% Mizar ND problem: t2_xreal_1,xreal_1,76,29 
fof(dh_c1_2__xreal_1,definition,
    ( ( v1_xreal_0(c1_2__xreal_1)
     => ! [A] : 
          ( v1_xreal_0(A)
         => ! [B] : 
              ( v1_xreal_0(B)
             => ( ( r1_xreal_0(c1_2__xreal_1,A)
                  & r1_xreal_0(A,B) )
               => r1_xreal_0(c1_2__xreal_1,B) ) ) ) )
   => ! [C] : 
        ( v1_xreal_0(C)
       => ! [D] : 
            ( v1_xreal_0(D)
           => ! [E] : 
                ( v1_xreal_0(E)
               => ( ( r1_xreal_0(C,D)
                    & r1_xreal_0(D,E) )
                 => r1_xreal_0(C,E) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_2__xreal_1),file(xreal_1,c1_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,c1_2__xreal_1)]).

fof(dh_c2_2__xreal_1,definition,
    ( ( v1_xreal_0(c2_2__xreal_1)
     => ! [A] : 
          ( v1_xreal_0(A)
         => ( ( r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1)
              & r1_xreal_0(c2_2__xreal_1,A) )
           => r1_xreal_0(c1_2__xreal_1,A) ) ) )
   => ! [B] : 
        ( v1_xreal_0(B)
       => ! [C] : 
            ( v1_xreal_0(C)
           => ( ( r1_xreal_0(c1_2__xreal_1,B)
                & r1_xreal_0(B,C) )
             => r1_xreal_0(c1_2__xreal_1,C) ) ) ) ),
    introduced(definition,[new_symbol(c2_2__xreal_1),file(xreal_1,c2_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,c2_2__xreal_1)]).

fof(dh_c3_2__xreal_1,definition,
    ( ( v1_xreal_0(c3_2__xreal_1)
     => ( ( r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1)
          & r1_xreal_0(c2_2__xreal_1,c3_2__xreal_1) )
       => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ) )
   => ! [A] : 
        ( v1_xreal_0(A)
       => ( ( r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1)
            & r1_xreal_0(c2_2__xreal_1,A) )
         => r1_xreal_0(c1_2__xreal_1,A) ) ) ),
    introduced(definition,[new_symbol(c3_2__xreal_1),file(xreal_1,c3_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,c3_2__xreal_1)]).

fof(e1_2__xreal_1,assumption,(
    r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1) ),
    introduced(assumption,[file(xreal_1,e1_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,e1_2__xreal_1)]).

fof(e2_2__xreal_1,assumption,(
    r1_xreal_0(c2_2__xreal_1,c3_2__xreal_1) ),
    introduced(assumption,[file(xreal_1,e2_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,e2_2__xreal_1)]).

fof(e1_2_1_1__xreal_1,assumption,(
    r2_hidden(c1_2__xreal_1,k2_arytm_2) ),
    introduced(assumption,[file(xreal_1,e1_2_1_1__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e1_2_1_1__xreal_1)]).

fof(e2_2_1_1__xreal_1,assumption,(
    r2_hidden(c2_2__xreal_1,k2_arytm_2) ),
    introduced(assumption,[file(xreal_1,e2_2_1_1__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e2_2_1_1__xreal_1)]).

fof(e3_2_1_1__xreal_1,assumption,(
    r2_hidden(c3_2__xreal_1,k2_arytm_2) ),
    introduced(assumption,[file(xreal_1,e3_2_1_1__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e3_2_1_1__xreal_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(cc1_arytm_3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_ordinal1(B)
            & v2_ordinal1(B)
            & v3_ordinal1(B) ) ) ) ),
    file(arytm_3,cc1_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]).

fof(cc1_xreal_0,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => v1_xreal_0(A) ) ),
    file(xreal_0,cc1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]).

fof(cc2_arytm_3,theorem,(
    ! [A] : 
      ( ( v1_xboole_0(A)
        & v3_ordinal1(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A) ) ) ),
    file(arytm_3,cc2_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]).

fof(cc2_xcmplx_0,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => v1_xcmplx_0(A) ) ),
    file(xcmplx_0,cc2_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,cc2_xcmplx_0)]).

fof(rc1_arytm_3,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A) ) ),
    file(arytm_3,rc1_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]).

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

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_k5_ordinal2,axiom,(
    $true ),
    file(ordinal2,k5_ordinal2),
    [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]).

fof(cc3_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_ordinal2)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A) ) ) ),
    file(arytm_3,cc3_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]).

fof(cc4_xreal_0,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v2_xreal_0(A) ) ) ),
    file(xreal_0,cc4_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]).

fof(cc5_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xreal_0(A)
        & v3_xreal_0(A) )
     => ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A) ) ) ),
    file(xreal_0,cc5_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]).

fof(cc8_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(xreal_0,cc8_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]).

fof(fc1_ordinal2,theorem,
    ( v1_ordinal1(k5_ordinal2)
    & v2_ordinal1(k5_ordinal2)
    & v3_ordinal1(k5_ordinal2)
    & ~ v1_xboole_0(k5_ordinal2) ),
    file(ordinal2,fc1_ordinal2),
    [interesting(0.9),axiom,file(ordinal2,fc1_ordinal2)]).

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(rc2_xreal_0,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v2_xreal_0(A)
      & ~ v3_xreal_0(A) ) ),
    file(xreal_0,rc2_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]).

fof(rc3_xreal_0,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A)
      & v3_xreal_0(A) ) ),
    file(xreal_0,rc3_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]).

fof(rc4_xreal_0,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A)
      & ~ v3_xreal_0(A) ) ),
    file(xreal_0,rc4_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]).

fof(t2_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(A,B)
              & v3_xreal_0(B) )
           => v3_xreal_0(A) ) ) ) ),
    file(real,t2_real),
    [interesting(0.9),axiom,file(real,t2_real)]).

fof(t3_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( r1_xreal_0(A,B)
              & ~ v3_xreal_0(A)
              & v3_xreal_0(B) ) ) ) ),
    file(real,t3_real),
    [interesting(0.9),axiom,file(real,t3_real)]).

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(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_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ( v1_xboole_0(B)
              | v3_xreal_0(A)
              | v2_xreal_0(B) ) ) ) ) ),
    file(real,t5_real),
    [interesting(0.9),axiom,file(real,t5_real)]).

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(t6_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ( v1_xboole_0(A)
              | v2_xreal_0(B)
              | v3_xreal_0(A) ) ) ) ) ),
    file(real,t6_real),
    [interesting(0.9),axiom,file(real,t6_real)]).

fof(t7_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,B)
              & ~ v2_xreal_0(A)
              & ~ v3_xreal_0(B) ) ) ) ),
    file(real,t7_real),
    [interesting(0.9),axiom,file(real,t7_real)]).

fof(t8_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,B)
              & ~ v3_xreal_0(B)
              & ~ v2_xreal_0(A) ) ) ) ),
    file(real,t8_real),
    [interesting(0.9),axiom,file(real,t8_real)]).

fof(existence_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m2_subset_1(C,A,B) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(redefinition_k5_numbers,definition,(
    k5_numbers = k5_ordinal2 ),
    file(numbers,k5_numbers),
    [interesting(0.9),axiom,file(numbers,k5_numbers)]).

fof(redefinition_m2_subset_1,definition,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(dt_k1_numbers,axiom,(
    $true ),
    file(numbers,k1_numbers),
    [interesting(0.9),axiom,file(numbers,k1_numbers)]).

fof(dt_k5_numbers,axiom,(
    m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ),
    file(numbers,k5_numbers),
    [interesting(0.9),axiom,file(numbers,k5_numbers)]).

fof(dt_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
         => m1_subset_1(C,A) ) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(cc1_xcmplx_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => v1_xcmplx_0(A) ) ),
    file(xcmplx_0,cc1_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,cc1_xcmplx_0)]).

fof(cc3_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(xreal_0,cc3_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]).

fof(cc6_xreal_0,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v3_xreal_0(A) ) ) ),
    file(xreal_0,cc6_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]).

fof(cc7_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xboole_0(A)
        & v1_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(xreal_0,cc7_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]).

fof(fc1_numbers,theorem,(
    ~ v1_xboole_0(k1_numbers) ),
    file(numbers,fc1_numbers),
    [interesting(0.9),axiom,file(numbers,fc1_numbers)]).

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(rc1_xcmplx_0,theorem,(
    ? [A] : v1_xcmplx_0(A) ),
    file(xcmplx_0,rc1_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,rc1_xcmplx_0)]).

fof(rc1_xreal_0,theorem,(
    ? [A] : 
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(xreal_0,rc1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc1_xreal_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(rc2_xcmplx_0,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A) ) ),
    file(xcmplx_0,rc2_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,rc2_xcmplx_0)]).

fof(t1_numerals,theorem,(
    m1_subset_1(0,k5_numbers) ),
    file(numerals,t1_numerals),
    [interesting(0.9),axiom,file(numerals,t1_numerals)]).

fof(t1_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(A,B)
              & v2_xreal_0(A) )
           => v2_xreal_0(B) ) ) ) ),
    file(real,t1_real),
    [interesting(0.9),axiom,file(real,t1_real)]).

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(t4_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( r1_xreal_0(A,B)
              & ~ v2_xreal_0(B)
              & v2_xreal_0(A) ) ) ) ),
    file(real,t4_real),
    [interesting(0.9),axiom,file(real,t4_real)]).

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(connectedness_r1_arytm_2,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => ( r1_arytm_2(A,B)
        | r1_arytm_2(B,A) ) ) ),
    file(arytm_2,r1_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,r1_arytm_2)]).

fof(reflexivity_r1_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => r1_xreal_0(A,A) ) ),
    file(xreal_0,r1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]).

fof(connectedness_r1_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( r1_xreal_0(A,B)
        | r1_xreal_0(B,A) ) ) ),
    file(xreal_0,r1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,r1_xreal_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(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_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    [interesting(0.9),axiom,file(tarski,k1_tarski)]).

fof(dt_k2_arytm_2,axiom,(
    $true ),
    file(arytm_2,k2_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,k2_arytm_2)]).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    [interesting(0.9),axiom,file(tarski,k4_tarski)]).

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

fof(dt_c1_2__xreal_1,assumption,(
    v1_xreal_0(c1_2__xreal_1) ),
    introduced(assumption,[file(xreal_1,c1_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,c1_2__xreal_1)]).

fof(dh_c1_2_1_1__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & ? [B] : 
            ( m1_subset_1(B,k2_arytm_2)
            & c1_2__xreal_1 = A
            & c2_2__xreal_1 = B
            & r1_arytm_2(A,B) ) )
   => ( m1_subset_1(c1_2_1_1__xreal_1,k2_arytm_2)
      & ? [C] : 
          ( m1_subset_1(C,k2_arytm_2)
          & c1_2__xreal_1 = c1_2_1_1__xreal_1
          & c2_2__xreal_1 = C
          & r1_arytm_2(c1_2_1_1__xreal_1,C) ) ) ),
    introduced(definition,[new_symbol(c1_2_1_1__xreal_1),file(xreal_1,c1_2_1_1__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c1_2_1_1__xreal_1)]).

fof(dt_c2_2__xreal_1,assumption,(
    v1_xreal_0(c2_2__xreal_1) ),
    introduced(assumption,[file(xreal_1,c2_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,c2_2__xreal_1)]).

fof(cc2_xreal_0,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => v1_xcmplx_0(A) ) ),
    file(xreal_0,cc2_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]).

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(fc2_arytm_2,theorem,(
    ~ v1_xboole_0(k2_arytm_2) ),
    file(arytm_2,fc2_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,fc2_arytm_2)]).

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(spc0_boole,theorem,(
    v1_xboole_0(0) ),
    file(boole,spc0_boole),
    [interesting(0.9),axiom,file(boole,spc0_boole)]).

fof(spc0_numerals,theorem,
    ( v2_xreal_0(0)
    & m2_subset_1(0,k1_numbers,k5_numbers)
    & m1_subset_1(0,k5_numbers)
    & m1_subset_1(0,k1_numbers) ),
    file(numerals,spc0_numerals),
    [interesting(0.9),axiom,file(numerals,spc0_numerals)]).

fof(d2_xreal_0,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( ( r2_hidden(A,k2_arytm_2)
                & r2_hidden(B,k2_arytm_2) )
             => ( r1_xreal_0(A,B)
              <=> ? [C] : 
                    ( m1_subset_1(C,k2_arytm_2)
                    & ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & A = C
                        & B = D
                        & r1_arytm_2(C,D) ) ) ) )
            & ( ( r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
                & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
             => ( r1_xreal_0(A,B)
              <=> ? [C] : 
                    ( m1_subset_1(C,k2_arytm_2)
                    & ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & A = k4_tarski(0,C)
                        & B = k4_tarski(0,D)
                        & r1_arytm_2(D,C) ) ) ) )
            & ~ ( ~ ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_arytm_2) )
                & ~ ( r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                & ~ ( r1_xreal_0(A,B)
                  <=> ( r2_hidden(B,k2_arytm_2)
                      & r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ) ) ) ) ) ) ),
    file(xreal_0,d2_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,d2_xreal_0)]).

fof(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,(
    r1_xreal_0(0,0) ),
    file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0),
    [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]).

fof(e4_2_1_1__xreal_1,plain,(
    ? [A] : 
      ( m1_subset_1(A,k2_arytm_2)
      & ? [B] : 
          ( m1_subset_1(B,k2_arytm_2)
          & c1_2__xreal_1 = A
          & c2_2__xreal_1 = B
          & r1_arytm_2(A,B) ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,e2_2_1_1__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e1_2__xreal_1,e1_2_1_1__xreal_1,e2_2_1_1__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e4_2_1_1__xreal_1),[file(xreal_1,e4_2_1_1__xreal_1)]]).

fof(dt_c1_2_1_1__xreal_1,plain,(
    m1_subset_1(c1_2_1_1__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,e2_2_1_1__xreal_1])],[dh_c1_2_1_1__xreal_1,e4_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,c1_2_1_1__xreal_1),[file(xreal_1,c1_2_1_1__xreal_1)]]).

fof(dt_c3_2__xreal_1,assumption,(
    v1_xreal_0(c3_2__xreal_1) ),
    introduced(assumption,[file(xreal_1,c3_2__xreal_1)]),
    [interesting(0.8),axiom,file(xreal_1,c3_2__xreal_1)]).

fof(dh_c3_2_1_1__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & ? [B] : 
            ( m1_subset_1(B,k2_arytm_2)
            & c2_2__xreal_1 = A
            & c3_2__xreal_1 = B
            & r1_arytm_2(A,B) ) )
   => ( m1_subset_1(c3_2_1_1__xreal_1,k2_arytm_2)
      & ? [C] : 
          ( m1_subset_1(C,k2_arytm_2)
          & c2_2__xreal_1 = c3_2_1_1__xreal_1
          & c3_2__xreal_1 = C
          & r1_arytm_2(c3_2_1_1__xreal_1,C) ) ) ),
    introduced(definition,[new_symbol(c3_2_1_1__xreal_1),file(xreal_1,c3_2_1_1__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c3_2_1_1__xreal_1)]).

fof(dh_c4_2_1_1__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & c2_2__xreal_1 = c3_2_1_1__xreal_1
        & c3_2__xreal_1 = A
        & r1_arytm_2(c3_2_1_1__xreal_1,A) )
   => ( m1_subset_1(c4_2_1_1__xreal_1,k2_arytm_2)
      & c2_2__xreal_1 = c3_2_1_1__xreal_1
      & c3_2__xreal_1 = c4_2_1_1__xreal_1
      & r1_arytm_2(c3_2_1_1__xreal_1,c4_2_1_1__xreal_1) ) ),
    introduced(definition,[new_symbol(c4_2_1_1__xreal_1),file(xreal_1,c4_2_1_1__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c4_2_1_1__xreal_1)]).

fof(e8_2_1_1__xreal_1,plain,(
    ? [A] : 
      ( m1_subset_1(A,k2_arytm_2)
      & ? [B] : 
          ( m1_subset_1(B,k2_arytm_2)
          & c2_2__xreal_1 = A
          & c3_2__xreal_1 = B
          & r1_arytm_2(A,B) ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e8_2_1_1__xreal_1),[file(xreal_1,e8_2_1_1__xreal_1)]]).

fof(dt_c4_2_1_1__xreal_1,plain,(
    m1_subset_1(c4_2_1_1__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[dh_c3_2_1_1__xreal_1,dh_c4_2_1_1__xreal_1,e8_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,c4_2_1_1__xreal_1),[file(xreal_1,c4_2_1_1__xreal_1)]]).

fof(dh_c2_2_1_1__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & c1_2__xreal_1 = c1_2_1_1__xreal_1
        & c2_2__xreal_1 = A
        & r1_arytm_2(c1_2_1_1__xreal_1,A) )
   => ( m1_subset_1(c2_2_1_1__xreal_1,k2_arytm_2)
      & c1_2__xreal_1 = c1_2_1_1__xreal_1
      & c2_2__xreal_1 = c2_2_1_1__xreal_1
      & r1_arytm_2(c1_2_1_1__xreal_1,c2_2_1_1__xreal_1) ) ),
    introduced(definition,[new_symbol(c2_2_1_1__xreal_1),file(xreal_1,c2_2_1_1__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c2_2_1_1__xreal_1)]).

fof(dt_c2_2_1_1__xreal_1,plain,(
    m1_subset_1(c2_2_1_1__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,e2_2_1_1__xreal_1])],[dh_c1_2_1_1__xreal_1,dh_c2_2_1_1__xreal_1,e4_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,c2_2_1_1__xreal_1),[file(xreal_1,c2_2_1_1__xreal_1)]]).

fof(dt_c3_2_1_1__xreal_1,plain,(
    m1_subset_1(c3_2_1_1__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[dh_c3_2_1_1__xreal_1,e8_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,c3_2_1_1__xreal_1),[file(xreal_1,c3_2_1_1__xreal_1)]]).

fof(e6_2_1_1__xreal_1,plain,(
    c2_2__xreal_1 = c2_2_1_1__xreal_1 ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,e2_2_1_1__xreal_1])],[dh_c1_2_1_1__xreal_1,dh_c2_2_1_1__xreal_1,e4_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,e6_2_1_1__xreal_1),[file(xreal_1,e6_2_1_1__xreal_1)]]).

fof(e7_2_1_1__xreal_1,plain,(
    r1_arytm_2(c1_2_1_1__xreal_1,c2_2_1_1__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,e2_2_1_1__xreal_1])],[dh_c1_2_1_1__xreal_1,dh_c2_2_1_1__xreal_1,e4_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,e7_2_1_1__xreal_1),[file(xreal_1,e7_2_1_1__xreal_1)]]).

fof(e9_2_1_1__xreal_1,plain,(
    c2_2__xreal_1 = c3_2_1_1__xreal_1 ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[dh_c3_2_1_1__xreal_1,dh_c4_2_1_1__xreal_1,e8_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,e9_2_1_1__xreal_1),[file(xreal_1,e9_2_1_1__xreal_1)]]).

fof(e11_2_1_1__xreal_1,plain,(
    r1_arytm_2(c3_2_1_1__xreal_1,c4_2_1_1__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[dh_c3_2_1_1__xreal_1,dh_c4_2_1_1__xreal_1,e8_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,e11_2_1_1__xreal_1),[file(xreal_1,e11_2_1_1__xreal_1)]]).

fof(t3_arytm_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => ( ( r1_arytm_2(A,B)
                  & r1_arytm_2(B,C) )
               => r1_arytm_2(A,C) ) ) ) ) ),
    file(arytm_1,t3_arytm_1),
    [interesting(0.9),axiom,file(arytm_1,t3_arytm_1)]).

fof(e12_2_1_1__xreal_1,plain,(
    r1_arytm_2(c1_2_1_1__xreal_1,c4_2_1_1__xreal_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[antisymmetry_r2_hidden,dt_k1_xboole_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xcmplx_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,cc2_xreal_0,cc7_xreal_0,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,connectedness_r1_arytm_2,existence_m1_subset_1,dt_k2_arytm_2,dt_m1_subset_1,dt_c1_2_1_1__xreal_1,dt_c2_2__xreal_1,dt_c2_2_1_1__xreal_1,dt_c3_2_1_1__xreal_1,dt_c4_2_1_1__xreal_1,fc2_arytm_2,e6_2_1_1__xreal_1,e7_2_1_1__xreal_1,e9_2_1_1__xreal_1,e11_2_1_1__xreal_1,t3_arytm_1]),
    [interesting(0.5),file(xreal_1,e12_2_1_1__xreal_1),[file(xreal_1,e12_2_1_1__xreal_1)]]).

fof(e5_2_1_1__xreal_1,plain,(
    c1_2__xreal_1 = c1_2_1_1__xreal_1 ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,e2_2_1_1__xreal_1])],[dh_c1_2_1_1__xreal_1,dh_c2_2_1_1__xreal_1,e4_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,e5_2_1_1__xreal_1),[file(xreal_1,e5_2_1_1__xreal_1)]]).

fof(e10_2_1_1__xreal_1,plain,(
    c3_2__xreal_1 = c4_2_1_1__xreal_1 ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[dh_c3_2_1_1__xreal_1,dh_c4_2_1_1__xreal_1,e8_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,e10_2_1_1__xreal_1),[file(xreal_1,e10_2_1_1__xreal_1)]]).

fof(e13_2_1_1__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c1_2__xreal_1,dt_c1_2_1_1__xreal_1,dt_c3_2__xreal_1,dt_c4_2_1_1__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e12_2_1_1__xreal_1,e5_2_1_1__xreal_1,e10_2_1_1__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e13_2_1_1__xreal_1),[file(xreal_1,e13_2_1_1__xreal_1)]]).

fof(i2_2_1_1__xreal_1,theorem,(
    $true ),
    introduced(tautology,[file(xreal_1,i2_2_1_1__xreal_1)]),
    [interesting(0.5),trivial,file(xreal_1,i2_2_1_1__xreal_1)]).

fof(i1_2_1_1__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,e1_2_1_1__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[e13_2_1_1__xreal_1,i2_2_1_1__xreal_1]),
    [interesting(0.5),file(xreal_1,i1_2_1_1__xreal_1),[file(xreal_1,i1_2_1_1__xreal_1)]]).

fof(i1_2_1__xreal_1,plain,
    ( ( r2_hidden(c1_2__xreal_1,k2_arytm_2)
      & r2_hidden(c2_2__xreal_1,k2_arytm_2)
      & r2_hidden(c3_2__xreal_1,k2_arytm_2) )
   => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1]),discharge_asm(discharge,[e1_2_1_1__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1])],[e1_2_1_1__xreal_1,e2_2_1_1__xreal_1,e3_2_1_1__xreal_1,i1_2_1_1__xreal_1]),
    [interesting(0.65),file(xreal_1,i1_2_1__xreal_1),[file(xreal_1,i1_2_1__xreal_1)]]).

fof(e1_2_1_2__xreal_1,assumption,
    ( r2_hidden(c1_2__xreal_1,k2_arytm_2)
    & r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ),
    introduced(assumption,[file(xreal_1,e1_2_1_2__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e1_2_1_2__xreal_1)]).

fof(cc4_arytm_3,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ( v3_ordinal1(A)
       => ( v1_ordinal1(A)
          & v2_ordinal1(A)
          & v3_ordinal1(A)
          & v4_ordinal2(A) ) ) ) ),
    file(arytm_3,cc4_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,cc4_arytm_3)]).

fof(rc2_arytm_3,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k6_arytm_3)
      & ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(arytm_3,rc2_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,rc2_arytm_3)]).

fof(rc3_arytm_3,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k6_arytm_3)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A) ) ),
    file(arytm_3,rc3_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,rc3_arytm_3)]).

fof(dt_k6_arytm_3,axiom,(
    $true ),
    file(arytm_3,k6_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,k6_arytm_3)]).

fof(fc8_arytm_3,theorem,(
    ~ v1_xboole_0(k6_arytm_3) ),
    file(arytm_3,fc8_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,fc8_arytm_3)]).

fof(symmetry_r1_subset_1,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ( r1_subset_1(A,B)
       => r1_subset_1(B,A) ) ) ),
    file(subset_1,r1_subset_1),
    [interesting(0.9),axiom,file(subset_1,r1_subset_1)]).

fof(irreflexivity_r1_subset_1,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ r1_subset_1(A,A) ) ),
    file(subset_1,r1_subset_1),
    [interesting(0.9),axiom,file(subset_1,r1_subset_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(redefinition_k12_arytm_3,definition,(
    k12_arytm_3 = k1_xboole_0 ),
    file(arytm_3,k12_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,k12_arytm_3)]).

fof(redefinition_r1_subset_1,definition,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ( r1_subset_1(A,B)
      <=> r1_xboole_0(A,B) ) ) ),
    file(subset_1,r1_subset_1),
    [interesting(0.9),axiom,file(subset_1,r1_subset_1)]).

fof(dt_k12_arytm_3,axiom,
    ( v1_xboole_0(k12_arytm_3)
    & m1_subset_1(k12_arytm_3,k6_arytm_3) ),
    file(arytm_3,k12_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,k12_arytm_3)]).

fof(t5_arytm_0,theorem,(
    r1_subset_1(k2_arytm_2,k2_zfmisc_1(k1_tarski(k12_arytm_3),k2_arytm_2)) ),
    file(arytm_0,t5_arytm_0),
    [interesting(0.9),axiom,file(arytm_0,t5_arytm_0)]).

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(e2_2_1_2__xreal_1,plain,
    ( ~ ( r2_hidden(c1_2__xreal_1,k2_arytm_2)
        & r2_hidden(c2_2__xreal_1,k2_arytm_2) )
    & ~ ( r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
        & r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2_1_2__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,cc4_arytm_3,rc1_arytm_3,rc2_arytm_3,rc3_arytm_3,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,rc1_xcmplx_0,rc1_xreal_0,rc2_xcmplx_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_xboole_0,dt_k5_numbers,dt_k6_arytm_3,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,fc1_xboole_0,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t1_numerals,t2_subset,t6_boole,t8_boole,symmetry_r1_subset_1,irreflexivity_r1_subset_1,symmetry_r1_xboole_0,antisymmetry_r2_hidden,redefinition_k12_arytm_3,redefinition_r1_subset_1,dt_k12_arytm_3,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e1_2_1_2__xreal_1,t5_arytm_0,t3_xboole_0]),
    [interesting(0.5),file(xreal_1,e2_2_1_2__xreal_1),[file(xreal_1,e2_2_1_2__xreal_1)]]).

fof(e3_2_1_2__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c3_2__xreal_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_2__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e2_2_1_2__xreal_1,e1_2__xreal_1,e1_2_1_2__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e3_2_1_2__xreal_1),[file(xreal_1,e3_2_1_2__xreal_1)]]).

fof(i2_2_1_2__xreal_1,theorem,(
    $true ),
    introduced(tautology,[file(xreal_1,i2_2_1_2__xreal_1)]),
    [interesting(0.5),trivial,file(xreal_1,i2_2_1_2__xreal_1)]).

fof(i1_2_1_2__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c3_2__xreal_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_2__xreal_1])],[e3_2_1_2__xreal_1,i2_2_1_2__xreal_1]),
    [interesting(0.5),file(xreal_1,i1_2_1_2__xreal_1),[file(xreal_1,i1_2_1_2__xreal_1)]]).

fof(i2_2_1__xreal_1,plain,
    ( ( r2_hidden(c1_2__xreal_1,k2_arytm_2)
      & r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
   => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c3_2__xreal_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1]),discharge_asm(discharge,[e1_2_1_2__xreal_1])],[e1_2_1_2__xreal_1,i1_2_1_2__xreal_1]),
    [interesting(0.65),file(xreal_1,i2_2_1__xreal_1),[file(xreal_1,i2_2_1__xreal_1)]]).

fof(e1_2_1_3__xreal_1,assumption,
    ( r2_hidden(c2_2__xreal_1,k2_arytm_2)
    & r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ),
    introduced(assumption,[file(xreal_1,e1_2_1_3__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e1_2_1_3__xreal_1)]).

fof(e2_2_1_3__xreal_1,plain,
    ( ~ ( r2_hidden(c3_2__xreal_1,k2_arytm_2)
        & r2_hidden(c2_2__xreal_1,k2_arytm_2) )
    & ~ ( r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
        & r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e1_2_1_3__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,cc4_arytm_3,rc1_arytm_3,rc2_arytm_3,rc3_arytm_3,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,rc1_xcmplx_0,rc1_xreal_0,rc2_xcmplx_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_xboole_0,dt_k5_numbers,dt_k6_arytm_3,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,fc1_xboole_0,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t1_numerals,t2_subset,t6_boole,t8_boole,symmetry_r1_subset_1,irreflexivity_r1_subset_1,symmetry_r1_xboole_0,antisymmetry_r2_hidden,redefinition_k12_arytm_3,redefinition_r1_subset_1,dt_k12_arytm_3,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e1_2_1_3__xreal_1,t5_arytm_0,t3_xboole_0]),
    [interesting(0.5),file(xreal_1,e2_2_1_3__xreal_1),[file(xreal_1,e2_2_1_3__xreal_1)]]).

fof(e3_2_1_3__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e1_2_1_3__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e2_2_1_3__xreal_1,e2_2__xreal_1,e1_2_1_3__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e3_2_1_3__xreal_1),[file(xreal_1,e3_2_1_3__xreal_1)]]).

fof(i2_2_1_3__xreal_1,theorem,(
    $true ),
    introduced(tautology,[file(xreal_1,i2_2_1_3__xreal_1)]),
    [interesting(0.5),trivial,file(xreal_1,i2_2_1_3__xreal_1)]).

fof(i1_2_1_3__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e1_2_1_3__xreal_1])],[e3_2_1_3__xreal_1,i2_2_1_3__xreal_1]),
    [interesting(0.5),file(xreal_1,i1_2_1_3__xreal_1),[file(xreal_1,i1_2_1_3__xreal_1)]]).

fof(i3_2_1__xreal_1,plain,
    ( ( r2_hidden(c2_2__xreal_1,k2_arytm_2)
      & r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
   => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1]),discharge_asm(discharge,[e1_2_1_3__xreal_1])],[e1_2_1_3__xreal_1,i1_2_1_3__xreal_1]),
    [interesting(0.65),file(xreal_1,i3_2_1__xreal_1),[file(xreal_1,i3_2_1__xreal_1)]]).

fof(e1_2_1_4__xreal_1,assumption,(
    r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ),
    introduced(assumption,[file(xreal_1,e1_2_1_4__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e1_2_1_4__xreal_1)]).

fof(e2_2_1_4__xreal_1,assumption,(
    r2_hidden(c3_2__xreal_1,k2_arytm_2) ),
    introduced(assumption,[file(xreal_1,e2_2_1_4__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e2_2_1_4__xreal_1)]).

fof(e3_2_1_4__xreal_1,plain,
    ( ~ ( r2_hidden(c1_2__xreal_1,k2_arytm_2)
        & r2_hidden(c3_2__xreal_1,k2_arytm_2) )
    & ~ ( r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
        & r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c3_2__xreal_1,e1_2_1_4__xreal_1,e2_2_1_4__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,cc4_arytm_3,rc1_arytm_3,rc2_arytm_3,rc3_arytm_3,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,rc1_xcmplx_0,rc1_xreal_0,rc2_xcmplx_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_xboole_0,dt_k5_numbers,dt_k6_arytm_3,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,fc1_xboole_0,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t1_numerals,t2_subset,t6_boole,t8_boole,symmetry_r1_subset_1,irreflexivity_r1_subset_1,symmetry_r1_xboole_0,antisymmetry_r2_hidden,redefinition_k12_arytm_3,redefinition_r1_subset_1,dt_k12_arytm_3,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_c1_2__xreal_1,dt_c3_2__xreal_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e1_2_1_4__xreal_1,e2_2_1_4__xreal_1,t5_arytm_0,t3_xboole_0]),
    [interesting(0.5),file(xreal_1,e3_2_1_4__xreal_1),[file(xreal_1,e3_2_1_4__xreal_1)]]).

fof(e4_2_1_4__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c3_2__xreal_1,e1_2_1_4__xreal_1,e2_2_1_4__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c1_2__xreal_1,dt_c3_2__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e3_2_1_4__xreal_1,e2_2_1_4__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e4_2_1_4__xreal_1),[file(xreal_1,e4_2_1_4__xreal_1)]]).

fof(i2_2_1_4__xreal_1,theorem,(
    $true ),
    introduced(tautology,[file(xreal_1,i2_2_1_4__xreal_1)]),
    [interesting(0.5),trivial,file(xreal_1,i2_2_1_4__xreal_1)]).

fof(i1_2_1_4__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c3_2__xreal_1,e1_2_1_4__xreal_1,e2_2_1_4__xreal_1])],[e4_2_1_4__xreal_1,i2_2_1_4__xreal_1]),
    [interesting(0.5),file(xreal_1,i1_2_1_4__xreal_1),[file(xreal_1,i1_2_1_4__xreal_1)]]).

fof(i4_2_1__xreal_1,plain,
    ( ( r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
      & r2_hidden(c3_2__xreal_1,k2_arytm_2) )
   => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c3_2__xreal_1]),discharge_asm(discharge,[e1_2_1_4__xreal_1,e2_2_1_4__xreal_1])],[e1_2_1_4__xreal_1,e2_2_1_4__xreal_1,i1_2_1_4__xreal_1]),
    [interesting(0.65),file(xreal_1,i4_2_1__xreal_1),[file(xreal_1,i4_2_1__xreal_1)]]).

fof(e1_2_1_5__xreal_1,assumption,(
    r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ),
    introduced(assumption,[file(xreal_1,e1_2_1_5__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e1_2_1_5__xreal_1)]).

fof(e2_2_1_5__xreal_1,assumption,(
    r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ),
    introduced(assumption,[file(xreal_1,e2_2_1_5__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e2_2_1_5__xreal_1)]).

fof(e3_2_1_5__xreal_1,assumption,(
    r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ),
    introduced(assumption,[file(xreal_1,e3_2_1_5__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,e3_2_1_5__xreal_1)]).

fof(dh_c1_2_1_5__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & ? [B] : 
            ( m1_subset_1(B,k2_arytm_2)
            & c1_2__xreal_1 = k4_tarski(0,A)
            & c2_2__xreal_1 = k4_tarski(0,B)
            & r1_arytm_2(B,A) ) )
   => ( m1_subset_1(c1_2_1_5__xreal_1,k2_arytm_2)
      & ? [C] : 
          ( m1_subset_1(C,k2_arytm_2)
          & c1_2__xreal_1 = k4_tarski(0,c1_2_1_5__xreal_1)
          & c2_2__xreal_1 = k4_tarski(0,C)
          & r1_arytm_2(C,c1_2_1_5__xreal_1) ) ) ),
    introduced(definition,[new_symbol(c1_2_1_5__xreal_1),file(xreal_1,c1_2_1_5__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c1_2_1_5__xreal_1)]).

fof(e4_2_1_5__xreal_1,plain,(
    ? [A] : 
      ( m1_subset_1(A,k2_arytm_2)
      & ? [B] : 
          ( m1_subset_1(B,k2_arytm_2)
          & c1_2__xreal_1 = k4_tarski(0,A)
          & c2_2__xreal_1 = k4_tarski(0,B)
          & r1_arytm_2(B,A) ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,e2_2_1_5__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e1_2__xreal_1,e1_2_1_5__xreal_1,e2_2_1_5__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e4_2_1_5__xreal_1),[file(xreal_1,e4_2_1_5__xreal_1)]]).

fof(dt_c1_2_1_5__xreal_1,plain,(
    m1_subset_1(c1_2_1_5__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,e2_2_1_5__xreal_1])],[dh_c1_2_1_5__xreal_1,e4_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,c1_2_1_5__xreal_1),[file(xreal_1,c1_2_1_5__xreal_1)]]).

fof(dh_c3_2_1_5__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & ? [B] : 
            ( m1_subset_1(B,k2_arytm_2)
            & c2_2__xreal_1 = k4_tarski(0,A)
            & c3_2__xreal_1 = k4_tarski(0,B)
            & r1_arytm_2(B,A) ) )
   => ( m1_subset_1(c3_2_1_5__xreal_1,k2_arytm_2)
      & ? [C] : 
          ( m1_subset_1(C,k2_arytm_2)
          & c2_2__xreal_1 = k4_tarski(0,c3_2_1_5__xreal_1)
          & c3_2__xreal_1 = k4_tarski(0,C)
          & r1_arytm_2(C,c3_2_1_5__xreal_1) ) ) ),
    introduced(definition,[new_symbol(c3_2_1_5__xreal_1),file(xreal_1,c3_2_1_5__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c3_2_1_5__xreal_1)]).

fof(dh_c4_2_1_5__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & c2_2__xreal_1 = k4_tarski(0,c3_2_1_5__xreal_1)
        & c3_2__xreal_1 = k4_tarski(0,A)
        & r1_arytm_2(A,c3_2_1_5__xreal_1) )
   => ( m1_subset_1(c4_2_1_5__xreal_1,k2_arytm_2)
      & c2_2__xreal_1 = k4_tarski(0,c3_2_1_5__xreal_1)
      & c3_2__xreal_1 = k4_tarski(0,c4_2_1_5__xreal_1)
      & r1_arytm_2(c4_2_1_5__xreal_1,c3_2_1_5__xreal_1) ) ),
    introduced(definition,[new_symbol(c4_2_1_5__xreal_1),file(xreal_1,c4_2_1_5__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c4_2_1_5__xreal_1)]).

fof(e8_2_1_5__xreal_1,plain,(
    ? [A] : 
      ( m1_subset_1(A,k2_arytm_2)
      & ? [B] : 
          ( m1_subset_1(B,k2_arytm_2)
          & c2_2__xreal_1 = k4_tarski(0,A)
          & c3_2__xreal_1 = k4_tarski(0,B)
          & r1_arytm_2(B,A) ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e8_2_1_5__xreal_1),[file(xreal_1,e8_2_1_5__xreal_1)]]).

fof(dt_c4_2_1_5__xreal_1,plain,(
    m1_subset_1(c4_2_1_5__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[dh_c3_2_1_5__xreal_1,dh_c4_2_1_5__xreal_1,e8_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,c4_2_1_5__xreal_1),[file(xreal_1,c4_2_1_5__xreal_1)]]).

fof(dh_c2_2_1_5__xreal_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & c1_2__xreal_1 = k4_tarski(0,c1_2_1_5__xreal_1)
        & c2_2__xreal_1 = k4_tarski(0,A)
        & r1_arytm_2(A,c1_2_1_5__xreal_1) )
   => ( m1_subset_1(c2_2_1_5__xreal_1,k2_arytm_2)
      & c1_2__xreal_1 = k4_tarski(0,c1_2_1_5__xreal_1)
      & c2_2__xreal_1 = k4_tarski(0,c2_2_1_5__xreal_1)
      & r1_arytm_2(c2_2_1_5__xreal_1,c1_2_1_5__xreal_1) ) ),
    introduced(definition,[new_symbol(c2_2_1_5__xreal_1),file(xreal_1,c2_2_1_5__xreal_1)]),
    [interesting(0.5),axiom,file(xreal_1,c2_2_1_5__xreal_1)]).

fof(dt_c2_2_1_5__xreal_1,plain,(
    m1_subset_1(c2_2_1_5__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,e2_2_1_5__xreal_1])],[dh_c1_2_1_5__xreal_1,dh_c2_2_1_5__xreal_1,e4_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,c2_2_1_5__xreal_1),[file(xreal_1,c2_2_1_5__xreal_1)]]).

fof(dt_c3_2_1_5__xreal_1,plain,(
    m1_subset_1(c3_2_1_5__xreal_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[dh_c3_2_1_5__xreal_1,e8_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,c3_2_1_5__xreal_1),[file(xreal_1,c3_2_1_5__xreal_1)]]).

fof(e6_2_1_5__xreal_1,plain,(
    c2_2__xreal_1 = k4_tarski(0,c2_2_1_5__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,e2_2_1_5__xreal_1])],[dh_c1_2_1_5__xreal_1,dh_c2_2_1_5__xreal_1,e4_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,e6_2_1_5__xreal_1),[file(xreal_1,e6_2_1_5__xreal_1)]]).

fof(e9_2_1_5__xreal_1,plain,(
    c2_2__xreal_1 = k4_tarski(0,c3_2_1_5__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[dh_c3_2_1_5__xreal_1,dh_c4_2_1_5__xreal_1,e8_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,e9_2_1_5__xreal_1),[file(xreal_1,e9_2_1_5__xreal_1)]]).

fof(t33_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( k4_tarski(A,B) = k4_tarski(C,D)
     => ( A = C
        & B = D ) ) ),
    file(zfmisc_1,t33_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,t33_zfmisc_1)]).

fof(e12_2_1_5__xreal_1,plain,(
    c2_2_1_5__xreal_1 = c3_2_1_5__xreal_1 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xcmplx_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_arytm_2,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,fc2_arytm_2,rc1_xboole_0,rc2_xboole_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,dt_k4_tarski,dt_c2_2__xreal_1,dt_c2_2_1_5__xreal_1,dt_c3_2_1_5__xreal_1,fc1_zfmisc_1,spc0_boole,spc0_numerals,e6_2_1_5__xreal_1,e9_2_1_5__xreal_1,t33_zfmisc_1]),
    [interesting(0.5),file(xreal_1,e12_2_1_5__xreal_1),[file(xreal_1,e12_2_1_5__xreal_1)]]).

fof(e7_2_1_5__xreal_1,plain,(
    r1_arytm_2(c2_2_1_5__xreal_1,c1_2_1_5__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,e2_2_1_5__xreal_1])],[dh_c1_2_1_5__xreal_1,dh_c2_2_1_5__xreal_1,e4_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,e7_2_1_5__xreal_1),[file(xreal_1,e7_2_1_5__xreal_1)]]).

fof(e11_2_1_5__xreal_1,plain,(
    r1_arytm_2(c4_2_1_5__xreal_1,c3_2_1_5__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[dh_c3_2_1_5__xreal_1,dh_c4_2_1_5__xreal_1,e8_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,e11_2_1_5__xreal_1),[file(xreal_1,e11_2_1_5__xreal_1)]]).

fof(e13_2_1_5__xreal_1,plain,(
    r1_arytm_2(c4_2_1_5__xreal_1,c1_2_1_5__xreal_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[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,connectedness_r1_arytm_2,existence_m1_subset_1,dt_k2_arytm_2,dt_m1_subset_1,dt_c1_2_1_5__xreal_1,dt_c2_2_1_5__xreal_1,dt_c3_2_1_5__xreal_1,dt_c4_2_1_5__xreal_1,fc2_arytm_2,e12_2_1_5__xreal_1,e7_2_1_5__xreal_1,e11_2_1_5__xreal_1,t3_arytm_1]),
    [interesting(0.5),file(xreal_1,e13_2_1_5__xreal_1),[file(xreal_1,e13_2_1_5__xreal_1)]]).

fof(e5_2_1_5__xreal_1,plain,(
    c1_2__xreal_1 = k4_tarski(0,c1_2_1_5__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,e2_2_1_5__xreal_1])],[dh_c1_2_1_5__xreal_1,dh_c2_2_1_5__xreal_1,e4_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,e5_2_1_5__xreal_1),[file(xreal_1,e5_2_1_5__xreal_1)]]).

fof(e10_2_1_5__xreal_1,plain,(
    c3_2__xreal_1 = k4_tarski(0,c4_2_1_5__xreal_1) ),
    inference(consider,[status(thm),assumptions([dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[dh_c3_2_1_5__xreal_1,dh_c4_2_1_5__xreal_1,e8_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,e10_2_1_5__xreal_1),[file(xreal_1,e10_2_1_5__xreal_1)]]).

fof(e14_2_1_5__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_xcmplx_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,connectedness_r1_arytm_2,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_m1_subset_1,dt_c1_2__xreal_1,dt_c1_2_1_5__xreal_1,dt_c3_2__xreal_1,dt_c4_2_1_5__xreal_1,cc2_xreal_0,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e13_2_1_5__xreal_1,e1_2_1_5__xreal_1,e3_2_1_5__xreal_1,e5_2_1_5__xreal_1,e10_2_1_5__xreal_1,d2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(xreal_1,e14_2_1_5__xreal_1),[file(xreal_1,e14_2_1_5__xreal_1)]]).

fof(i2_2_1_5__xreal_1,theorem,(
    $true ),
    introduced(tautology,[file(xreal_1,i2_2_1_5__xreal_1)]),
    [interesting(0.5),trivial,file(xreal_1,i2_2_1_5__xreal_1)]).

fof(i1_2_1_5__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,e1_2_1_5__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[e14_2_1_5__xreal_1,i2_2_1_5__xreal_1]),
    [interesting(0.5),file(xreal_1,i1_2_1_5__xreal_1),[file(xreal_1,i1_2_1_5__xreal_1)]]).

fof(i5_2_1__xreal_1,plain,
    ( ( r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
      & r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
      & r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
   => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__xreal_1,e1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,e2_2__xreal_1]),discharge_asm(discharge,[e1_2_1_5__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1])],[e1_2_1_5__xreal_1,e2_2_1_5__xreal_1,e3_2_1_5__xreal_1,i1_2_1_5__xreal_1]),
    [interesting(0.65),file(xreal_1,i5_2_1__xreal_1),[file(xreal_1,i5_2_1__xreal_1)]]).

fof(t1_boole,theorem,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    [interesting(0.9),axiom,file(boole,t1_boole)]).

fof(fc2_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc2_xboole_0)]).

fof(fc3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc3_xboole_0)]).

fof(commutativity_k2_xboole_0,theorem,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(idempotence_k2_xboole_0,theorem,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(t3_boole,theorem,(
    ! [A] : k4_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t3_boole),
    [interesting(0.9),axiom,file(boole,t3_boole)]).

fof(t4_boole,theorem,(
    ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ),
    file(boole,t4_boole),
    [interesting(0.9),axiom,file(boole,t4_boole)]).

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k4_xboole_0)]).

fof(d1_xreal_0,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
    <=> r2_hidden(A,k1_numbers) ) ),
    file(xreal_0,d1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,d1_xreal_0)]).

fof(e3_2__xreal_1,plain,
    ( r2_hidden(c1_2__xreal_1,k1_numbers)
    & r2_hidden(c2_2__xreal_1,k1_numbers)
    & r2_hidden(c3_2__xreal_1,k1_numbers) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1])],[dt_k1_xboole_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t8_boole,existence_m1_subset_1,dt_m1_subset_1,cc1_xcmplx_0,cc7_xreal_0,rc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xboole_0,rc2_xcmplx_0,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_k1_numbers,dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,cc2_xreal_0,fc1_numbers,t1_subset,t7_boole,d1_xreal_0]),
    [interesting(0.8),file(xreal_1,e3_2__xreal_1),[file(xreal_1,e3_2__xreal_1)]]).

fof(d1_numbers,definition,(
    k1_numbers = k4_xboole_0(k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)),k1_tarski(k4_tarski(0,0))) ),
    file(numbers,d1_numbers),
    [interesting(0.9),axiom,file(numbers,d1_numbers)]).

fof(d4_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d4_xboole_0)]).

fof(e4_2__xreal_1,plain,
    ( r2_hidden(c1_2__xreal_1,k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)))
    & r2_hidden(c2_2__xreal_1,k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)))
    & r2_hidden(c3_2__xreal_1,k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xcmplx_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_boole,t3_boole,t3_subset,t4_boole,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,t1_numerals,t2_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k1_numbers,dt_k1_tarski,dt_k2_arytm_2,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k4_tarski,dt_k4_xboole_0,dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,fc1_numbers,fc1_zfmisc_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e3_2__xreal_1,d1_numbers,d4_xboole_0]),
    [interesting(0.8),file(xreal_1,e4_2__xreal_1),[file(xreal_1,e4_2__xreal_1)]]).

fof(d2_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d2_xboole_0)]).

fof(e1_2_1__xreal_1,plain,(
    ~ ( ~ ( r2_hidden(c1_2__xreal_1,k2_arytm_2)
          & r2_hidden(c2_2__xreal_1,k2_arytm_2)
          & r2_hidden(c3_2__xreal_1,k2_arytm_2) )
      & ~ ( r2_hidden(c1_2__xreal_1,k2_arytm_2)
          & r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
      & ~ ( r2_hidden(c2_2__xreal_1,k2_arytm_2)
          & r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
      & ~ ( r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
          & r2_hidden(c3_2__xreal_1,k2_arytm_2) )
      & ~ ( r2_hidden(c1_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
          & r2_hidden(c2_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2))
          & r2_hidden(c3_2__xreal_1,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xcmplx_0,rc1_arytm_3,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,rc1_xcmplx_0,rc1_xreal_0,rc2_xcmplx_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_boole,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_numbers,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,t1_numerals,t2_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k1_tarski,dt_k2_arytm_2,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1,fc2_arytm_2,t1_subset,t7_boole,spc0_boole,spc0_numerals,e4_2__xreal_1,d2_xboole_0]),
    [interesting(0.65),file(xreal_1,e1_2_1__xreal_1),[file(xreal_1,e1_2_1__xreal_1)]]).

fof(i4_2__xreal_1,plain,(
    r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(percases,[status(thm),assumptions([e1_2__xreal_1,e2_2__xreal_1,dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1])],[i1_2_1__xreal_1,i2_2_1__xreal_1,i3_2_1__xreal_1,i4_2_1__xreal_1,i5_2_1__xreal_1,e1_2_1__xreal_1]),
    [interesting(0.8),file(xreal_1,i4_2__xreal_1),[file(xreal_1,i4_2__xreal_1)]]).

fof(i3_2__xreal_1,plain,
    ( ( r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1)
      & r1_xreal_0(c2_2__xreal_1,c3_2__xreal_1) )
   => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1,dt_c3_2__xreal_1]),discharge_asm(discharge,[e1_2__xreal_1,e2_2__xreal_1])],[e1_2__xreal_1,e2_2__xreal_1,i4_2__xreal_1]),
    [interesting(0.8),file(xreal_1,i3_2__xreal_1),[file(xreal_1,i3_2__xreal_1)]]).

fof(i3_2_tmp__xreal_1,plain,
    ( v1_xreal_0(c3_2__xreal_1)
   => ( ( r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1)
        & r1_xreal_0(c2_2__xreal_1,c3_2__xreal_1) )
     => r1_xreal_0(c1_2__xreal_1,c3_2__xreal_1) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1]),discharge_asm(discharge,[dt_c3_2__xreal_1])],[dt_c3_2__xreal_1,i3_2__xreal_1]),
    [interesting(0.8),i2_2__xreal_1]).

fof(i2_2__xreal_1,plain,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ( ( r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1)
          & r1_xreal_0(c2_2__xreal_1,A) )
       => r1_xreal_0(c1_2__xreal_1,A) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_2__xreal_1,dt_c2_2__xreal_1])],[i3_2_tmp__xreal_1,dh_c3_2__xreal_1]),
    [interesting(0.8),file(xreal_1,i2_2__xreal_1),[file(xreal_1,i2_2__xreal_1)]]).

fof(i2_2_tmp__xreal_1,plain,
    ( v1_xreal_0(c2_2__xreal_1)
   => ! [A] : 
        ( v1_xreal_0(A)
       => ( ( r1_xreal_0(c1_2__xreal_1,c2_2__xreal_1)
            & r1_xreal_0(c2_2__xreal_1,A) )
         => r1_xreal_0(c1_2__xreal_1,A) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__xreal_1]),discharge_asm(discharge,[dt_c2_2__xreal_1])],[dt_c2_2__xreal_1,i2_2__xreal_1]),
    [interesting(0.8),i1_2__xreal_1]).

fof(i1_2__xreal_1,plain,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(c1_2__xreal_1,A)
              & r1_xreal_0(A,B) )
           => r1_xreal_0(c1_2__xreal_1,B) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_2__xreal_1])],[i2_2_tmp__xreal_1,dh_c2_2__xreal_1]),
    [interesting(0.8),file(xreal_1,i1_2__xreal_1),[file(xreal_1,i1_2__xreal_1)]]).

fof(i1_2_tmp__xreal_1,plain,
    ( v1_xreal_0(c1_2__xreal_1)
   => ! [A] : 
        ( v1_xreal_0(A)
       => ! [B] : 
            ( v1_xreal_0(B)
           => ( ( r1_xreal_0(c1_2__xreal_1,A)
                & r1_xreal_0(A,B) )
             => r1_xreal_0(c1_2__xreal_1,B) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__xreal_1])],[dt_c1_2__xreal_1,i1_2__xreal_1]),
    [interesting(1),t2_xreal_1]).

fof(t2_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => r1_xreal_0(A,C) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_2_tmp__xreal_1,dh_c1_2__xreal_1]),
    [interesting(1),file(xreal_1,t2_xreal_1),[file(xreal_1,t2_xreal_1)]]).