% Mizar ND problem: t6_altcat_2,altcat_2,243,25 
fof(dh_c1_10__altcat_2,definition,
    ( ( ~ v1_xboole_0(c1_10__altcat_2)
     => ! [A] : 
          ( ~ v1_xboole_0(A)
         => ! [B] : 
              ( m1_pboole(B,k2_zfmisc_1(c1_10__altcat_2,A))
             => ! [C] : 
                  ( m1_subset_1(C,c1_10__altcat_2)
                 => ! [D] : 
                      ( m1_subset_1(D,A)
                     => k1_binop_1(k18_pralg_1(c1_10__altcat_2,A,B),D,C) = k1_binop_1(B,C,D) ) ) ) ) )
   => ! [E] : 
        ( ~ v1_xboole_0(E)
       => ! [F] : 
            ( ~ v1_xboole_0(F)
           => ! [G] : 
                ( m1_pboole(G,k2_zfmisc_1(E,F))
               => ! [H] : 
                    ( m1_subset_1(H,E)
                   => ! [I] : 
                        ( m1_subset_1(I,F)
                       => k1_binop_1(k18_pralg_1(E,F,G),I,H) = k1_binop_1(G,H,I) ) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_10__altcat_2),file(altcat_2,c1_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c1_10__altcat_2)]).

fof(dh_c2_10__altcat_2,definition,
    ( ( ~ v1_xboole_0(c2_10__altcat_2)
     => ! [A] : 
          ( m1_pboole(A,k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2))
         => ! [B] : 
              ( m1_subset_1(B,c1_10__altcat_2)
             => ! [C] : 
                  ( m1_subset_1(C,c2_10__altcat_2)
                 => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,A),C,B) = k1_binop_1(A,B,C) ) ) ) )
   => ! [D] : 
        ( ~ v1_xboole_0(D)
       => ! [E] : 
            ( m1_pboole(E,k2_zfmisc_1(c1_10__altcat_2,D))
           => ! [F] : 
                ( m1_subset_1(F,c1_10__altcat_2)
               => ! [G] : 
                    ( m1_subset_1(G,D)
                   => k1_binop_1(k18_pralg_1(c1_10__altcat_2,D,E),G,F) = k1_binop_1(E,F,G) ) ) ) ) ),
    introduced(definition,[new_symbol(c2_10__altcat_2),file(altcat_2,c2_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c2_10__altcat_2)]).

fof(dh_c3_10__altcat_2,definition,
    ( ( m1_pboole(c3_10__altcat_2,k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2))
     => ! [A] : 
          ( m1_subset_1(A,c1_10__altcat_2)
         => ! [B] : 
              ( m1_subset_1(B,c2_10__altcat_2)
             => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),B,A) = k1_binop_1(c3_10__altcat_2,A,B) ) ) )
   => ! [C] : 
        ( m1_pboole(C,k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2))
       => ! [D] : 
            ( m1_subset_1(D,c1_10__altcat_2)
           => ! [E] : 
                ( m1_subset_1(E,c2_10__altcat_2)
               => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,C),E,D) = k1_binop_1(C,D,E) ) ) ) ),
    introduced(definition,[new_symbol(c3_10__altcat_2),file(altcat_2,c3_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c3_10__altcat_2)]).

fof(dh_c4_10__altcat_2,definition,
    ( ( m1_subset_1(c4_10__altcat_2,c1_10__altcat_2)
     => ! [A] : 
          ( m1_subset_1(A,c2_10__altcat_2)
         => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),A,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,A) ) )
   => ! [B] : 
        ( m1_subset_1(B,c1_10__altcat_2)
       => ! [C] : 
            ( m1_subset_1(C,c2_10__altcat_2)
           => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),C,B) = k1_binop_1(c3_10__altcat_2,B,C) ) ) ),
    introduced(definition,[new_symbol(c4_10__altcat_2),file(altcat_2,c4_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c4_10__altcat_2)]).

fof(dh_c5_10__altcat_2,definition,
    ( ( m1_subset_1(c5_10__altcat_2,c2_10__altcat_2)
     => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),c5_10__altcat_2,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,c5_10__altcat_2) )
   => ! [A] : 
        ( m1_subset_1(A,c2_10__altcat_2)
       => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),A,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,A) ) ),
    introduced(definition,[new_symbol(c5_10__altcat_2),file(altcat_2,c5_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c5_10__altcat_2)]).

fof(cc1_pboole,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_pboole(B,A)
         => ( v2_relat_1(B)
           => ~ v3_relat_1(B) ) ) ) ),
    file(pboole,cc1_pboole),
    [interesting(0.9),axiom,file(pboole,cc1_pboole)]).

fof(fc2_pboole,theorem,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & v2_relat_1(B)
        & m1_pboole(B,A)
        & m1_subset_1(C,A) )
     => ~ v1_xboole_0(k1_funct_1(B,C)) ) ),
    file(pboole,fc2_pboole),
    [interesting(0.9),axiom,file(pboole,fc2_pboole)]).

fof(rc3_pboole,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v2_relat_1(B)
      & v1_funct_1(B) ) ),
    file(pboole,rc3_pboole),
    [interesting(0.9),axiom,file(pboole,rc3_pboole)]).

fof(rc5_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v2_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc5_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc5_funct_1)]).

fof(cc2_pboole,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_pboole(B,A)
         => ( v3_relat_1(B)
           => ~ v2_relat_1(B) ) ) ) ),
    file(pboole,cc2_pboole),
    [interesting(0.9),axiom,file(pboole,cc2_pboole)]).

fof(rc1_pboole,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(pboole,rc1_pboole),
    [interesting(0.9),axiom,file(pboole,rc1_pboole)]).

fof(rc2_pboole,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v3_relat_1(B)
      & v1_funct_1(B) ) ),
    file(pboole,rc2_pboole),
    [interesting(0.9),axiom,file(pboole,rc2_pboole)]).

fof(rc3_relat_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A) ) ),
    file(relat_1,rc3_relat_1),
    [interesting(0.9),axiom,file(relat_1,rc3_relat_1)]).

fof(rc4_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(cc1_realset1,theorem,(
    ! [A] : 
      ( ~ v1_realset1(A)
     => ~ v1_xboole_0(A) ) ),
    file(realset1,cc1_realset1),
    [interesting(0.9),axiom,file(realset1,cc1_realset1)]).

fof(fc12_relat_1,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0) ),
    file(relat_1,fc12_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc12_relat_1)]).

fof(fc4_relat_1,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0) ),
    file(relat_1,fc4_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc4_relat_1)]).

fof(rc1_realset1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_realset1(A) ) ),
    file(realset1,rc1_realset1),
    [interesting(0.9),axiom,file(realset1,rc1_realset1)]).

fof(rc2_realset1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & ~ v1_realset1(A) ) ),
    file(realset1,rc2_realset1),
    [interesting(0.9),axiom,file(realset1,rc2_realset1)]).

fof(rc3_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]).

fof(commutativity_k2_tarski,theorem,(
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ),
    file(tarski,k2_tarski),
    [interesting(0.9),axiom,file(tarski,k2_tarski)]).

fof(existence_m1_pboole,axiom,(
    ! [A] : 
    ? [B] : m1_pboole(B,A) ),
    file(pboole,m1_pboole),
    [interesting(0.9),axiom,file(pboole,m1_pboole)]).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    [interesting(0.9),axiom,file(tarski,k1_tarski)]).

fof(dt_k2_tarski,axiom,(
    $true ),
    file(tarski,k2_tarski),
    [interesting(0.9),axiom,file(tarski,k2_tarski)]).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]).

fof(dt_m1_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v1_relat_1(B)
        & v1_funct_1(B) ) ) ),
    file(pboole,m1_pboole),
    [interesting(0.9),axiom,file(pboole,m1_pboole)]).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(cc1_funct_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]).

fof(cc1_relat_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_relat_1(A) ) ),
    file(relat_1,cc1_relat_1),
    [interesting(0.9),axiom,file(relat_1,cc1_relat_1)]).

fof(cc2_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) ),
    file(funct_1,cc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]).

fof(fc2_altcat_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ~ v1_xboole_0(k1_tarski(A))
        & v1_finset_1(k1_tarski(A))
        & v1_fraenkel(k1_tarski(A))
        & v1_realset1(k1_tarski(A)) ) ) ),
    file(altcat_1,fc2_altcat_1),
    [interesting(0.9),axiom,file(altcat_1,fc2_altcat_1)]).

fof(fc2_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]).

fof(fc3_realset1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A))
      & v1_realset1(k1_tarski(A)) ) ),
    file(realset1,fc3_realset1),
    [interesting(0.9),axiom,file(realset1,fc3_realset1)]).

fof(fc3_subset_1,theorem,(
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ),
    file(subset_1,fc3_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc3_subset_1)]).

fof(fc4_subset_1,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]).

fof(fc5_relat_1,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A) )
     => ~ v1_xboole_0(k1_relat_1(A)) ) ),
    file(relat_1,fc5_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc5_relat_1)]).

fof(fc7_relat_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_xboole_0(k1_relat_1(A))
        & v1_relat_1(k1_relat_1(A)) ) ) ),
    file(relat_1,fc7_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc7_relat_1)]).

fof(rc1_relat_1,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc1_relat_1),
    [interesting(0.9),axiom,file(relat_1,rc1_relat_1)]).

fof(rc2_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]).

fof(rc2_relat_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc2_relat_1),
    [interesting(0.9),axiom,file(relat_1,rc2_relat_1)]).

fof(t2_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    [interesting(0.9),axiom,file(subset,t2_subset)]).

fof(t6_boole,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    [interesting(0.9),axiom,file(boole,t6_boole)]).

fof(t8_boole,theorem,(
    ! [A,B] : 
      ~ ( v1_xboole_0(A)
        & A != B
        & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    [interesting(0.9),axiom,file(boole,t8_boole)]).

fof(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(redefinition_k18_pralg_1,definition,(
    ! [A,B,C] : 
      ( m1_pboole(C,k2_zfmisc_1(A,B))
     => k18_pralg_1(A,B,C) = k2_funct_4(C) ) ),
    file(pralg_1,k18_pralg_1),
    [interesting(0.9),axiom,file(pralg_1,k18_pralg_1)]).

fof(dt_k18_pralg_1,axiom,(
    ! [A,B,C] : 
      ( m1_pboole(C,k2_zfmisc_1(A,B))
     => m1_pboole(k18_pralg_1(A,B,C),k2_zfmisc_1(B,A)) ) ),
    file(pralg_1,k18_pralg_1),
    [interesting(0.9),axiom,file(pralg_1,k18_pralg_1)]).

fof(dt_k1_binop_1,axiom,(
    $true ),
    file(binop_1,k1_binop_1),
    [interesting(0.9),axiom,file(binop_1,k1_binop_1)]).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    [interesting(0.9),axiom,file(funct_1,k1_funct_1)]).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    [interesting(0.9),axiom,file(relat_1,k1_relat_1)]).

fof(dt_k2_funct_4,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k2_funct_4(A))
        & v1_funct_1(k2_funct_4(A)) ) ) ),
    file(funct_4,k2_funct_4),
    [interesting(0.9),axiom,file(funct_4,k2_funct_4)]).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    [interesting(0.9),axiom,file(tarski,k4_tarski)]).

fof(dt_c1_10__altcat_2,assumption,(
    ~ v1_xboole_0(c1_10__altcat_2) ),
    introduced(assumption,[file(altcat_2,c1_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c1_10__altcat_2)]).

fof(dt_c2_10__altcat_2,assumption,(
    ~ v1_xboole_0(c2_10__altcat_2) ),
    introduced(assumption,[file(altcat_2,c2_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c2_10__altcat_2)]).

fof(dt_c3_10__altcat_2,assumption,(
    m1_pboole(c3_10__altcat_2,k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2)) ),
    introduced(assumption,[file(altcat_2,c3_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c3_10__altcat_2)]).

fof(dt_c4_10__altcat_2,assumption,(
    m1_subset_1(c4_10__altcat_2,c1_10__altcat_2) ),
    introduced(assumption,[file(altcat_2,c4_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c4_10__altcat_2)]).

fof(dt_c5_10__altcat_2,assumption,(
    m1_subset_1(c5_10__altcat_2,c2_10__altcat_2) ),
    introduced(assumption,[file(altcat_2,c5_10__altcat_2)]),
    [interesting(0.8),axiom,file(altcat_2,c5_10__altcat_2)]).

fof(rc1_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]).

fof(t1_subset,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    [interesting(0.9),axiom,file(subset,t1_subset)]).

fof(t7_boole,theorem,(
    ! [A,B] : 
      ~ ( r2_hidden(A,B)
        & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    [interesting(0.9),axiom,file(boole,t7_boole)]).

fof(d1_binop_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : k1_binop_1(A,B,C) = k1_funct_1(A,k4_tarski(B,C)) ) ),
    file(binop_1,d1_binop_1),
    [interesting(0.9),axiom,file(binop_1,d1_binop_1)]).

fof(d5_tarski,definition,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    [interesting(0.9),axiom,file(tarski,d5_tarski)]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,t106_zfmisc_1)]).

fof(e1_10__altcat_2,plain,(
    r2_hidden(k4_tarski(c4_10__altcat_2,c5_10__altcat_2),k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c4_10__altcat_2,dt_c5_10__altcat_2])],[rc1_pboole,rc3_funct_1,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,cc1_realset1,cc2_funct_1,fc12_relat_1,fc2_altcat_1,fc4_relat_1,rc1_funct_1,rc1_realset1,rc1_relat_1,rc2_funct_1,rc2_realset1,rc2_relat_1,commutativity_k2_tarski,existence_m1_subset_1,dt_k1_tarski,dt_k2_tarski,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,fc2_subset_1,fc3_realset1,fc3_subset_1,fc4_subset_1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k2_zfmisc_1,dt_k4_tarski,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c4_10__altcat_2,dt_c5_10__altcat_2,t1_subset,t7_boole,d5_tarski,t106_zfmisc_1]),
    [interesting(0.8),file(altcat_2,e1_10__altcat_2),[file(altcat_2,e1_10__altcat_2)]]).

fof(d3_pboole,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( m1_pboole(B,A)
      <=> k1_relat_1(B) = A ) ) ),
    file(pboole,d3_pboole),
    [interesting(0.9),axiom,file(pboole,d3_pboole)]).

fof(e2_10__altcat_2,plain,(
    r2_hidden(k4_tarski(c4_10__altcat_2,c5_10__altcat_2),k1_relat_1(c3_10__altcat_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c3_10__altcat_2,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c4_10__altcat_2,dt_c5_10__altcat_2])],[cc1_pboole,rc3_pboole,rc5_funct_1,cc2_pboole,rc1_pboole,rc2_pboole,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,cc1_realset1,fc12_relat_1,fc4_relat_1,rc1_realset1,rc2_realset1,rc3_funct_1,commutativity_k2_tarski,existence_m1_subset_1,dt_k1_tarski,dt_k2_tarski,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc2_altcat_1,fc2_subset_1,fc3_realset1,fc3_subset_1,fc4_subset_1,fc5_relat_1,fc7_relat_1,rc1_relat_1,rc2_funct_1,rc2_relat_1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_pboole,dt_k1_relat_1,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_pboole,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c3_10__altcat_2,dt_c4_10__altcat_2,dt_c5_10__altcat_2,rc1_funct_1,t1_subset,t7_boole,d5_tarski,e1_10__altcat_2,d3_pboole]),
    [interesting(0.8),file(altcat_2,e2_10__altcat_2),[file(altcat_2,e2_10__altcat_2)]]).

fof(d2_funct_4,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k2_funct_4(A)
          <=> ( ! [C] : 
                  ( r2_hidden(C,k1_relat_1(B))
                <=> ? [D,E] : 
                      ( C = k4_tarski(E,D)
                      & r2_hidden(k4_tarski(D,E),k1_relat_1(A)) ) )
              & ! [C,D] : 
                  ( r2_hidden(k4_tarski(C,D),k1_relat_1(A))
                 => k1_funct_1(B,k4_tarski(D,C)) = k1_funct_1(A,k4_tarski(C,D)) ) ) ) ) ) ),
    file(funct_4,d2_funct_4),
    [interesting(0.9),axiom,file(funct_4,d2_funct_4)]).

fof(e3_10__altcat_2,plain,(
    k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),c5_10__altcat_2,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,c5_10__altcat_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c3_10__altcat_2,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c4_10__altcat_2,dt_c5_10__altcat_2])],[cc1_pboole,fc2_pboole,rc3_pboole,rc5_funct_1,cc2_pboole,rc1_pboole,rc2_pboole,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,cc1_realset1,fc12_relat_1,fc4_relat_1,rc1_realset1,rc2_realset1,rc3_funct_1,commutativity_k2_tarski,existence_m1_pboole,existence_m1_subset_1,dt_k1_tarski,dt_k2_tarski,dt_k2_zfmisc_1,dt_m1_pboole,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc2_altcat_1,fc2_subset_1,fc3_realset1,fc3_subset_1,fc4_subset_1,fc5_relat_1,fc7_relat_1,rc1_relat_1,rc2_funct_1,rc2_relat_1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k18_pralg_1,dt_k18_pralg_1,dt_k1_binop_1,dt_k1_funct_1,dt_k1_relat_1,dt_k2_funct_4,dt_k4_tarski,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c3_10__altcat_2,dt_c4_10__altcat_2,dt_c5_10__altcat_2,rc1_funct_1,t1_subset,t7_boole,d1_binop_1,d5_tarski,e2_10__altcat_2,d2_funct_4]),
    [interesting(0.8),file(altcat_2,e3_10__altcat_2),[file(altcat_2,e3_10__altcat_2)]]).

fof(i5_10__altcat_2,theorem,(
    $true ),
    introduced(tautology,[file(altcat_2,i5_10__altcat_2)]),
    [interesting(0.8),trivial,file(altcat_2,i5_10__altcat_2)]).

fof(i4_10__altcat_2,plain,(
    k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),c5_10__altcat_2,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,c5_10__altcat_2) ),
    inference(conclusion,[status(thm),assumptions([dt_c3_10__altcat_2,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c4_10__altcat_2,dt_c5_10__altcat_2])],[e3_10__altcat_2,i5_10__altcat_2]),
    [interesting(0.8),file(altcat_2,i4_10__altcat_2),[file(altcat_2,i4_10__altcat_2)]]).

fof(i4_10_tmp__altcat_2,plain,
    ( m1_subset_1(c5_10__altcat_2,c2_10__altcat_2)
   => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),c5_10__altcat_2,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,c5_10__altcat_2) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c3_10__altcat_2,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c4_10__altcat_2]),discharge_asm(discharge,[dt_c5_10__altcat_2])],[dt_c5_10__altcat_2,i4_10__altcat_2]),
    [interesting(0.8),i3_10__altcat_2]).

fof(i3_10__altcat_2,plain,(
    ! [A] : 
      ( m1_subset_1(A,c2_10__altcat_2)
     => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),A,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,A) ) ),
    inference(let,[status(thm),assumptions([dt_c3_10__altcat_2,dt_c1_10__altcat_2,dt_c2_10__altcat_2,dt_c4_10__altcat_2])],[i4_10_tmp__altcat_2,dh_c5_10__altcat_2]),
    [interesting(0.8),file(altcat_2,i3_10__altcat_2),[file(altcat_2,i3_10__altcat_2)]]).

fof(i3_10_tmp__altcat_2,plain,
    ( m1_subset_1(c4_10__altcat_2,c1_10__altcat_2)
   => ! [A] : 
        ( m1_subset_1(A,c2_10__altcat_2)
       => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),A,c4_10__altcat_2) = k1_binop_1(c3_10__altcat_2,c4_10__altcat_2,A) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c3_10__altcat_2,dt_c1_10__altcat_2,dt_c2_10__altcat_2]),discharge_asm(discharge,[dt_c4_10__altcat_2])],[dt_c4_10__altcat_2,i3_10__altcat_2]),
    [interesting(0.8),i2_10__altcat_2]).

fof(i2_10__altcat_2,plain,(
    ! [A] : 
      ( m1_subset_1(A,c1_10__altcat_2)
     => ! [B] : 
          ( m1_subset_1(B,c2_10__altcat_2)
         => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),B,A) = k1_binop_1(c3_10__altcat_2,A,B) ) ) ),
    inference(let,[status(thm),assumptions([dt_c3_10__altcat_2,dt_c1_10__altcat_2,dt_c2_10__altcat_2])],[i3_10_tmp__altcat_2,dh_c4_10__altcat_2]),
    [interesting(0.8),file(altcat_2,i2_10__altcat_2),[file(altcat_2,i2_10__altcat_2)]]).

fof(i2_10_tmp__altcat_2,plain,
    ( m1_pboole(c3_10__altcat_2,k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2))
   => ! [A] : 
        ( m1_subset_1(A,c1_10__altcat_2)
       => ! [B] : 
            ( m1_subset_1(B,c2_10__altcat_2)
           => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,c3_10__altcat_2),B,A) = k1_binop_1(c3_10__altcat_2,A,B) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_10__altcat_2,dt_c2_10__altcat_2]),discharge_asm(discharge,[dt_c3_10__altcat_2])],[dt_c3_10__altcat_2,i2_10__altcat_2]),
    [interesting(0.8),i1_10__altcat_2]).

fof(i1_10__altcat_2,plain,(
    ! [A] : 
      ( m1_pboole(A,k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2))
     => ! [B] : 
          ( m1_subset_1(B,c1_10__altcat_2)
         => ! [C] : 
              ( m1_subset_1(C,c2_10__altcat_2)
             => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,A),C,B) = k1_binop_1(A,B,C) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_10__altcat_2,dt_c2_10__altcat_2])],[i2_10_tmp__altcat_2,dh_c3_10__altcat_2]),
    [interesting(0.8),file(altcat_2,i1_10__altcat_2),[file(altcat_2,i1_10__altcat_2)]]).

fof(i1_10_tmp__altcat_2,plain,
    ( ( ~ v1_xboole_0(c1_10__altcat_2)
      & ~ v1_xboole_0(c2_10__altcat_2) )
   => ! [A] : 
        ( m1_pboole(A,k2_zfmisc_1(c1_10__altcat_2,c2_10__altcat_2))
       => ! [B] : 
            ( m1_subset_1(B,c1_10__altcat_2)
           => ! [C] : 
                ( m1_subset_1(C,c2_10__altcat_2)
               => k1_binop_1(k18_pralg_1(c1_10__altcat_2,c2_10__altcat_2,A),C,B) = k1_binop_1(A,B,C) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_10__altcat_2,dt_c2_10__altcat_2])],[dt_c1_10__altcat_2,dt_c2_10__altcat_2,i1_10__altcat_2]),
    [interesting(1),t6_altcat_2]).

fof(t6_altcat_2,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( m1_pboole(C,k2_zfmisc_1(A,B))
             => ! [D] : 
                  ( m1_subset_1(D,A)
                 => ! [E] : 
                      ( m1_subset_1(E,B)
                     => k1_binop_1(k18_pralg_1(A,B,C),E,D) = k1_binop_1(C,D,E) ) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_10_tmp__altcat_2,dh_c1_10__altcat_2,dh_c2_10__altcat_2]),
    [interesting(1),file(altcat_2,t6_altcat_2),[file(altcat_2,t6_altcat_2)]]).