% Mizar ND problem: t4_classes2,classes2,101,55 
fof(dh_c1_6__classes2,definition,
    ( ! [A,B] : 
        ( ( v2_classes1(c1_6__classes2)
          & r2_hidden(A,c1_6__classes2)
          & r2_hidden(B,c1_6__classes2) )
       => r2_hidden(k4_tarski(A,B),c1_6__classes2) )
   => ! [C,D,E] : 
        ( ( v2_classes1(C)
          & r2_hidden(D,C)
          & r2_hidden(E,C) )
       => r2_hidden(k4_tarski(D,E),C) ) ),
    introduced(definition,[new_symbol(c1_6__classes2),file(classes2,c1_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,c1_6__classes2)]).

fof(dh_c2_6__classes2,definition,
    ( ! [A] : 
        ( ( v2_classes1(c1_6__classes2)
          & r2_hidden(c2_6__classes2,c1_6__classes2)
          & r2_hidden(A,c1_6__classes2) )
       => r2_hidden(k4_tarski(c2_6__classes2,A),c1_6__classes2) )
   => ! [B,C] : 
        ( ( v2_classes1(c1_6__classes2)
          & r2_hidden(B,c1_6__classes2)
          & r2_hidden(C,c1_6__classes2) )
       => r2_hidden(k4_tarski(B,C),c1_6__classes2) ) ),
    introduced(definition,[new_symbol(c2_6__classes2),file(classes2,c2_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,c2_6__classes2)]).

fof(dh_c3_6__classes2,definition,
    ( ( ( v2_classes1(c1_6__classes2)
        & r2_hidden(c2_6__classes2,c1_6__classes2)
        & r2_hidden(c3_6__classes2,c1_6__classes2) )
     => r2_hidden(k4_tarski(c2_6__classes2,c3_6__classes2),c1_6__classes2) )
   => ! [A] : 
        ( ( v2_classes1(c1_6__classes2)
          & r2_hidden(c2_6__classes2,c1_6__classes2)
          & r2_hidden(A,c1_6__classes2) )
       => r2_hidden(k4_tarski(c2_6__classes2,A),c1_6__classes2) ) ),
    introduced(definition,[new_symbol(c3_6__classes2),file(classes2,c3_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,c3_6__classes2)]).

fof(e1_6__classes2,assumption,(
    v2_classes1(c1_6__classes2) ),
    introduced(assumption,[file(classes2,e1_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,e1_6__classes2)]).

fof(e2_6__classes2,assumption,
    ( r2_hidden(c2_6__classes2,c1_6__classes2)
    & r2_hidden(c3_6__classes2,c1_6__classes2) ),
    introduced(assumption,[file(classes2,e2_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,e2_6__classes2)]).

fof(rc1_partfun1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A) ) ),
    file(partfun1,rc1_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc1_partfun1)]).

fof(rc2_ordinal1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc2_ordinal1)]).

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_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A) ) ) ),
    file(ordinal1,cc1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc1_ordinal1)]).

fof(cc2_ordinal1,theorem,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_ordinal1(A) )
     => v3_ordinal1(A) ) ),
    file(ordinal1,cc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc2_ordinal1)]).

fof(fc2_ordinal1,theorem,
    ( v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_xboole_0(k1_xboole_0)
    & v1_ordinal1(k1_xboole_0)
    & v2_ordinal1(k1_xboole_0)
    & v3_ordinal1(k1_xboole_0) ),
    file(ordinal1,fc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,fc2_ordinal1)]).

fof(rc1_ordinal1,theorem,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc1_ordinal1)]).

fof(rc3_ordinal1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc3_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc3_ordinal1)]).

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_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(cc3_ordinal1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(ordinal1,cc3_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc3_ordinal1)]).

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(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(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(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_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    [interesting(0.9),axiom,file(tarski,k4_tarski)]).

fof(dt_c1_6__classes2,assumption,(
    $true ),
    introduced(assumption,[file(classes2,c1_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,c1_6__classes2)]).

fof(dt_c2_6__classes2,assumption,(
    $true ),
    introduced(assumption,[file(classes2,c2_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,c2_6__classes2)]).

fof(dt_c3_6__classes2,assumption,(
    $true ),
    introduced(assumption,[file(classes2,c3_6__classes2)]),
    [interesting(0.8),axiom,file(classes2,c3_6__classes2)]).

fof(cc1_classes2,theorem,(
    ! [A] : 
      ( v2_classes1(A)
     => v1_classes1(A) ) ),
    file(classes2,cc1_classes2),
    [interesting(0.9),axiom,file(classes2,cc1_classes2)]).

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_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(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(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(t3_classes2,theorem,(
    ! [A,B,C] : 
      ( ( v2_classes1(A)
        & r2_hidden(B,A)
        & r2_hidden(C,A) )
     => ( r2_hidden(k1_tarski(B),A)
        & r2_hidden(k2_tarski(B,C),A) ) ) ),
    file(classes2,t3_classes2),
    [interesting(0.9),axiom,file(classes2,t3_classes2)]).

fof(e3_6__classes2,plain,
    ( r2_hidden(k1_tarski(c2_6__classes2),c1_6__classes2)
    & r2_hidden(k2_tarski(c2_6__classes2,c3_6__classes2),c1_6__classes2)
    & k4_tarski(c2_6__classes2,c3_6__classes2) = k2_tarski(k2_tarski(c2_6__classes2,c3_6__classes2),k1_tarski(c2_6__classes2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_6__classes2,dt_c2_6__classes2,dt_c3_6__classes2,e2_6__classes2,e1_6__classes2])],[rc1_partfun1,rc2_ordinal1,dt_k1_xboole_0,cc1_ordinal1,cc2_ordinal1,fc2_ordinal1,rc1_ordinal1,rc3_ordinal1,existence_m1_subset_1,dt_m1_subset_1,cc3_ordinal1,t2_subset,t6_boole,t8_boole,commutativity_k2_tarski,antisymmetry_r2_hidden,dt_k1_tarski,dt_k2_tarski,dt_k4_tarski,dt_c1_6__classes2,dt_c2_6__classes2,dt_c3_6__classes2,cc1_classes2,fc2_subset_1,fc3_subset_1,t1_subset,t7_boole,d5_tarski,e2_6__classes2,e1_6__classes2,t3_classes2]),
    [interesting(0.8),file(classes2,e3_6__classes2),[file(classes2,e3_6__classes2)]]).

fof(e4_6__classes2,plain,(
    r2_hidden(k4_tarski(c2_6__classes2,c3_6__classes2),c1_6__classes2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_6__classes2,dt_c2_6__classes2,dt_c3_6__classes2,e2_6__classes2,e1_6__classes2])],[rc1_partfun1,rc2_ordinal1,dt_k1_xboole_0,cc1_ordinal1,cc2_ordinal1,fc2_ordinal1,rc1_ordinal1,rc3_ordinal1,existence_m1_subset_1,dt_m1_subset_1,cc3_ordinal1,t2_subset,t6_boole,t8_boole,commutativity_k2_tarski,antisymmetry_r2_hidden,dt_k1_tarski,dt_k2_tarski,dt_k4_tarski,dt_c1_6__classes2,dt_c2_6__classes2,dt_c3_6__classes2,cc1_classes2,fc2_subset_1,fc3_subset_1,t1_subset,t7_boole,d5_tarski,e3_6__classes2,e1_6__classes2,t3_classes2]),
    [interesting(0.8),file(classes2,e4_6__classes2),[file(classes2,e4_6__classes2)]]).

fof(i6_6__classes2,theorem,(
    $true ),
    introduced(tautology,[file(classes2,i6_6__classes2)]),
    [interesting(0.8),trivial,file(classes2,i6_6__classes2)]).

fof(i5_6__classes2,plain,(
    r2_hidden(k4_tarski(c2_6__classes2,c3_6__classes2),c1_6__classes2) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_6__classes2,dt_c2_6__classes2,dt_c3_6__classes2,e2_6__classes2,e1_6__classes2])],[e4_6__classes2,i6_6__classes2]),
    [interesting(0.8),file(classes2,i5_6__classes2),[file(classes2,i5_6__classes2)]]).

fof(i4_6__classes2,plain,(
    ~ ( r2_hidden(c2_6__classes2,c1_6__classes2)
      & r2_hidden(c3_6__classes2,c1_6__classes2)
      & ~ r2_hidden(k4_tarski(c2_6__classes2,c3_6__classes2),c1_6__classes2) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__classes2,dt_c2_6__classes2,dt_c3_6__classes2,e1_6__classes2]),discharge_asm(discharge,[e2_6__classes2])],[e2_6__classes2,i5_6__classes2]),
    [interesting(0.8),file(classes2,i4_6__classes2),[file(classes2,i4_6__classes2)]]).

fof(i3_6__classes2,plain,
    ( ( v2_classes1(c1_6__classes2)
      & r2_hidden(c2_6__classes2,c1_6__classes2)
      & r2_hidden(c3_6__classes2,c1_6__classes2) )
   => r2_hidden(k4_tarski(c2_6__classes2,c3_6__classes2),c1_6__classes2) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__classes2,dt_c2_6__classes2,dt_c3_6__classes2]),discharge_asm(discharge,[e1_6__classes2])],[e1_6__classes2,i4_6__classes2]),
    [interesting(0.8),file(classes2,i3_6__classes2),[file(classes2,i3_6__classes2)]]).

fof(i3_6_tmp__classes2,plain,
    ( ( v2_classes1(c1_6__classes2)
      & r2_hidden(c2_6__classes2,c1_6__classes2)
      & r2_hidden(c3_6__classes2,c1_6__classes2) )
   => r2_hidden(k4_tarski(c2_6__classes2,c3_6__classes2),c1_6__classes2) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__classes2,dt_c2_6__classes2]),discharge_asm(discharge,[dt_c3_6__classes2])],[dt_c3_6__classes2,i3_6__classes2]),
    [interesting(0.8),i2_6__classes2]).

fof(i2_6__classes2,plain,(
    ! [A] : 
      ( ( v2_classes1(c1_6__classes2)
        & r2_hidden(c2_6__classes2,c1_6__classes2)
        & r2_hidden(A,c1_6__classes2) )
     => r2_hidden(k4_tarski(c2_6__classes2,A),c1_6__classes2) ) ),
    inference(let,[status(thm),assumptions([dt_c1_6__classes2,dt_c2_6__classes2])],[i3_6_tmp__classes2,dh_c3_6__classes2]),
    [interesting(0.8),file(classes2,i2_6__classes2),[file(classes2,i2_6__classes2)]]).

fof(i2_6_tmp__classes2,plain,(
    ! [A] : 
      ( ( v2_classes1(c1_6__classes2)
        & r2_hidden(c2_6__classes2,c1_6__classes2)
        & r2_hidden(A,c1_6__classes2) )
     => r2_hidden(k4_tarski(c2_6__classes2,A),c1_6__classes2) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__classes2]),discharge_asm(discharge,[dt_c2_6__classes2])],[dt_c2_6__classes2,i2_6__classes2]),
    [interesting(0.8),i1_6__classes2]).

fof(i1_6__classes2,plain,(
    ! [A,B] : 
      ( ( v2_classes1(c1_6__classes2)
        & r2_hidden(A,c1_6__classes2)
        & r2_hidden(B,c1_6__classes2) )
     => r2_hidden(k4_tarski(A,B),c1_6__classes2) ) ),
    inference(let,[status(thm),assumptions([dt_c1_6__classes2])],[i2_6_tmp__classes2,dh_c2_6__classes2]),
    [interesting(0.8),file(classes2,i1_6__classes2),[file(classes2,i1_6__classes2)]]).

fof(i1_6_tmp__classes2,plain,(
    ! [A,B] : 
      ( ( v2_classes1(c1_6__classes2)
        & r2_hidden(A,c1_6__classes2)
        & r2_hidden(B,c1_6__classes2) )
     => r2_hidden(k4_tarski(A,B),c1_6__classes2) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_6__classes2])],[dt_c1_6__classes2,i1_6__classes2]),
    [interesting(1),t4_classes2]).

fof(t4_classes2,theorem,(
    ! [A,B,C] : 
      ( ( v2_classes1(A)
        & r2_hidden(B,A)
        & r2_hidden(C,A) )
     => r2_hidden(k4_tarski(B,C),A) ) ),
    inference(let,[status(thm),assumptions([])],[i1_6_tmp__classes2,dh_c1_6__classes2]),
    [interesting(1),file(classes2,t4_classes2),[file(classes2,t4_classes2)]]).