fof(t16_binop_1,theorem,(
    ! [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)
         => ( v1_binop_1(B,A)
           => ( r1_binop_1(A,C,B)
            <=> r2_binop_1(A,C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t14_binop_1,t15_binop_1]),
    [file(binop_1,t16_binop_1),interesting(1.00)]).

fof(t28_binop_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(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] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( v1_binop_1(C,A)
               => ( r5_binop_1(A,C,B)
                <=> r4_binop_1(A,C,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t26_binop_1,t27_binop_1]),
    [file(binop_1,t28_binop_1),interesting(0.86)]).

fof(t14_binop_1,theorem,(
    ! [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)
         => ( v1_binop_1(B,A)
           => ( r3_binop_1(A,C,B)
            <=> r1_binop_1(A,C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_binop_1,t12_binop_1]),
    [file(binop_1,t14_binop_1),interesting(0.85)]).

fof(t15_binop_1,theorem,(
    ! [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)
         => ( v1_binop_1(B,A)
           => ( r3_binop_1(A,C,B)
            <=> r2_binop_1(A,C,B) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_binop_1,t13_binop_1]),
    [file(binop_1,t15_binop_1),interesting(0.85)]).

fof(t18_binop_1,theorem,(
    ! [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)
         => ! [D] : 
              ( m1_subset_1(D,A)
             => ( ( r3_binop_1(A,C,B)
                  & r3_binop_1(A,D,B) )
               => C = D ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_binop_1,t17_binop_1]),
    [file(binop_1,t18_binop_1),interesting(0.80)]).

fof(t26_binop_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(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] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( v1_binop_1(C,A)
               => ( r6_binop_1(A,C,B)
                <=> r4_binop_1(A,C,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_binop_1,t24_binop_1]),
    [file(binop_1,t26_binop_1),interesting(0.74)]).

fof(t27_binop_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(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] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( v1_binop_1(C,A)
               => ( r6_binop_1(A,C,B)
                <=> r5_binop_1(A,C,B) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d10_binop_1,t25_binop_1]),
    [file(binop_1,t27_binop_1),interesting(0.74)]).

fof(t11_binop_1,theorem,(
    ! [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)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => ( k1_binop_1(B,C,D) = D
                  & k1_binop_1(B,D,C) = D ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_binop_1,d5_binop_1,d6_binop_1,d5_binop_1,d6_binop_1,d7_binop_1]),
    [file(binop_1,t11_binop_1),interesting(0.64)]).

fof(t17_binop_1,theorem,(
    ! [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)
         => ! [D] : 
              ( m1_subset_1(D,A)
             => ( ( r1_binop_1(A,C,B)
                  & r2_binop_1(A,D,B) )
               => C = D ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_binop_1,d5_binop_1]),
    [file(binop_1,t17_binop_1),interesting(0.64)]).

fof(t12_binop_1,theorem,(
    ! [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)
         => ( v1_binop_1(B,A)
           => ( r3_binop_1(A,C,B)
            <=> ! [D] : 
                  ( m1_subset_1(D,A)
                 => k1_binop_1(B,C,D) = D ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_binop_1,t11_binop_1]),
    [file(binop_1,t12_binop_1),interesting(0.61)]).

fof(t13_binop_1,theorem,(
    ! [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)
         => ( v1_binop_1(B,A)
           => ( r3_binop_1(A,C,B)
            <=> ! [D] : 
                  ( m1_subset_1(D,A)
                 => k1_binop_1(B,D,C) = D ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_binop_1,t11_binop_1]),
    [file(binop_1,t13_binop_1),interesting(0.61)]).

fof(t23_binop_1,theorem,(
    ! [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] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r6_binop_1(A,B,C)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => ! [E] : 
                    ( m1_subset_1(E,A)
                   => ! [F] : 
                        ( m1_subset_1(F,A)
                       => ( k1_binop_1(B,D,k1_binop_1(C,E,F)) = k1_binop_1(C,k1_binop_1(B,D,E),k1_binop_1(B,D,F))
                          & k1_binop_1(B,k1_binop_1(C,D,E),F) = k1_binop_1(C,k1_binop_1(B,D,F),k1_binop_1(B,E,F)) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d11_binop_1,d9_binop_1,d10_binop_1,d9_binop_1,d10_binop_1,d11_binop_1]),
    [file(binop_1,t23_binop_1),interesting(0.38)]).

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 ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t120_funct_2]),
    [file(binop_1,t2_binop_1),interesting(0.26)]).

fof(t24_binop_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(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] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( v1_binop_1(C,A)
               => ( r6_binop_1(A,C,B)
                <=> ! [D] : 
                      ( m1_subset_1(D,A)
                     => ! [E] : 
                          ( m1_subset_1(E,A)
                         => ! [F] : 
                              ( m1_subset_1(F,A)
                             => k2_binop_1(A,A,A,C,D,k2_binop_1(A,A,A,B,E,F)) = k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,C,D,E),k2_binop_1(A,A,A,C,D,F)) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_binop_1,d2_binop_1,d2_binop_1,t23_binop_1]),
    [file(binop_1,t24_binop_1),interesting(0.18)]).

fof(t25_binop_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(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] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(A,A),A)
                & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
             => ( v1_binop_1(C,A)
               => ( r6_binop_1(A,C,B)
                <=> ! [D] : 
                      ( m1_subset_1(D,A)
                     => ! [E] : 
                          ( m1_subset_1(E,A)
                         => ! [F] : 
                              ( m1_subset_1(F,A)
                             => k2_binop_1(A,A,A,C,k2_binop_1(A,A,A,B,D,E),F) = k2_binop_1(A,A,A,B,k2_binop_1(A,A,A,C,D,F),k2_binop_1(A,A,A,C,E,F)) ) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_binop_1,d2_binop_1,d2_binop_1,t23_binop_1]),
    [file(binop_1,t25_binop_1),interesting(0.18)]).

fof(s1_binop_1,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s1_binop_1)
       => ! [B] : 
            ( m1_subset_1(B,f1_s1_binop_1)
           => ? [C] : 
                ( m1_subset_1(C,f1_s1_binop_1)
                & p1_s1_binop_1(A,B,C) ) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(f1_s1_binop_1,f1_s1_binop_1),f1_s1_binop_1)
        & m2_relset_1(A,k2_zfmisc_1(f1_s1_binop_1,f1_s1_binop_1),f1_s1_binop_1)
        & ! [B] : 
            ( m1_subset_1(B,f1_s1_binop_1)
           => ! [C] : 
                ( m1_subset_1(C,f1_s1_binop_1)
               => p1_s1_binop_1(B,C,k2_binop_1(f1_s1_binop_1,f1_s1_binop_1,f1_s1_binop_1,A,B,C)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s7_funct_2]),
    [file(binop_1,s1_binop_1),interesting(0.11)]).

fof(s7_funct_2,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s7_funct_2)
       => ! [B] : 
            ( m1_subset_1(B,f2_s7_funct_2)
           => ? [C] : 
                ( m1_subset_1(C,f3_s7_funct_2)
                & p1_s7_funct_2(A,B,C) ) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,k2_zfmisc_1(f1_s7_funct_2,f2_s7_funct_2),f3_s7_funct_2)
        & m2_relset_1(A,k2_zfmisc_1(f1_s7_funct_2,f2_s7_funct_2),f3_s7_funct_2)
        & ! [B] : 
            ( m1_subset_1(B,f1_s7_funct_2)
           => ! [C] : 
                ( m1_subset_1(C,f2_s7_funct_2)
               => p1_s7_funct_2(B,C,k1_funct_1(A,k4_tarski(B,C))) ) ) ) ),
    file(funct_2,s7_funct_2),
    [interesting(0.00)]).

fof(s8_funct_2,theorem,(
    ? [A] : 
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(f1_s8_funct_2,f2_s8_funct_2),f3_s8_funct_2)
      & m2_relset_1(A,k2_zfmisc_1(f1_s8_funct_2,f2_s8_funct_2),f3_s8_funct_2)
      & ! [B] : 
          ( m1_subset_1(B,f1_s8_funct_2)
         => ! [C] : 
              ( m1_subset_1(C,f2_s8_funct_2)
             => k1_funct_1(A,k4_tarski(B,C)) = f4_s8_funct_2(B,C) ) ) ) ),
    file(funct_2,s8_funct_2),
    [interesting(0.00)]).

fof(s2_binop_1,theorem,(
    ? [A] : 
      ( v1_funct_1(A)
      & v1_funct_2(A,k2_zfmisc_1(f1_s2_binop_1,f1_s2_binop_1),f1_s2_binop_1)
      & m2_relset_1(A,k2_zfmisc_1(f1_s2_binop_1,f1_s2_binop_1),f1_s2_binop_1)
      & ! [B] : 
          ( m1_subset_1(B,f1_s2_binop_1)
         => ! [C] : 
              ( m1_subset_1(C,f1_s2_binop_1)
             => k2_binop_1(f1_s2_binop_1,f1_s2_binop_1,f1_s2_binop_1,A,B,C) = f2_s2_binop_1(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s8_funct_2]),
    [file(binop_1,s2_binop_1),interesting(0.00)]).

fof(t10_binop_1,theorem,(
    $true ),
    file(binop_1,t10_binop_1),
    [interesting(0.00)]).

fof(d5_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) )
         => ( r1_binop_1(A,B,C)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => k1_binop_1(C,B,D) = D ) ) ) ) ),
    file(binop_1,d5_binop_1),
    [interesting(0.00)]).

fof(d2_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) )
     => ( v1_binop_1(B,A)
      <=> ! [C] : 
            ( m1_subset_1(C,A)
           => ! [D] : 
                ( m1_subset_1(D,A)
               => k1_binop_1(B,C,D) = k1_binop_1(B,D,C) ) ) ) ) ),
    file(binop_1,d2_binop_1),
    [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(d6_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) )
         => ( r2_binop_1(A,B,C)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => k1_binop_1(C,D,B) = D ) ) ) ) ),
    file(binop_1,d6_binop_1),
    [interesting(0.00)]).

fof(t19_binop_1,theorem,(
    $true ),
    file(binop_1,t19_binop_1),
    [interesting(0.00)]).

fof(t1_binop_1,theorem,(
    $true ),
    file(binop_1,t1_binop_1),
    [interesting(0.00)]).

fof(t20_binop_1,theorem,(
    $true ),
    file(binop_1,t20_binop_1),
    [interesting(0.00)]).

fof(t21_binop_1,theorem,(
    $true ),
    file(binop_1,t21_binop_1),
    [interesting(0.00)]).

fof(t22_binop_1,theorem,(
    $true ),
    file(binop_1,t22_binop_1),
    [interesting(0.00)]).

fof(d9_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] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r4_binop_1(A,B,C)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => ! [E] : 
                    ( m1_subset_1(E,A)
                   => ! [F] : 
                        ( m1_subset_1(F,A)
                       => k1_binop_1(B,D,k1_binop_1(C,E,F)) = k1_binop_1(C,k1_binop_1(B,D,E),k1_binop_1(B,D,F)) ) ) ) ) ) ) ),
    file(binop_1,d9_binop_1),
    [interesting(0.00)]).

fof(d11_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] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r6_binop_1(A,B,C)
          <=> ( r4_binop_1(A,B,C)
              & r5_binop_1(A,B,C) ) ) ) ) ),
    file(binop_1,d11_binop_1),
    [interesting(0.00)]).

fof(d10_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] : 
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k2_zfmisc_1(A,A),A)
            & m2_relset_1(C,k2_zfmisc_1(A,A),A) )
         => ( r5_binop_1(A,B,C)
          <=> ! [D] : 
                ( m1_subset_1(D,A)
               => ! [E] : 
                    ( m1_subset_1(E,A)
                   => ! [F] : 
                        ( m1_subset_1(F,A)
                       => k1_binop_1(B,k1_binop_1(C,D,E),F) = k1_binop_1(C,k1_binop_1(B,D,F),k1_binop_1(B,E,F)) ) ) ) ) ) ) ),
    file(binop_1,d10_binop_1),
    [interesting(0.00)]).

fof(t120_funct_2,theorem,(
    ! [A,B,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)
                   => k1_funct_1(D,k4_tarski(F,G)) = k1_funct_1(E,k4_tarski(F,G)) ) )
           => D = E ) ) ) ),
    file(funct_2,t120_funct_2),
    [interesting(0.00)]).

fof(t3_binop_1,theorem,(
    $true ),
    file(binop_1,t3_binop_1),
    [interesting(0.00)]).

fof(t4_binop_1,theorem,(
    $true ),
    file(binop_1,t4_binop_1),
    [interesting(0.00)]).

fof(t5_binop_1,theorem,(
    $true ),
    file(binop_1,t5_binop_1),
    [interesting(0.00)]).

fof(t6_binop_1,theorem,(
    $true ),
    file(binop_1,t6_binop_1),
    [interesting(0.00)]).

fof(t7_binop_1,theorem,(
    $true ),
    file(binop_1,t7_binop_1),
    [interesting(0.00)]).

fof(t8_binop_1,theorem,(
    $true ),
    file(binop_1,t8_binop_1),
    [interesting(0.00)]).

fof(t9_binop_1,theorem,(
    $true ),
    file(binop_1,t9_binop_1),
    [interesting(0.00)]).