% Mizar ND problem: t5_relat_1,relat_1,65,35 
fof(dh_c1_6__relat_1,definition,
    ( ! [A,B,C] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,A),k4_tarski(B,C)))
   => ! [D,E,F,G] : v1_relat_1(k2_tarski(k4_tarski(D,E),k4_tarski(F,G))) ),
    introduced(definition,[new_symbol(c1_6__relat_1),file(relat_1,c1_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c1_6__relat_1)]).

fof(dh_c2_6__relat_1,definition,
    ( ! [A,B] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(A,B)))
   => ! [C,D,E] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,C),k4_tarski(D,E))) ),
    introduced(definition,[new_symbol(c2_6__relat_1),file(relat_1,c2_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c2_6__relat_1)]).

fof(dh_c3_6__relat_1,definition,
    ( ! [A] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,A)))
   => ! [B,C] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(B,C))) ),
    introduced(definition,[new_symbol(c3_6__relat_1),file(relat_1,c3_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c3_6__relat_1)]).

fof(dh_c4_6__relat_1,definition,
    ( v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1)))
   => ! [A] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,A))) ),
    introduced(definition,[new_symbol(c4_6__relat_1),file(relat_1,c4_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c4_6__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(rc1_xboole_0,theorem,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]).

fof(rc2_xboole_0,theorem,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]).

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_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__relat_1,assumption,(
    $true ),
    introduced(assumption,[file(relat_1,c1_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c1_6__relat_1)]).

fof(dt_c2_6__relat_1,assumption,(
    $true ),
    introduced(assumption,[file(relat_1,c2_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c2_6__relat_1)]).

fof(dt_c3_6__relat_1,assumption,(
    $true ),
    introduced(assumption,[file(relat_1,c3_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c3_6__relat_1)]).

fof(dt_c4_6__relat_1,assumption,(
    $true ),
    introduced(assumption,[file(relat_1,c4_6__relat_1)]),
    [interesting(0.8),axiom,file(relat_1,c4_6__relat_1)]).

fof(fc1_zfmisc_1,theorem,(
    ! [A,B] : ~ v1_xboole_0(k4_tarski(A,B)) ),
    file(zfmisc_1,fc1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,fc1_zfmisc_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(d1_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
    <=> ! [B] : 
          ~ ( r2_hidden(B,A)
            & ! [C,D] : B != k4_tarski(C,D) ) ) ),
    file(relat_1,d1_relat_1),
    [interesting(0.9),axiom,file(relat_1,d1_relat_1)]).

fof(dh_c1_6_1__relat_1,definition,
    ( ~ ( r2_hidden(c1_6_1__relat_1,k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1)))
        & ! [A,B] : c1_6_1__relat_1 != k4_tarski(A,B) )
   => ! [C] : 
        ~ ( r2_hidden(C,k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1)))
          & ! [D,E] : C != k4_tarski(D,E) ) ),
    introduced(definition,[new_symbol(c1_6_1__relat_1),file(relat_1,c1_6_1__relat_1)]),
    [interesting(0.65),axiom,file(relat_1,c1_6_1__relat_1)]).

fof(e1_6_1__relat_1,assumption,(
    r2_hidden(c1_6_1__relat_1,k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1))) ),
    introduced(assumption,[file(relat_1,e1_6_1__relat_1)]),
    [interesting(0.65),axiom,file(relat_1,e1_6_1__relat_1)]).

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

fof(fc1_xboole_0,theorem,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]).

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

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(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(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(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(dt_c1_6_1__relat_1,assumption,(
    $true ),
    introduced(assumption,[file(relat_1,c1_6_1__relat_1)]),
    [interesting(0.65),axiom,file(relat_1,c1_6_1__relat_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(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(0.9),axiom,file(tarski,d2_tarski)]).

fof(e2_6_1__relat_1,plain,
    ( c1_6_1__relat_1 = k4_tarski(c1_6__relat_1,c2_6__relat_1)
    | c1_6_1__relat_1 = k4_tarski(c3_6__relat_1,c4_6__relat_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_6__relat_1,dt_c1_6_1__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1,e1_6_1__relat_1])],[dt_k1_xboole_0,fc1_xboole_0,existence_m1_subset_1,dt_k1_tarski,dt_m1_subset_1,fc2_subset_1,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t8_boole,commutativity_k2_tarski,antisymmetry_r2_hidden,dt_k2_tarski,dt_k4_tarski,dt_c1_6__relat_1,dt_c1_6_1__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1,fc1_zfmisc_1,fc3_subset_1,t1_subset,t7_boole,d5_tarski,e1_6_1__relat_1,d2_tarski]),
    [interesting(0.65),file(relat_1,e2_6_1__relat_1),[file(relat_1,e2_6_1__relat_1)]]).

fof(e3_6_1__relat_1,plain,(
    ? [A,B] : c1_6_1__relat_1 = k4_tarski(A,B) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_6__relat_1,dt_c1_6_1__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1,e1_6_1__relat_1])],[existence_m1_subset_1,dt_m1_subset_1,t2_subset,antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,commutativity_k2_tarski,dt_k1_tarski,dt_k2_tarski,fc2_subset_1,fc3_subset_1,rc1_xboole_0,rc2_xboole_0,t6_boole,t7_boole,t8_boole,dt_k4_tarski,dt_c1_6__relat_1,dt_c1_6_1__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1,fc1_zfmisc_1,d5_tarski,e2_6_1__relat_1]),
    [interesting(0.65),file(relat_1,e3_6_1__relat_1),[file(relat_1,e3_6_1__relat_1)]]).

fof(i3_6_1__relat_1,theorem,(
    $true ),
    introduced(tautology,[file(relat_1,i3_6_1__relat_1)]),
    [interesting(0.65),trivial,file(relat_1,i3_6_1__relat_1)]).

fof(i2_6_1__relat_1,plain,(
    ? [A,B] : c1_6_1__relat_1 = k4_tarski(A,B) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_6__relat_1,dt_c1_6_1__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1,e1_6_1__relat_1])],[e3_6_1__relat_1,i3_6_1__relat_1]),
    [interesting(0.65),file(relat_1,i2_6_1__relat_1),[file(relat_1,i2_6_1__relat_1)]]).

fof(i1_6_1__relat_1,plain,(
    ~ ( r2_hidden(c1_6_1__relat_1,k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1)))
      & ! [A,B] : c1_6_1__relat_1 != k4_tarski(A,B) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__relat_1,dt_c1_6_1__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1]),discharge_asm(discharge,[e1_6_1__relat_1])],[e1_6_1__relat_1,i2_6_1__relat_1]),
    [interesting(0.65),file(relat_1,i1_6_1__relat_1),[file(relat_1,i1_6_1__relat_1)]]).

fof(i1_6_1_tmp__relat_1,plain,(
    ~ ( r2_hidden(c1_6_1__relat_1,k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1)))
      & ! [A,B] : c1_6_1__relat_1 != k4_tarski(A,B) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1]),discharge_asm(discharge,[dt_c1_6_1__relat_1])],[dt_c1_6_1__relat_1,i1_6_1__relat_1]),
    [interesting(0.8),e1_6__relat_1]).

fof(e1_6__relat_1,plain,(
    ! [A] : 
      ~ ( r2_hidden(A,k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1)))
        & ! [B,C] : A != k4_tarski(B,C) ) ),
    inference(let,[status(thm),assumptions([dt_c1_6__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1])],[i1_6_1_tmp__relat_1,dh_c1_6_1__relat_1]),
    [interesting(0.8),file(relat_1,e1_6__relat_1),[file(relat_1,e1_6__relat_1)]]).

fof(i5_6__relat_1,theorem,(
    $true ),
    introduced(tautology,[file(relat_1,i5_6__relat_1)]),
    [interesting(0.8),trivial,file(relat_1,i5_6__relat_1)]).

fof(i4_6__relat_1,plain,(
    v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1))) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_6__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1])],[rc1_relat_1,rc1_xboole_0,rc2_xboole_0,commutativity_k2_tarski,antisymmetry_r2_hidden,dt_k2_tarski,dt_k4_tarski,dt_c1_6__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1,dt_c4_6__relat_1,fc1_zfmisc_1,fc3_subset_1,d1_relat_1,e1_6__relat_1,i5_6__relat_1]),
    [interesting(0.8),file(relat_1,i4_6__relat_1),[file(relat_1,i4_6__relat_1)]]).

fof(i4_6_tmp__relat_1,plain,(
    v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,c4_6__relat_1))) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1]),discharge_asm(discharge,[dt_c4_6__relat_1])],[dt_c4_6__relat_1,i4_6__relat_1]),
    [interesting(0.8),i3_6__relat_1]).

fof(i3_6__relat_1,plain,(
    ! [A] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,A))) ),
    inference(let,[status(thm),assumptions([dt_c1_6__relat_1,dt_c2_6__relat_1,dt_c3_6__relat_1])],[i4_6_tmp__relat_1,dh_c4_6__relat_1]),
    [interesting(0.8),file(relat_1,i3_6__relat_1),[file(relat_1,i3_6__relat_1)]]).

fof(i3_6_tmp__relat_1,plain,(
    ! [A] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(c3_6__relat_1,A))) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__relat_1,dt_c2_6__relat_1]),discharge_asm(discharge,[dt_c3_6__relat_1])],[dt_c3_6__relat_1,i3_6__relat_1]),
    [interesting(0.8),i2_6__relat_1]).

fof(i2_6__relat_1,plain,(
    ! [A,B] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(A,B))) ),
    inference(let,[status(thm),assumptions([dt_c1_6__relat_1,dt_c2_6__relat_1])],[i3_6_tmp__relat_1,dh_c3_6__relat_1]),
    [interesting(0.8),file(relat_1,i2_6__relat_1),[file(relat_1,i2_6__relat_1)]]).

fof(i2_6_tmp__relat_1,plain,(
    ! [A,B] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,c2_6__relat_1),k4_tarski(A,B))) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__relat_1]),discharge_asm(discharge,[dt_c2_6__relat_1])],[dt_c2_6__relat_1,i2_6__relat_1]),
    [interesting(0.8),i1_6__relat_1]).

fof(i1_6__relat_1,plain,(
    ! [A,B,C] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,A),k4_tarski(B,C))) ),
    inference(let,[status(thm),assumptions([dt_c1_6__relat_1])],[i2_6_tmp__relat_1,dh_c2_6__relat_1]),
    [interesting(0.8),file(relat_1,i1_6__relat_1),[file(relat_1,i1_6__relat_1)]]).

fof(i1_6_tmp__relat_1,plain,(
    ! [A,B,C] : v1_relat_1(k2_tarski(k4_tarski(c1_6__relat_1,A),k4_tarski(B,C))) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_6__relat_1])],[dt_c1_6__relat_1,i1_6__relat_1]),
    [interesting(1),t5_relat_1]).

fof(t5_relat_1,theorem,(
    ! [A,B,C,D] : v1_relat_1(k2_tarski(k4_tarski(A,B),k4_tarski(C,D))) ),
    inference(let,[status(thm),assumptions([])],[i1_6_tmp__relat_1,dh_c1_6__relat_1]),
    [interesting(1),file(relat_1,t5_relat_1),[file(relat_1,t5_relat_1)]]).