% Mizar ND problem: t1_mcart_1,mcart_1,31,43 
fof(dh_c1_1__mcart_1,definition,
    ( ~ ( c1_1__mcart_1 != k1_xboole_0
        & ! [A] : 
            ~ ( r2_hidden(A,c1_1__mcart_1)
              & r1_xboole_0(A,c1_1__mcart_1) ) )
   => ! [B] : 
        ~ ( B != k1_xboole_0
          & ! [C] : 
              ~ ( r2_hidden(C,B)
                & r1_xboole_0(C,B) ) ) ),
    introduced(definition,[new_symbol(c1_1__mcart_1),file(mcart_1,c1_1__mcart_1)]),
    [interesting(0.8),axiom,file(mcart_1,c1_1__mcart_1)]).

fof(e1_1__mcart_1,assumption,(
    c1_1__mcart_1 != k1_xboole_0 ),
    introduced(assumption,[file(mcart_1,e1_1__mcart_1)]),
    [interesting(0.8),axiom,file(mcart_1,e1_1__mcart_1)]).

fof(symmetry_r1_xboole_0,theorem,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(B,A) ) ),
    file(xboole_0,r1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,r1_xboole_0)]).

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_c1_1__mcart_1,assumption,(
    $true ),
    introduced(assumption,[file(mcart_1,c1_1__mcart_1)]),
    [interesting(0.8),axiom,file(mcart_1,c1_1__mcart_1)]).

fof(dh_c3_1__mcart_1,definition,
    ( ? [A] : 
        ( r2_hidden(A,c1_1__mcart_1)
        & ! [B] : 
            ~ ( r2_hidden(B,c1_1__mcart_1)
              & r2_hidden(B,A) ) )
   => ( r2_hidden(c3_1__mcart_1,c1_1__mcart_1)
      & ! [C] : 
          ~ ( r2_hidden(C,c1_1__mcart_1)
            & r2_hidden(C,c3_1__mcart_1) ) ) ),
    introduced(definition,[new_symbol(c3_1__mcart_1),file(mcart_1,c3_1__mcart_1)]),
    [interesting(0.8),axiom,file(mcart_1,c3_1__mcart_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(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(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_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(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(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(dh_c2_1__mcart_1,definition,
    ( ? [A] : m1_subset_1(A,c1_1__mcart_1)
   => m1_subset_1(c2_1__mcart_1,c1_1__mcart_1) ),
    introduced(definition,[new_symbol(c2_1__mcart_1),file(mcart_1,c2_1__mcart_1)]),
    [interesting(0.8),axiom,file(mcart_1,c2_1__mcart_1)]).

fof(e2_1__mcart_1,plain,(
    ? [A] : m1_subset_1(A,c1_1__mcart_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__mcart_1])],[existence_m1_subset_1,dt_m1_subset_1,dt_c1_1__mcart_1]),
    [interesting(0.8),file(mcart_1,e2_1__mcart_1),[file(mcart_1,e2_1__mcart_1)]]).

fof(dt_c2_1__mcart_1,plain,(
    m1_subset_1(c2_1__mcart_1,c1_1__mcart_1) ),
    inference(consider,[status(thm),assumptions([dt_c1_1__mcart_1])],[dh_c2_1__mcart_1,e2_1__mcart_1]),
    [interesting(0.8),file(mcart_1,c2_1__mcart_1),[file(mcart_1,c2_1__mcart_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(e3_1__mcart_1,plain,(
    r2_hidden(c2_1__mcart_1,c1_1__mcart_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__mcart_1,e1_1__mcart_1])],[existence_m1_subset_1,dt_m1_subset_1,rc1_xboole_0,rc2_xboole_0,t2_subset,t8_boole,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c1_1__mcart_1,dt_c2_1__mcart_1,fc1_xboole_0,t1_subset,t6_boole,t7_boole,e1_1__mcart_1]),
    [interesting(0.8),file(mcart_1,e3_1__mcart_1),[file(mcart_1,e3_1__mcart_1)]]).

fof(t7_tarski,theorem,(
    ! [A,B] : 
      ~ ( r2_hidden(A,B)
        & ! [C] : 
            ~ ( r2_hidden(C,B)
              & ! [D] : 
                  ~ ( r2_hidden(D,B)
                    & r2_hidden(D,C) ) ) ) ),
    file(tarski,t7_tarski),
    [interesting(0.9),axiom,file(tarski,t7_tarski)]).

fof(e4_1__mcart_1,plain,(
    ? [A] : 
      ( r2_hidden(A,c1_1__mcart_1)
      & ! [B] : 
          ~ ( r2_hidden(B,c1_1__mcart_1)
            & r2_hidden(B,A) ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__mcart_1,e1_1__mcart_1])],[dt_k1_xboole_0,fc1_xboole_0,t8_boole,existence_m1_subset_1,dt_m1_subset_1,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_c1_1__mcart_1,dt_c2_1__mcart_1,t1_subset,t7_boole,e3_1__mcart_1,t7_tarski]),
    [interesting(0.8),file(mcart_1,e4_1__mcart_1),[file(mcart_1,e4_1__mcart_1)]]).

fof(dt_c3_1__mcart_1,plain,(
    $true ),
    inference(consider,[status(thm),assumptions([dt_c1_1__mcart_1,e1_1__mcart_1])],[dh_c3_1__mcart_1,e4_1__mcart_1]),
    [interesting(0.8),file(mcart_1,c3_1__mcart_1),[file(mcart_1,c3_1__mcart_1)]]).

fof(e5_1__mcart_1,plain,
    ( r2_hidden(c3_1__mcart_1,c1_1__mcart_1)
    & ! [A] : 
        ~ ( r2_hidden(A,c1_1__mcart_1)
          & r2_hidden(A,c3_1__mcart_1) ) ),
    inference(consider,[status(thm),assumptions([dt_c1_1__mcart_1,e1_1__mcart_1])],[dh_c3_1__mcart_1,e4_1__mcart_1]),
    [interesting(0.8),file(mcart_1,e5_1__mcart_1),[file(mcart_1,e5_1__mcart_1)]]).

fof(t3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : 
              ~ ( r2_hidden(C,A)
                & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,t3_xboole_0)]).

fof(e6_1__mcart_1,plain,
    ( r2_hidden(c3_1__mcart_1,c1_1__mcart_1)
    & r1_xboole_0(c3_1__mcart_1,c1_1__mcart_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__mcart_1,e1_1__mcart_1])],[dt_k1_xboole_0,fc1_xboole_0,t8_boole,existence_m1_subset_1,dt_m1_subset_1,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,symmetry_r1_xboole_0,antisymmetry_r2_hidden,dt_c1_1__mcart_1,dt_c3_1__mcart_1,t1_subset,t7_boole,e5_1__mcart_1,t3_xboole_0]),
    [interesting(0.8),file(mcart_1,e6_1__mcart_1),[file(mcart_1,e6_1__mcart_1)]]).

fof(i4_1__mcart_1,theorem,(
    $true ),
    introduced(tautology,[file(mcart_1,i4_1__mcart_1)]),
    [interesting(0.8),trivial,file(mcart_1,i4_1__mcart_1)]).

fof(i3_1__mcart_1,plain,
    ( r2_hidden(c3_1__mcart_1,c1_1__mcart_1)
    & r1_xboole_0(c3_1__mcart_1,c1_1__mcart_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_1__mcart_1,e1_1__mcart_1])],[e6_1__mcart_1,i4_1__mcart_1]),
    [interesting(0.8),file(mcart_1,i3_1__mcart_1),[file(mcart_1,i3_1__mcart_1)]]).

fof(i2_1__mcart_1,plain,(
    ? [A] : 
      ( r2_hidden(A,c1_1__mcart_1)
      & r1_xboole_0(A,c1_1__mcart_1) ) ),
    inference(take,[status(thm),assumptions([dt_c1_1__mcart_1,e1_1__mcart_1])],[symmetry_r1_xboole_0,antisymmetry_r2_hidden,dt_c1_1__mcart_1,dt_c3_1__mcart_1,i3_1__mcart_1]),
    [interesting(0.8),file(mcart_1,i2_1__mcart_1),[file(mcart_1,i2_1__mcart_1)]]).

fof(i1_1__mcart_1,plain,(
    ~ ( c1_1__mcart_1 != k1_xboole_0
      & ! [A] : 
          ~ ( r2_hidden(A,c1_1__mcart_1)
            & r1_xboole_0(A,c1_1__mcart_1) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_1__mcart_1]),discharge_asm(discharge,[e1_1__mcart_1])],[e1_1__mcart_1,i2_1__mcart_1]),
    [interesting(0.8),file(mcart_1,i1_1__mcart_1),[file(mcart_1,i1_1__mcart_1)]]).

fof(i1_1_tmp__mcart_1,plain,(
    ~ ( c1_1__mcart_1 != k1_xboole_0
      & ! [A] : 
          ~ ( r2_hidden(A,c1_1__mcart_1)
            & r1_xboole_0(A,c1_1__mcart_1) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_1__mcart_1])],[dt_c1_1__mcart_1,i1_1__mcart_1]),
    [interesting(1),t1_mcart_1]).

fof(t1_mcart_1,theorem,(
    ! [A] : 
      ~ ( A != k1_xboole_0
        & ! [B] : 
            ~ ( r2_hidden(B,A)
              & r1_xboole_0(B,A) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_1_tmp__mcart_1,dh_c1_1__mcart_1]),
    [interesting(1),file(mcart_1,t1_mcart_1),[file(mcart_1,t1_mcart_1)]]).