d22_binop_2),
    [interesting(0.00)]).