% Mizar ND problem: t4_arithm,arithm,93,7 
fof(dh_c1_7__arithm,definition,
    ( ( v1_xcmplx_0(c1_7__arithm)
     => k6_xcmplx_0(c1_7__arithm,0) = c1_7__arithm )
   => ! [A] : 
        ( v1_xcmplx_0(A)
       => k6_xcmplx_0(A,0) = A ) ),
    introduced(definition,[new_symbol(c1_7__arithm),file(arithm,c1_7__arithm)]),
    [interesting(0.8),axiom,file(arithm,c1_7__arithm)]).

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(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(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(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(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(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_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(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(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_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

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(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(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(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(commutativity_k2_xcmplx_0,theorem,(
    ! [A,B] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ),
    file(xcmplx_0,k2_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]).

fof(dt_k2_xcmplx_0,axiom,(
    $true ),
    file(xcmplx_0,k2_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]).

fof(dt_k6_xcmplx_0,axiom,(
    $true ),
    file(xcmplx_0,k6_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]).

fof(dt_c1_7__arithm,assumption,(
    v1_xcmplx_0(c1_7__arithm) ),
    introduced(assumption,[file(arithm,c1_7__arithm)]),
    [interesting(0.8),axiom,file(arithm,c1_7__arithm)]).

fof(fc2_xcmplx_0,theorem,(
    ! [A,B] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => v1_xcmplx_0(k2_xcmplx_0(A,B)) ) ),
    file(xcmplx_0,fc2_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,fc2_xcmplx_0)]).

fof(fc4_xcmplx_0,theorem,(
    ! [A,B] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => v1_xcmplx_0(k6_xcmplx_0(A,B)) ) ),
    file(xcmplx_0,fc4_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,fc4_xcmplx_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(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(spc0_boole,theorem,(
    v1_xboole_0(0) ),
    file(boole,spc0_boole),
    [interesting(0.9),axiom,file(boole,spc0_boole)]).

fof(fc6_xcmplx_0,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A) )
     => ( ~ v1_xboole_0(k4_xcmplx_0(A))
        & v1_xcmplx_0(k4_xcmplx_0(A)) ) ) ),
    file(xcmplx_0,fc6_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,fc6_xcmplx_0)]).

fof(involutiveness_k4_xcmplx_0,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ),
    file(xcmplx_0,k4_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]).

fof(dt_k4_xcmplx_0,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => v1_xcmplx_0(k4_xcmplx_0(A)) ) ),
    file(xcmplx_0,k4_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]).

fof(l2_arithm,plain,(
    k4_xcmplx_0(0) = 0 ),
    file(arithm,l2_arithm),
    [interesting(0.9),axiom,file(arithm,l2_arithm)]).

fof(d8_xcmplx_0,definition,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => k6_xcmplx_0(A,B) = k2_xcmplx_0(A,k4_xcmplx_0(B)) ) ) ),
    file(xcmplx_0,d8_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,d8_xcmplx_0)]).

fof(e1_7__arithm,plain,(
    k6_xcmplx_0(c1_7__arithm,0) = k2_xcmplx_0(c1_7__arithm,0) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_7__arithm])],[reflexivity_r1_tarski,cc1_arytm_3,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,fc1_xboole_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_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,fc1_numbers,fc6_xcmplx_0,rc1_xboole_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_7__arithm,fc2_xcmplx_0,fc4_xcmplx_0,rc1_xcmplx_0,spc0_numerals,spc0_boole,l2_arithm,d8_xcmplx_0]),
    [interesting(0.8),file(arithm,e1_7__arithm),[file(arithm,e1_7__arithm)]]).

fof(t1_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k2_xcmplx_0(A,0) = A ) ),
    file(arithm,t1_arithm),
    [interesting(0.9),axiom,file(arithm,t1_arithm)]).

fof(e2_7__arithm,plain,(
    k6_xcmplx_0(c1_7__arithm,0) = c1_7__arithm ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_7__arithm])],[reflexivity_r1_tarski,cc1_arytm_3,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,fc1_xboole_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_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_xcmplx_0,fc1_numbers,rc1_xboole_0,rc2_xboole_0,rc2_xcmplx_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xcmplx_0,dt_k2_xcmplx_0,dt_k6_xcmplx_0,dt_c1_7__arithm,fc2_xcmplx_0,fc4_xcmplx_0,rc1_xcmplx_0,spc0_numerals,spc0_boole,e1_7__arithm,t1_arithm]),
    [interesting(0.8),file(arithm,e2_7__arithm),[file(arithm,e2_7__arithm)]]).

fof(i2_7__arithm,theorem,(
    $true ),
    introduced(tautology,[file(arithm,i2_7__arithm)]),
    [interesting(0.8),trivial,file(arithm,i2_7__arithm)]).

fof(i1_7__arithm,plain,(
    k6_xcmplx_0(c1_7__arithm,0) = c1_7__arithm ),
    inference(conclusion,[status(thm),assumptions([dt_c1_7__arithm])],[e2_7__arithm,i2_7__arithm]),
    [interesting(0.8),file(arithm,i1_7__arithm),[file(arithm,i1_7__arithm)]]).

fof(i1_7_tmp__arithm,plain,
    ( v1_xcmplx_0(c1_7__arithm)
   => k6_xcmplx_0(c1_7__arithm,0) = c1_7__arithm ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_7__arithm])],[dt_c1_7__arithm,i1_7__arithm]),
    [interesting(1),t4_arithm]).

fof(t4_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k6_xcmplx_0(A,0) = A ) ),
    inference(let,[status(thm),assumptions([])],[i1_7_tmp__arithm,dh_c1_7__arithm]),
    [interesting(1),file(arithm,t4_arithm),[file(arithm,t4_arithm)]]).