% Mizar ND problem: t3_afproj,afproj,68,66 
fof(dh_c1_3__afproj,definition,
    ( ( ( ~ v3_struct_0(c1_3__afproj)
        & ~ v3_realset2(c1_3__afproj)
        & v1_diraf(c1_3__afproj)
        & l1_analoaf(c1_3__afproj) )
     => ! [A] : 
          ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
         => ! [B] : 
              ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
             => ( ( ~ v3_struct_0(c1_3__afproj)
                  & ~ v3_realset2(c1_3__afproj)
                  & v1_diraf(c1_3__afproj)
                  & v2_diraf(c1_3__afproj)
                  & l1_analoaf(c1_3__afproj)
                  & v1_aff_4(A,c1_3__afproj)
                  & v1_aff_4(B,c1_3__afproj) )
               => A = B ) ) ) )
   => ! [C] : 
        ( ( ~ v3_struct_0(C)
          & ~ v3_realset2(C)
          & v1_diraf(C)
          & l1_analoaf(C) )
       => ! [D] : 
            ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
           => ! [E] : 
                ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(C)))
               => ( ( ~ v3_struct_0(C)
                    & ~ v3_realset2(C)
                    & v1_diraf(C)
                    & v2_diraf(C)
                    & l1_analoaf(C)
                    & v1_aff_4(D,C)
                    & v1_aff_4(E,C) )
                 => D = E ) ) ) ) ),
    introduced(definition,[new_symbol(c1_3__afproj),file(afproj,c1_3__afproj)]),
    [interesting(0.8),axiom,file(afproj,c1_3__afproj)]).

fof(dh_c2_3__afproj,definition,
    ( ( m1_subset_1(c2_3__afproj,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
     => ! [A] : 
          ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
         => ( ( ~ v3_struct_0(c1_3__afproj)
              & ~ v3_realset2(c1_3__afproj)
              & v1_diraf(c1_3__afproj)
              & v2_diraf(c1_3__afproj)
              & l1_analoaf(c1_3__afproj)
              & v1_aff_4(c2_3__afproj,c1_3__afproj)
              & v1_aff_4(A,c1_3__afproj) )
           => c2_3__afproj = A ) ) )
   => ! [B] : 
        ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
       => ! [C] : 
            ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
           => ( ( ~ v3_struct_0(c1_3__afproj)
                & ~ v3_realset2(c1_3__afproj)
                & v1_diraf(c1_3__afproj)
                & v2_diraf(c1_3__afproj)
                & l1_analoaf(c1_3__afproj)
                & v1_aff_4(B,c1_3__afproj)
                & v1_aff_4(C,c1_3__afproj) )
             => B = C ) ) ) ),
    introduced(definition,[new_symbol(c2_3__afproj),file(afproj,c2_3__afproj)]),
    [interesting(0.8),axiom,file(afproj,c2_3__afproj)]).

fof(dh_c3_3__afproj,definition,
    ( ( m1_subset_1(c3_3__afproj,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
     => ( ( ~ v3_struct_0(c1_3__afproj)
          & ~ v3_realset2(c1_3__afproj)
          & v1_diraf(c1_3__afproj)
          & v2_diraf(c1_3__afproj)
          & l1_analoaf(c1_3__afproj)
          & v1_aff_4(c2_3__afproj,c1_3__afproj)
          & v1_aff_4(c3_3__afproj,c1_3__afproj) )
       => c2_3__afproj = c3_3__afproj ) )
   => ! [A] : 
        ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
       => ( ( ~ v3_struct_0(c1_3__afproj)
            & ~ v3_realset2(c1_3__afproj)
            & v1_diraf(c1_3__afproj)
            & v2_diraf(c1_3__afproj)
            & l1_analoaf(c1_3__afproj)
            & v1_aff_4(c2_3__afproj,c1_3__afproj)
            & v1_aff_4(A,c1_3__afproj) )
         => c2_3__afproj = A ) ) ),
    introduced(definition,[new_symbol(c3_3__afproj),file(afproj,c3_3__afproj)]),
    [interesting(0.8),axiom,file(afproj,c3_3__afproj)]).

fof(e1_3__afproj,assumption,
    ( ~ v3_struct_0(c1_3__afproj)
    & ~ v3_realset2(c1_3__afproj)
    & v1_diraf(c1_3__afproj)
    & v2_diraf(c1_3__afproj)
    & l1_analoaf(c1_3__afproj)
    & v1_aff_4(c2_3__afproj,c1_3__afproj)
    & v1_aff_4(c3_3__afproj,c1_3__afproj) ),
    introduced(assumption,[file(afproj,e1_3__afproj)]),
    [interesting(0.8),axiom,file(afproj,e1_3__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(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

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_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(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(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_3__afproj,assumption,
    ( ~ v3_struct_0(c1_3__afproj)
    & ~ v3_realset2(c1_3__afproj)
    & v1_diraf(c1_3__afproj)
    & l1_analoaf(c1_3__afproj) ),
    introduced(assumption,[file(afproj,c1_3__afproj)]),
    [interesting(0.8),axiom,file(afproj,c1_3__afproj)]).

fof(dt_c2_3__afproj,assumption,(
    m1_subset_1(c2_3__afproj,k1_zfmisc_1(u1_struct_0(c1_3__afproj))) ),
    introduced(assumption,[file(afproj,c2_3__afproj)]),
    [interesting(0.8),axiom,file(afproj,c2_3__afproj)]).

fof(dt_c3_3__afproj,assumption,(
    m1_subset_1(c3_3__afproj,k1_zfmisc_1(u1_struct_0(c1_3__afproj))) ),
    introduced(assumption,[file(afproj,c3_3__afproj)]),
    [interesting(0.8),axiom,file(afproj,c3_3__afproj)]).

fof(t2_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)
              & v1_aff_4(B,A) )
           => B = u1_struct_0(A) ) ) ) ),
    file(afproj,t2_afproj),
    [interesting(0.9),axiom,file(afproj,t2_afproj)]).

fof(e2_3__afproj,plain,
    ( c2_3__afproj = u1_struct_0(c1_3__afproj)
    & c3_3__afproj = u1_struct_0(c1_3__afproj) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__afproj,dt_c2_3__afproj,dt_c3_3__afproj,e1_3__afproj])],[antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,existence_l1_struct_0,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,t6_boole,t7_boole,t8_boole,existence_l1_analoaf,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_analoaf,dt_m1_subset_1,dt_u1_struct_0,dt_c1_3__afproj,dt_c2_3__afproj,dt_c3_3__afproj,fc1_subset_1,t3_subset,e1_3__afproj,t2_afproj]),
    [interesting(0.8),file(afproj,e2_3__afproj),[file(afproj,e2_3__afproj)]]).

fof(e3_3__afproj,plain,(
    c2_3__afproj = c3_3__afproj ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__afproj,dt_c2_3__afproj,dt_c3_3__afproj,e1_3__afproj])],[antisymmetry_r2_hidden,dt_k1_xboole_0,fc1_xboole_0,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,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_zfmisc_1,dt_l1_analoaf,dt_l1_struct_0,dt_m1_subset_1,fc1_struct_0,fc1_subset_1,rc3_struct_0,t3_subset,dt_u1_struct_0,dt_c1_3__afproj,dt_c2_3__afproj,dt_c3_3__afproj,e2_3__afproj]),
    [interesting(0.8),file(afproj,e3_3__afproj),[file(afproj,e3_3__afproj)]]).

fof(i5_3__afproj,theorem,(
    $true ),
    introduced(tautology,[file(afproj,i5_3__afproj)]),
    [interesting(0.8),trivial,file(afproj,i5_3__afproj)]).

fof(i4_3__afproj,plain,(
    c2_3__afproj = c3_3__afproj ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3__afproj,dt_c2_3__afproj,dt_c3_3__afproj,e1_3__afproj])],[e3_3__afproj,i5_3__afproj]),
    [interesting(0.8),file(afproj,i4_3__afproj),[file(afproj,i4_3__afproj)]]).

fof(i3_3__afproj,plain,
    ( ( ~ v3_struct_0(c1_3__afproj)
      & ~ v3_realset2(c1_3__afproj)
      & v1_diraf(c1_3__afproj)
      & v2_diraf(c1_3__afproj)
      & l1_analoaf(c1_3__afproj)
      & v1_aff_4(c2_3__afproj,c1_3__afproj)
      & v1_aff_4(c3_3__afproj,c1_3__afproj) )
   => c2_3__afproj = c3_3__afproj ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__afproj,dt_c2_3__afproj,dt_c3_3__afproj]),discharge_asm(discharge,[e1_3__afproj])],[e1_3__afproj,i4_3__afproj]),
    [interesting(0.8),file(afproj,i3_3__afproj),[file(afproj,i3_3__afproj)]]).

fof(i3_3_tmp__afproj,plain,
    ( m1_subset_1(c3_3__afproj,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
   => ( ( ~ v3_struct_0(c1_3__afproj)
        & ~ v3_realset2(c1_3__afproj)
        & v1_diraf(c1_3__afproj)
        & v2_diraf(c1_3__afproj)
        & l1_analoaf(c1_3__afproj)
        & v1_aff_4(c2_3__afproj,c1_3__afproj)
        & v1_aff_4(c3_3__afproj,c1_3__afproj) )
     => c2_3__afproj = c3_3__afproj ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__afproj,dt_c2_3__afproj]),discharge_asm(discharge,[dt_c3_3__afproj])],[dt_c3_3__afproj,i3_3__afproj]),
    [interesting(0.8),i2_3__afproj]).

fof(i2_3__afproj,plain,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
     => ( ( ~ v3_struct_0(c1_3__afproj)
          & ~ v3_realset2(c1_3__afproj)
          & v1_diraf(c1_3__afproj)
          & v2_diraf(c1_3__afproj)
          & l1_analoaf(c1_3__afproj)
          & v1_aff_4(c2_3__afproj,c1_3__afproj)
          & v1_aff_4(A,c1_3__afproj) )
       => c2_3__afproj = A ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__afproj,dt_c2_3__afproj])],[i3_3_tmp__afproj,dh_c3_3__afproj]),
    [interesting(0.8),file(afproj,i2_3__afproj),[file(afproj,i2_3__afproj)]]).

fof(i2_3_tmp__afproj,plain,
    ( m1_subset_1(c2_3__afproj,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
   => ! [A] : 
        ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
       => ( ( ~ v3_struct_0(c1_3__afproj)
            & ~ v3_realset2(c1_3__afproj)
            & v1_diraf(c1_3__afproj)
            & v2_diraf(c1_3__afproj)
            & l1_analoaf(c1_3__afproj)
            & v1_aff_4(c2_3__afproj,c1_3__afproj)
            & v1_aff_4(A,c1_3__afproj) )
         => c2_3__afproj = A ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__afproj]),discharge_asm(discharge,[dt_c2_3__afproj])],[dt_c2_3__afproj,i2_3__afproj]),
    [interesting(0.8),i1_3__afproj]).

fof(i1_3__afproj,plain,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
         => ( ( ~ v3_struct_0(c1_3__afproj)
              & ~ v3_realset2(c1_3__afproj)
              & v1_diraf(c1_3__afproj)
              & v2_diraf(c1_3__afproj)
              & l1_analoaf(c1_3__afproj)
              & v1_aff_4(A,c1_3__afproj)
              & v1_aff_4(B,c1_3__afproj) )
           => A = B ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__afproj])],[i2_3_tmp__afproj,dh_c2_3__afproj]),
    [interesting(0.8),file(afproj,i1_3__afproj),[file(afproj,i1_3__afproj)]]).

fof(i1_3_tmp__afproj,plain,
    ( ( ~ v3_struct_0(c1_3__afproj)
      & ~ v3_realset2(c1_3__afproj)
      & v1_diraf(c1_3__afproj)
      & l1_analoaf(c1_3__afproj) )
   => ! [A] : 
        ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
       => ! [B] : 
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_3__afproj)))
           => ( ( ~ v3_struct_0(c1_3__afproj)
                & ~ v3_realset2(c1_3__afproj)
                & v1_diraf(c1_3__afproj)
                & v2_diraf(c1_3__afproj)
                & l1_analoaf(c1_3__afproj)
                & v1_aff_4(A,c1_3__afproj)
                & v1_aff_4(B,c1_3__afproj) )
             => A = B ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_3__afproj])],[dt_c1_3__afproj,i1_3__afproj]),
    [interesting(1),t3_afproj]).

fof(t3_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)))
         => ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( ~ v3_struct_0(A)
                  & ~ v3_realset2(A)
                  & v1_diraf(A)
                  & v2_diraf(A)
                  & l1_analoaf(A)
                  & v1_aff_4(B,A)
                  & v1_aff_4(C,A) )
               => B = C ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_3_tmp__afproj,dh_c1_3__afproj]),
    [interesting(1),file(afproj,t3_afproj),[file(afproj,t3_afproj)]]).