% Mizar ND problem: t1_afproj,afproj,35,70 
fof(dh_c1_1__afproj,definition,
    ( ( ( ~ v3_struct_0(c1_1__afproj)
        & ~ v3_realset2(c1_1__afproj)
        & v1_diraf(c1_1__afproj)
        & l1_analoaf(c1_1__afproj) )
     => ! [A] : 
          ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_1__afproj)))
         => ( ( ~ v3_struct_0(c1_1__afproj)
              & ~ v3_realset2(c1_1__afproj)
              & v1_diraf(c1_1__afproj)
              & v2_diraf(c1_1__afproj)
              & l1_analoaf(c1_1__afproj)
              & A = u1_struct_0(c1_1__afproj) )
           => v1_aff_4(A,c1_1__afproj) ) ) )
   => ! [B] : 
        ( ( ~ v3_struct_0(B)
          & ~ v3_realset2(B)
          & v1_diraf(B)
          & l1_analoaf(B) )
       => ! [C] : 
            ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
           => ( ( ~ v3_struct_0(B)
                & ~ v3_realset2(B)
                & v1_diraf(B)
                & v2_diraf(B)
                & l1_analoaf(B)
                & C = u1_struct_0(B) )
             => v1_aff_4(C,B) ) ) ) ),
    introduced(definition,[new_symbol(c1_1__afproj),file(afproj,c1_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,c1_1__afproj)]).

fof(dh_c2_1__afproj,definition,
    ( ( m1_subset_1(c2_1__afproj,k1_zfmisc_1(u1_struct_0(c1_1__afproj)))
     => ( ( ~ v3_struct_0(c1_1__afproj)
          & ~ v3_realset2(c1_1__afproj)
          & v1_diraf(c1_1__afproj)
          & v2_diraf(c1_1__afproj)
          & l1_analoaf(c1_1__afproj)
          & c2_1__afproj = u1_struct_0(c1_1__afproj) )
       => v1_aff_4(c2_1__afproj,c1_1__afproj) ) )
   => ! [A] : 
        ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_1__afproj)))
       => ( ( ~ v3_struct_0(c1_1__afproj)
            & ~ v3_realset2(c1_1__afproj)
            & v1_diraf(c1_1__afproj)
            & v2_diraf(c1_1__afproj)
            & l1_analoaf(c1_1__afproj)
            & A = u1_struct_0(c1_1__afproj) )
         => v1_aff_4(A,c1_1__afproj) ) ) ),
    introduced(definition,[new_symbol(c2_1__afproj),file(afproj,c2_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,c2_1__afproj)]).

fof(e1_1__afproj,assumption,
    ( ~ v3_struct_0(c1_1__afproj)
    & ~ v3_realset2(c1_1__afproj)
    & v1_diraf(c1_1__afproj)
    & v2_diraf(c1_1__afproj)
    & l1_analoaf(c1_1__afproj) ),
    introduced(assumption,[file(afproj,e1_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,e1_1__afproj)]).

fof(e2_1__afproj,assumption,(
    c2_1__afproj = u1_struct_0(c1_1__afproj) ),
    introduced(assumption,[file(afproj,e2_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,e2_1__afproj)]).

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_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(t4_subset,theorem,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    [interesting(0.9),axiom,file(subset,t4_subset)]).

fof(t5_subset,theorem,(
    ! [A,B,C] : 
      ~ ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C))
        & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    [interesting(0.9),axiom,file(subset,t5_subset)]).

fof(rc1_subset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    [interesting(0.9),axiom,file(subset_1,rc1_subset_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_subset_1,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]).

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(rc5_struct_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B) ) ) ),
    file(struct_0,rc5_struct_0),
    [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]).

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(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(existence_l1_analoaf,axiom,(
    ? [A] : l1_analoaf(A) ),
    file(analoaf,l1_analoaf),
    [interesting(0.9),axiom,file(analoaf,l1_analoaf)]).

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

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_aff_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k1_aff_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ),
    file(aff_1,k1_aff_1),
    [interesting(0.9),axiom,file(aff_1,k1_aff_1)]).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]).

fof(dt_l1_analoaf,axiom,(
    ! [A] : 
      ( l1_analoaf(A)
     => l1_struct_0(A) ) ),
    file(analoaf,l1_analoaf),
    [interesting(0.9),axiom,file(analoaf,l1_analoaf)]).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

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

fof(fc1_struct_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(u1_struct_0(A)) ) ),
    file(struct_0,fc1_struct_0),
    [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]).

fof(fc1_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]).

fof(rc3_struct_0,theorem,(
    ? [A] : 
      ( l1_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc3_struct_0),
    [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]).

fof(commutativity_k2_aff_1,theorem,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k2_aff_1(A,B,C) = k2_aff_1(A,C,B) ) ),
    file(aff_1,k2_aff_1),
    [interesting(0.9),axiom,file(aff_1,k2_aff_1)]).

fof(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

fof(redefinition_k2_aff_1,definition,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => k2_aff_1(A,B,C) = k1_aff_1(A,B,C) ) ),
    file(aff_1,k2_aff_1),
    [interesting(0.9),axiom,file(aff_1,k2_aff_1)]).

fof(dt_k1_aff_4,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k1_aff_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ),
    file(aff_4,k1_aff_4),
    [interesting(0.9),axiom,file(aff_4,k1_aff_4)]).

fof(dt_k2_aff_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k2_aff_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ),
    file(aff_1,k2_aff_1),
    [interesting(0.9),axiom,file(aff_1,k2_aff_1)]).

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    [interesting(0.9),axiom,file(struct_0,u1_struct_0)]).

fof(dt_c1_1__afproj,assumption,
    ( ~ v3_struct_0(c1_1__afproj)
    & ~ v3_realset2(c1_1__afproj)
    & v1_diraf(c1_1__afproj)
    & l1_analoaf(c1_1__afproj) ),
    introduced(assumption,[file(afproj,c1_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,c1_1__afproj)]).

fof(dt_c2_1__afproj,assumption,(
    m1_subset_1(c2_1__afproj,k1_zfmisc_1(u1_struct_0(c1_1__afproj))) ),
    introduced(assumption,[file(afproj,c2_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,c2_1__afproj)]).

fof(dh_c3_1__afproj,definition,
    ( ~ ! [A] : 
          ( m1_subset_1(A,u1_struct_0(c1_1__afproj))
         => ! [B] : 
              ( m1_subset_1(B,u1_struct_0(c1_1__afproj))
             => ! [C] : 
                  ( m1_subset_1(C,u1_struct_0(c1_1__afproj))
                 => r1_aff_1(c1_1__afproj,A,B,C) ) ) )
   => ( m1_subset_1(c3_1__afproj,u1_struct_0(c1_1__afproj))
      & ~ ! [D] : 
            ( m1_subset_1(D,u1_struct_0(c1_1__afproj))
           => ! [E] : 
                ( m1_subset_1(E,u1_struct_0(c1_1__afproj))
               => r1_aff_1(c1_1__afproj,c3_1__afproj,D,E) ) ) ) ),
    introduced(definition,[new_symbol(c3_1__afproj),file(afproj,c3_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,c3_1__afproj)]).

fof(t21_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ~ ! [B] : 
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
               => ! [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                   => r1_aff_1(A,B,C,D) ) ) ) ) ),
    file(aff_1,t21_aff_1),
    [interesting(0.9),axiom,file(aff_1,t21_aff_1)]).

fof(e3_1__afproj,plain,(
    ~ ! [A] : 
        ( m1_subset_1(A,u1_struct_0(c1_1__afproj))
       => ! [B] : 
            ( m1_subset_1(B,u1_struct_0(c1_1__afproj))
           => ! [C] : 
                ( m1_subset_1(C,u1_struct_0(c1_1__afproj))
               => r1_aff_1(c1_1__afproj,A,B,C) ) ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__afproj])],[antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,t8_boole,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t7_boole,existence_l1_struct_0,dt_l1_struct_0,fc1_struct_0,rc3_struct_0,existence_l1_analoaf,existence_m1_subset_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,t21_aff_1]),
    [interesting(0.8),file(afproj,e3_1__afproj),[file(afproj,e3_1__afproj)]]).

fof(dt_c3_1__afproj,plain,(
    m1_subset_1(c3_1__afproj,u1_struct_0(c1_1__afproj)) ),
    inference(consider,[status(thm),assumptions([dt_c1_1__afproj])],[dh_c3_1__afproj,e3_1__afproj]),
    [interesting(0.8),file(afproj,c3_1__afproj),[file(afproj,c3_1__afproj)]]).

fof(dh_c4_1__afproj,definition,
    ( ~ ! [A] : 
          ( m1_subset_1(A,u1_struct_0(c1_1__afproj))
         => ! [B] : 
              ( m1_subset_1(B,u1_struct_0(c1_1__afproj))
             => r1_aff_1(c1_1__afproj,c3_1__afproj,A,B) ) )
   => ( m1_subset_1(c4_1__afproj,u1_struct_0(c1_1__afproj))
      & ~ ! [C] : 
            ( m1_subset_1(C,u1_struct_0(c1_1__afproj))
           => r1_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj,C) ) ) ),
    introduced(definition,[new_symbol(c4_1__afproj),file(afproj,c4_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,c4_1__afproj)]).

fof(dt_c4_1__afproj,plain,(
    m1_subset_1(c4_1__afproj,u1_struct_0(c1_1__afproj)) ),
    inference(consider,[status(thm),assumptions([dt_c1_1__afproj])],[dh_c3_1__afproj,dh_c4_1__afproj,e3_1__afproj]),
    [interesting(0.8),file(afproj,c4_1__afproj),[file(afproj,c4_1__afproj)]]).

fof(dh_c5_1__afproj,definition,
    ( ~ ! [A] : 
          ( m1_subset_1(A,u1_struct_0(c1_1__afproj))
         => r1_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj,A) )
   => ( m1_subset_1(c5_1__afproj,u1_struct_0(c1_1__afproj))
      & ~ r1_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj,c5_1__afproj) ) ),
    introduced(definition,[new_symbol(c5_1__afproj),file(afproj,c5_1__afproj)]),
    [interesting(0.8),axiom,file(afproj,c5_1__afproj)]).

fof(dt_c5_1__afproj,plain,(
    m1_subset_1(c5_1__afproj,u1_struct_0(c1_1__afproj)) ),
    inference(consider,[status(thm),assumptions([dt_c1_1__afproj])],[dh_c3_1__afproj,dh_c4_1__afproj,dh_c5_1__afproj,e3_1__afproj]),
    [interesting(0.8),file(afproj,c5_1__afproj),[file(afproj,c5_1__afproj)]]).

fof(t3_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    [interesting(0.9),axiom,file(subset,t3_subset)]).

fof(dh_c1_1_2__afproj,definition,
    ( ( r2_hidden(c1_1_2__afproj,c2_1__afproj)
     => r2_hidden(c1_1_2__afproj,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) )
   => ! [A] : 
        ( r2_hidden(A,c2_1__afproj)
       => r2_hidden(A,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) ) ),
    introduced(definition,[new_symbol(c1_1_2__afproj),file(afproj,c1_1_2__afproj)]),
    [interesting(0.65),axiom,file(afproj,c1_1_2__afproj)]).

fof(e1_1_2__afproj,assumption,(
    r2_hidden(c1_1_2__afproj,c2_1__afproj) ),
    introduced(assumption,[file(afproj,e1_1_2__afproj)]),
    [interesting(0.65),axiom,file(afproj,e1_1_2__afproj)]).

fof(dt_c1_1_2__afproj,assumption,(
    $true ),
    introduced(assumption,[file(afproj,c1_1_2__afproj)]),
    [interesting(0.65),axiom,file(afproj,c1_1_2__afproj)]).

fof(de_c2_1_2__afproj,definition,(
    c2_1_2__afproj = c1_1_2__afproj ),
    introduced(definition,[new_symbol(c2_1_2__afproj),file(afproj,c2_1_2__afproj)]),
    [interesting(0.65),axiom,file(afproj,c2_1_2__afproj)]).

fof(e2_1_2__afproj,plain,(
    m1_subset_1(c1_1_2__afproj,u1_struct_0(c1_1__afproj)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj])],[reflexivity_r1_tarski,dt_k1_xboole_0,fc1_xboole_0,t8_boole,existence_l1_analoaf,existence_l1_struct_0,dt_k1_zfmisc_1,dt_l1_analoaf,dt_l1_struct_0,fc1_struct_0,fc1_subset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,antisymmetry_r2_hidden,existence_m1_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,t1_subset,t7_boole,e1_1_2__afproj]),
    [interesting(0.65),file(afproj,e2_1_2__afproj),[file(afproj,e2_1_2__afproj)]]).

fof(dt_c2_1_2__afproj,plain,(
    m1_subset_1(c2_1_2__afproj,u1_struct_0(c1_1__afproj)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj])],[antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_analoaf,existence_l1_struct_0,dt_l1_analoaf,dt_l1_struct_0,fc1_struct_0,rc3_struct_0,existence_m1_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c1_1_2__afproj,de_c2_1_2__afproj,e2_1_2__afproj]),
    [interesting(0.65),file(afproj,c2_1_2__afproj),[file(afproj,c2_1_2__afproj)]]).

fof(t2_tarski,theorem,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    [interesting(0.9),axiom,file(tarski,t2_tarski)]).

fof(fraenkel_a_3_0_aff_4,definition,(
    ! [A,B,C,D] : 
      ( ( ~ v3_struct_0(B)
        & ~ v3_realset2(B)
        & v1_diraf(B)
        & l1_analoaf(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_0_aff_4(B,C,D))
      <=> ? [E] : 
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & ? [F] : 
                ( m1_subset_1(F,u1_struct_0(B))
                & r2_aff_1(B,E,F,C)
                & r2_hidden(F,D) ) ) ) ) ),
    file(aff_4,a_3_0_aff_4),
    [interesting(0.9),axiom,file(aff_4,a_3_0_aff_4)]).

fof(fraenkel_a_4_0_afproj,definition,(
    ! [A,B,C,D,E] : 
      ( ( ~ v3_struct_0(B)
        & ~ v3_realset2(B)
        & v1_diraf(B)
        & l1_analoaf(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B))
        & m1_subset_1(E,u1_struct_0(B)) )
     => ( r2_hidden(A,a_4_0_afproj(B,C,D,E))
      <=> ? [F] : 
            ( m1_subset_1(F,u1_struct_0(B))
            & A = F
            & ? [G] : 
                ( m1_subset_1(G,u1_struct_0(B))
                & r2_aff_1(B,F,G,k2_aff_1(B,C,E))
                & r2_hidden(G,k2_aff_1(B,C,D)) ) ) ) ) ),
    file(afproj,a_4_0_afproj),
    [interesting(0.9),axiom,file(afproj,a_4_0_afproj)]).

fof(dt_k2_aff_4,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => m1_subset_1(k2_aff_4(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ),
    file(aff_4,k2_aff_4),
    [interesting(0.9),axiom,file(aff_4,k2_aff_4)]).

fof(dh_c4_1_2__afproj,definition,
    ( ? [A] : 
        ( m1_subset_1(A,u1_struct_0(c1_1__afproj))
        & r2_hidden(A,k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)))
        & r2_hidden(A,k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) )
   => ( m1_subset_1(c4_1_2__afproj,u1_struct_0(c1_1__afproj))
      & r2_hidden(c4_1_2__afproj,k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)))
      & r2_hidden(c4_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) ) ),
    introduced(definition,[new_symbol(c4_1_2__afproj),file(afproj,c4_1_2__afproj)]),
    [interesting(0.65),axiom,file(afproj,c4_1_2__afproj)]).

fof(symmetry_r4_aff_1,theorem,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( r4_aff_1(A,B,C)
       => r4_aff_1(A,C,B) ) ) ),
    file(aff_1,r4_aff_1),
    [interesting(0.9),axiom,file(aff_1,r4_aff_1)]).

fof(redefinition_r4_aff_1,definition,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
     => ( r4_aff_1(A,B,C)
      <=> r3_aff_1(A,B,C) ) ) ),
    file(aff_1,r4_aff_1),
    [interesting(0.9),axiom,file(aff_1,r4_aff_1)]).

fof(e4_1__afproj,plain,(
    ~ r1_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj,c5_1__afproj) ),
    inference(consider,[status(thm),assumptions([dt_c1_1__afproj])],[dh_c3_1__afproj,dh_c4_1__afproj,dh_c5_1__afproj,e3_1__afproj]),
    [interesting(0.8),file(afproj,e4_1__afproj),[file(afproj,e4_1__afproj)]]).

fof(t16_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_aff_1(A,B,B,C)
                & r1_aff_1(A,B,C,C)
                & r1_aff_1(A,B,C,B) ) ) ) ) ),
    file(aff_1,t16_aff_1),
    [interesting(0.9),axiom,file(aff_1,t16_aff_1)]).

fof(e5_1__afproj,plain,
    ( c3_1__afproj != c4_1__afproj
    & c3_1__afproj != c5_1__afproj ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__afproj])],[antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_struct_0,dt_l1_struct_0,fc1_struct_0,rc3_struct_0,existence_l1_analoaf,existence_m1_subset_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,e4_1__afproj,t16_aff_1]),
    [interesting(0.8),file(afproj,e5_1__afproj),[file(afproj,e5_1__afproj)]]).

fof(t26_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(B,k2_aff_1(A,B,C))
                & r2_hidden(C,k2_aff_1(A,B,C)) ) ) ) ) ),
    file(aff_1,t26_aff_1),
    [interesting(0.9),axiom,file(aff_1,t26_aff_1)]).

fof(d3_aff_1,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_aff_1(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,u1_struct_0(A))
                & ? [D] : 
                    ( m1_subset_1(D,u1_struct_0(A))
                    & C != D
                    & B = k2_aff_1(A,C,D) ) ) ) ) ) ),
    file(aff_1,d3_aff_1),
    [interesting(0.9),axiom,file(aff_1,d3_aff_1)]).

fof(e6_1__afproj,plain,
    ( r2_hidden(c3_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))
    & r2_hidden(c3_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj))
    & r2_hidden(c4_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))
    & r2_hidden(c5_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj))
    & v1_aff_1(k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj),c1_1__afproj)
    & v1_aff_1(k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),c1_1__afproj) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,dt_k1_zfmisc_1,dt_k2_aff_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e5_1__afproj,t26_aff_1,d3_aff_1]),
    [interesting(0.8),file(afproj,e6_1__afproj),[file(afproj,e6_1__afproj)]]).

fof(t27_aff_4,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( v1_aff_1(C,A)
               => v1_aff_1(k2_aff_4(A,B,C),A) ) ) ) ) ),
    file(aff_4,t27_aff_4),
    [interesting(0.9),axiom,file(aff_4,t27_aff_4)]).

fof(d3_aff_4,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( v1_aff_1(C,A)
               => ! [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                   => ( D = k2_aff_4(A,B,C)
                    <=> ( r2_hidden(B,D)
                        & r4_aff_1(A,C,D) ) ) ) ) ) ) ) ),
    file(aff_4,d3_aff_4),
    [interesting(0.9),axiom,file(aff_4,d3_aff_4)]).

fof(e3_1_2__afproj,plain,
    ( v1_aff_1(k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)),c1_1__afproj)
    & r4_aff_1(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)))
    & r2_hidden(c2_1_2__afproj,k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,dt_c1_1_2__afproj,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,symmetry_r4_aff_1,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,redefinition_r4_aff_1,dt_k1_zfmisc_1,dt_k2_aff_1,dt_k2_aff_4,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c2_1_2__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,de_c2_1_2__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e6_1__afproj,t27_aff_4,d3_aff_4]),
    [interesting(0.65),file(afproj,e3_1_2__afproj),[file(afproj,e3_1_2__afproj)]]).

fof(e1_1_1__afproj,assumption,(
    r4_aff_1(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) ),
    introduced(assumption,[file(afproj,e1_1_1__afproj)]),
    [interesting(0.65),axiom,file(afproj,e1_1_1__afproj)]).

fof(t59_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ( r4_aff_1(A,C,D)
                      & r2_hidden(B,C)
                      & r2_hidden(B,D) )
                   => C = D ) ) ) ) ) ),
    file(aff_1,t59_aff_1),
    [interesting(0.9),axiom,file(aff_1,t59_aff_1)]).

fof(e2_1_1__afproj,plain,(
    r2_hidden(c5_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) ),
    inference(mizar_by,[status(thm),assumptions([e1_1_1__afproj,dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,symmetry_r4_aff_1,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,redefinition_r4_aff_1,dt_k1_zfmisc_1,dt_k2_aff_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e1_1_1__afproj,e6_1__afproj,t59_aff_1]),
    [interesting(0.65),file(afproj,e2_1_1__afproj),[file(afproj,e2_1_1__afproj)]]).

fof(t33_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( r1_aff_1(A,B,C,D)
                  <=> ? [E] : 
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                        & v1_aff_1(E,A)
                        & r2_hidden(B,E)
                        & r2_hidden(C,E)
                        & r2_hidden(D,E) ) ) ) ) ) ) ),
    file(aff_1,t33_aff_1),
    [interesting(0.9),axiom,file(aff_1,t33_aff_1)]).

fof(e3_1_1__afproj,plain,(
    ~ $true ),
    inference(mizar_by,[status(thm),assumptions([e1_1_1__afproj,dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,dt_k1_zfmisc_1,dt_k2_aff_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e2_1_1__afproj,e4_1__afproj,e6_1__afproj,t33_aff_1]),
    [interesting(0.65),file(afproj,e3_1_1__afproj),[file(afproj,e3_1_1__afproj)]]).

fof(i2_1_1__afproj,theorem,(
    $true ),
    introduced(tautology,[file(afproj,i2_1_1__afproj)]),
    [interesting(0.65),trivial,file(afproj,i2_1_1__afproj)]).

fof(i1_1_1__afproj,plain,(
    ~ $true ),
    inference(conclusion,[status(thm),assumptions([e1_1_1__afproj,dt_c1_1__afproj])],[e3_1_1__afproj,i2_1_1__afproj]),
    [interesting(0.65),file(afproj,i1_1_1__afproj),[file(afproj,i1_1_1__afproj)]]).

fof(e7_1__afproj,plain,(
    ~ r4_aff_1(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_1__afproj]),discharge_asm(discharge,[e1_1_1__afproj])],[e1_1_1__afproj,i1_1_1__afproj]),
    [interesting(0.8),file(afproj,e7_1__afproj),[file(afproj,e7_1__afproj)]]).

fof(t58_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( ~ ( ~ ( r4_aff_1(A,B,C)
                            & r4_aff_1(A,C,D) )
                        & ~ ( r4_aff_1(A,B,C)
                            & r4_aff_1(A,D,C) )
                        & ~ ( r4_aff_1(A,C,B)
                            & r4_aff_1(A,C,D) )
                        & ~ ( r4_aff_1(A,C,B)
                            & r4_aff_1(A,D,C) ) )
                   => r4_aff_1(A,B,D) ) ) ) ) ) ),
    file(aff_1,t58_aff_1),
    [interesting(0.9),axiom,file(aff_1,t58_aff_1)]).

fof(e4_1_2__afproj,plain,(
    ~ r4_aff_1(c1_1__afproj,k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,dt_c1_1_2__afproj,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,symmetry_r4_aff_1,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,redefinition_r4_aff_1,dt_k1_zfmisc_1,dt_k2_aff_1,dt_k2_aff_4,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c2_1_2__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,de_c2_1_2__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e3_1_2__afproj,e7_1__afproj,t58_aff_1]),
    [interesting(0.65),file(afproj,e4_1_2__afproj),[file(afproj,e4_1_2__afproj)]]).

fof(t72_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ~ ( v1_aff_1(B,A)
                  & v1_aff_1(C,A)
                  & ~ r4_aff_1(A,B,C)
                  & ! [D] : 
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r2_hidden(D,B)
                          & r2_hidden(D,C) ) ) ) ) ) ) ),
    file(aff_1,t72_aff_1),
    [interesting(0.9),axiom,file(aff_1,t72_aff_1)]).

fof(e5_1_2__afproj,plain,(
    ? [A] : 
      ( m1_subset_1(A,u1_struct_0(c1_1__afproj))
      & r2_hidden(A,k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)))
      & r2_hidden(A,k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) ) ),
    inference(mizar_by,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,dt_c1_1_2__afproj,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,symmetry_r4_aff_1,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,redefinition_r4_aff_1,dt_k1_zfmisc_1,dt_k2_aff_1,dt_k2_aff_4,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c2_1_2__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,de_c2_1_2__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e4_1_2__afproj,e1_1__afproj,e6_1__afproj,e3_1_2__afproj,t72_aff_1]),
    [interesting(0.65),file(afproj,e5_1_2__afproj),[file(afproj,e5_1_2__afproj)]]).

fof(dt_c4_1_2__afproj,plain,(
    m1_subset_1(c4_1_2__afproj,u1_struct_0(c1_1__afproj)) ),
    inference(consider,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[dh_c4_1_2__afproj,e5_1_2__afproj]),
    [interesting(0.65),file(afproj,c4_1_2__afproj),[file(afproj,c4_1_2__afproj)]]).

fof(e6_1_2__afproj,plain,
    ( r2_hidden(c4_1_2__afproj,k2_aff_4(c1_1__afproj,c2_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)))
    & r2_hidden(c4_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)) ),
    inference(consider,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[dh_c4_1_2__afproj,e5_1_2__afproj]),
    [interesting(0.65),file(afproj,e6_1_2__afproj),[file(afproj,e6_1_2__afproj)]]).

fof(t54_aff_1,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] : 
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [E] : 
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ( ( r2_hidden(B,D)
                          & r2_hidden(C,D)
                          & r3_aff_1(A,D,E) )
                       => r2_aff_1(A,B,C,E) ) ) ) ) ) ) ),
    file(aff_1,t54_aff_1),
    [interesting(0.9),axiom,file(aff_1,t54_aff_1)]).

fof(e7_1_2__afproj,plain,(
    r2_aff_1(c1_1__afproj,c2_1_2__afproj,c4_1_2__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj)) ),
    inference(mizar_by,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,dt_c1_1_2__afproj,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,symmetry_r4_aff_1,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,redefinition_r4_aff_1,dt_k1_zfmisc_1,dt_k2_aff_1,dt_k2_aff_4,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c2_1_2__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c4_1_2__afproj,dt_c5_1__afproj,de_c2_1_2__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e3_1_2__afproj,e6_1_2__afproj,t54_aff_1]),
    [interesting(0.65),file(afproj,e7_1_2__afproj),[file(afproj,e7_1_2__afproj)]]).

fof(e8_1_2__afproj,plain,(
    r2_hidden(c2_1_2__afproj,a_4_0_afproj(c1_1__afproj,c3_1__afproj,c4_1__afproj,c5_1__afproj)) ),
    inference(mizar_by,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_xboole_0,dt_l1_struct_0,fc1_struct_0,fc1_xboole_0,rc3_struct_0,rc5_struct_0,existence_l1_analoaf,existence_m1_subset_1,dt_k1_aff_1,dt_k1_zfmisc_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1_2__afproj,fc1_subset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,redefinition_k2_aff_1,dt_k2_aff_1,dt_k2_aff_4,dt_c1_1__afproj,dt_c2_1_2__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c4_1_2__afproj,dt_c5_1__afproj,de_c2_1_2__afproj,t1_subset,t7_boole,t2_tarski,fraenkel_a_4_0_afproj,e7_1_2__afproj,e6_1_2__afproj]),
    [interesting(0.65),file(afproj,e8_1_2__afproj),[file(afproj,e8_1_2__afproj)]]).

fof(d1_aff_4,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k1_aff_4(A,B,C) = a_3_0_aff_4(A,B,C) ) ) ) ),
    file(aff_4,d1_aff_4),
    [interesting(0.9),axiom,file(aff_4,d1_aff_4)]).

fof(e9_1_2__afproj,plain,(
    r2_hidden(c1_1_2__afproj,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) ),
    inference(mizar_by,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,dt_k1_aff_4,dt_k1_zfmisc_1,dt_k2_aff_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c1_1_2__afproj,dt_c2_1_2__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,de_c2_1_2__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,t2_tarski,fraenkel_a_3_0_aff_4,fraenkel_a_4_0_afproj,e8_1_2__afproj,d1_aff_4]),
    [interesting(0.65),file(afproj,e9_1_2__afproj),[file(afproj,e9_1_2__afproj)]]).

fof(i3_1_2__afproj,theorem,(
    $true ),
    introduced(tautology,[file(afproj,i3_1_2__afproj)]),
    [interesting(0.65),trivial,file(afproj,i3_1_2__afproj)]).

fof(i2_1_2__afproj,plain,(
    r2_hidden(c1_1_2__afproj,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) ),
    inference(conclusion,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,e1_1_2__afproj,dt_c1_1__afproj])],[e9_1_2__afproj,i3_1_2__afproj]),
    [interesting(0.65),file(afproj,i2_1_2__afproj),[file(afproj,i2_1_2__afproj)]]).

fof(i1_1_2__afproj,plain,
    ( r2_hidden(c1_1_2__afproj,c2_1__afproj)
   => r2_hidden(c1_1_2__afproj,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) ),
    inference(discharge_asm,[status(thm),assumptions([e1_1__afproj,dt_c1_1_2__afproj,dt_c2_1__afproj,dt_c1_1__afproj]),discharge_asm(discharge,[e1_1_2__afproj])],[e1_1_2__afproj,i2_1_2__afproj]),
    [interesting(0.65),file(afproj,i1_1_2__afproj),[file(afproj,i1_1_2__afproj)]]).

fof(i1_1_2_tmp__afproj,plain,
    ( r2_hidden(c1_1_2__afproj,c2_1__afproj)
   => r2_hidden(c1_1_2__afproj,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) ),
    inference(discharge_asm,[status(thm),assumptions([e1_1__afproj,dt_c2_1__afproj,dt_c1_1__afproj]),discharge_asm(discharge,[dt_c1_1_2__afproj])],[dt_c1_1_2__afproj,i1_1_2__afproj]),
    [interesting(0.8),e9_1__afproj]).

fof(e9_1__afproj,plain,(
    ! [A] : 
      ( r2_hidden(A,c2_1__afproj)
     => r2_hidden(A,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) ) ),
    inference(let,[status(thm),assumptions([e1_1__afproj,dt_c2_1__afproj,dt_c1_1__afproj])],[i1_1_2_tmp__afproj,dh_c1_1_2__afproj]),
    [interesting(0.8),file(afproj,e9_1__afproj),[file(afproj,e9_1__afproj)]]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.9),axiom,file(tarski,d3_tarski)]).

fof(e10_1__afproj,plain,(
    r1_tarski(c2_1__afproj,k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj))) ),
    inference(mizar_by,[status(thm),assumptions([e1_1__afproj,dt_c2_1__afproj,dt_c1_1__afproj])],[existence_l1_struct_0,dt_k1_xboole_0,dt_l1_struct_0,fc1_struct_0,fc1_xboole_0,rc3_struct_0,rc5_struct_0,existence_l1_analoaf,existence_m1_subset_1,dt_k1_aff_1,dt_k1_zfmisc_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,fc1_subset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k2_aff_1,dt_k1_aff_4,dt_k2_aff_1,dt_c1_1__afproj,dt_c2_1__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,t1_subset,t3_subset,t7_boole,e9_1__afproj,d3_tarski]),
    [interesting(0.8),file(afproj,e10_1__afproj),[file(afproj,e10_1__afproj)]]).

fof(d2_aff_4,definition,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v1_aff_4(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                & ? [D] : 
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                    & v1_aff_1(C,A)
                    & v1_aff_1(D,A)
                    & ~ r4_aff_1(A,C,D)
                    & B = k1_aff_4(A,C,D) ) ) ) ) ) ),
    file(aff_4,d2_aff_4),
    [interesting(0.9),axiom,file(aff_4,d2_aff_4)]).

fof(e8_1__afproj,plain,(
    v1_aff_4(k1_aff_4(c1_1__afproj,k2_aff_1(c1_1__afproj,c3_1__afproj,c5_1__afproj),k2_aff_1(c1_1__afproj,c3_1__afproj,c4_1__afproj)),c1_1__afproj) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__afproj])],[dt_k1_xboole_0,fc1_xboole_0,reflexivity_r1_tarski,existence_l1_struct_0,dt_k1_aff_1,dt_l1_struct_0,fc1_struct_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_aff_1,antisymmetry_r2_hidden,symmetry_r4_aff_1,existence_l1_analoaf,existence_m1_subset_1,redefinition_k2_aff_1,redefinition_r4_aff_1,dt_k1_aff_4,dt_k1_zfmisc_1,dt_k2_aff_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,fc1_subset_1,t1_subset,t3_subset,t4_subset,t7_boole,e6_1__afproj,e7_1__afproj,d2_aff_4]),
    [interesting(0.8),file(afproj,e8_1__afproj),[file(afproj,e8_1__afproj)]]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]).

fof(e11_1__afproj,plain,(
    v1_aff_4(c2_1__afproj,c1_1__afproj) ),
    inference(mizar_by,[status(thm),assumptions([e1_1__afproj,dt_c2_1__afproj,e2_1__afproj,dt_c1_1__afproj])],[antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,t4_subset,t5_subset,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_analoaf,existence_l1_struct_0,existence_m1_subset_1,dt_k1_aff_1,dt_k1_zfmisc_1,dt_l1_analoaf,dt_l1_struct_0,dt_m1_subset_1,fc1_struct_0,fc1_subset_1,rc3_struct_0,commutativity_k2_aff_1,reflexivity_r1_tarski,redefinition_k2_aff_1,dt_k1_aff_4,dt_k2_aff_1,dt_u1_struct_0,dt_c1_1__afproj,dt_c2_1__afproj,dt_c3_1__afproj,dt_c4_1__afproj,dt_c5_1__afproj,t3_subset,e10_1__afproj,e2_1__afproj,e8_1__afproj,d10_xboole_0]),
    [interesting(0.8),file(afproj,e11_1__afproj),[file(afproj,e11_1__afproj)]]).

fof(i4_1__afproj,theorem,(
    $true ),
    introduced(tautology,[file(afproj,i4_1__afproj)]),
    [interesting(0.8),trivial,file(afproj,i4_1__afproj)]).

fof(i3_1__afproj,plain,(
    v1_aff_4(c2_1__afproj,c1_1__afproj) ),
    inference(conclusion,[status(thm),assumptions([e1_1__afproj,dt_c2_1__afproj,e2_1__afproj,dt_c1_1__afproj])],[e11_1__afproj,i4_1__afproj]),
    [interesting(0.8),file(afproj,i3_1__afproj),[file(afproj,i3_1__afproj)]]).

fof(i2_1__afproj,plain,
    ( ( ~ v3_struct_0(c1_1__afproj)
      & ~ v3_realset2(c1_1__afproj)
      & v1_diraf(c1_1__afproj)
      & v2_diraf(c1_1__afproj)
      & l1_analoaf(c1_1__afproj)
      & c2_1__afproj = u1_struct_0(c1_1__afproj) )
   => v1_aff_4(c2_1__afproj,c1_1__afproj) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_1__afproj,dt_c1_1__afproj]),discharge_asm(discharge,[e1_1__afproj,e2_1__afproj])],[e1_1__afproj,e2_1__afproj,i3_1__afproj]),
    [interesting(0.8),file(afproj,i2_1__afproj),[file(afproj,i2_1__afproj)]]).

fof(i2_1_tmp__afproj,plain,
    ( m1_subset_1(c2_1__afproj,k1_zfmisc_1(u1_struct_0(c1_1__afproj)))
   => ( ( ~ v3_struct_0(c1_1__afproj)
        & ~ v3_realset2(c1_1__afproj)
        & v1_diraf(c1_1__afproj)
        & v2_diraf(c1_1__afproj)
        & l1_analoaf(c1_1__afproj)
        & c2_1__afproj = u1_struct_0(c1_1__afproj) )
     => v1_aff_4(c2_1__afproj,c1_1__afproj) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_1__afproj]),discharge_asm(discharge,[dt_c2_1__afproj])],[dt_c2_1__afproj,i2_1__afproj]),
    [interesting(0.8),i1_1__afproj]).

fof(i1_1__afproj,plain,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_1__afproj)))
     => ( ( ~ v3_struct_0(c1_1__afproj)
          & ~ v3_realset2(c1_1__afproj)
          & v1_diraf(c1_1__afproj)
          & v2_diraf(c1_1__afproj)
          & l1_analoaf(c1_1__afproj)
          & A = u1_struct_0(c1_1__afproj) )
       => v1_aff_4(A,c1_1__afproj) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_1__afproj])],[i2_1_tmp__afproj,dh_c2_1__afproj]),
    [interesting(0.8),file(afproj,i1_1__afproj),[file(afproj,i1_1__afproj)]]).

fof(i1_1_tmp__afproj,plain,
    ( ( ~ v3_struct_0(c1_1__afproj)
      & ~ v3_realset2(c1_1__afproj)
      & v1_diraf(c1_1__afproj)
      & l1_analoaf(c1_1__afproj) )
   => ! [A] : 
        ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_1__afproj)))
       => ( ( ~ v3_struct_0(c1_1__afproj)
            & ~ v3_realset2(c1_1__afproj)
            & v1_diraf(c1_1__afproj)
            & v2_diraf(c1_1__afproj)
            & l1_analoaf(c1_1__afproj)
            & A = u1_struct_0(c1_1__afproj) )
         => v1_aff_4(A,c1_1__afproj) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_1__afproj])],[dt_c1_1__afproj,i1_1__afproj]),
    [interesting(1),t1_afproj]).

fof(t1_afproj,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & l1_analoaf(A) )
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( ( ~ v3_struct_0(A)
              & ~ v3_realset2(A)
              & v1_diraf(A)
              & v2_diraf(A)
              & l1_analoaf(A)
              & B = u1_struct_0(A) )
           => v1_aff_4(B,A) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_1_tmp__afproj,dh_c1_1__afproj]),
    [interesting(1),file(afproj,t1_afproj),[file(afproj,t1_afproj)]]).