% Mizar problem: t26_arytm_0,arytm_0,2366,30 
fof(t26_arytm_0,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( r2_hidden(k5_arytm_0(A,B),k1_numbers)
           => B = 0 ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_arytm_3,cc1_funct_2,cc2_arytm_3,cc4_arytm_3,cc5_funct_2,cc6_funct_2,commutativity_k2_tarski,d7_arytm_0,dt_k13_arytm_3,dt_k1_numbers,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_numbers,dt_k2_tarski,dt_k2_zfmisc_1,dt_k4_funct_4,dt_k4_ordinal2,dt_k5_arytm_0,dt_k5_funct_4,dt_k6_arytm_3,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc1_arytm_3,fc1_funct_4,fc1_numbers,fc1_subset_1,fc1_xboole_0,fc2_numbers,fc3_subset_1,fc4_subset_1,fc8_arytm_3,rc1_arytm_3,rc1_funct_2,rc1_subset_1,rc1_xboole_0,rc2_arytm_3,rc2_subset_1,rc2_xboole_0,rc3_arytm_3,redefinition_k13_arytm_3,redefinition_k5_funct_4,redefinition_m2_relset_1,reflexivity_r1_tarski,spc0_boole,t10_arytm_0,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(arytm_0,t26_arytm_0)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc1_arytm_3,axiom,(
    ! [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),
    []).

fof(cc1_funct_2,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => ( ( v1_funct_1(C)
          & v1_partfun1(C,A,B) )
       => ( v1_funct_1(C)
          & v1_funct_2(C,A,B) ) ) ) ),
    file(funct_2,cc1_funct_2),
    []).

fof(cc2_arytm_3,axiom,(
    ! [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),
    []).

fof(cc4_arytm_3,axiom,(
    ! [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),
    []).

fof(cc5_funct_2,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ! [C] : 
          ( m1_relset_1(C,A,B)
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,A,B) )
           => ( v1_funct_1(C)
              & v1_partfun1(C,A,B)
              & v1_funct_2(C,A,B) ) ) ) ) ),
    file(funct_2,cc5_funct_2),
    []).

fof(cc6_funct_2,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ! [C] : 
          ( m1_relset_1(C,A,B)
         => ( ( v1_funct_1(C)
              & v1_funct_2(C,A,B) )
           => ( v1_funct_1(C)
              & ~ v1_xboole_0(C)
              & v1_partfun1(C,A,B)
              & v1_funct_2(C,A,B) ) ) ) ) ),
    file(funct_2,cc6_funct_2),
    []).

fof(commutativity_k2_tarski,axiom,(
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ),
    file(tarski,k2_tarski),
    []).

fof(d7_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( ( B = 0
             => k5_arytm_0(A,B) = A )
            & ( B != 0
             => k5_arytm_0(A,B) = k5_funct_4(k1_numbers,0,k13_arytm_3,A,B) ) ) ) ) ),
    file(arytm_0,d7_arytm_0),
    []).

fof(dt_k13_arytm_3,axiom,
    ( ~ v1_xboole_0(k13_arytm_3)
    & v3_ordinal1(k13_arytm_3)
    & m1_subset_1(k13_arytm_3,k6_arytm_3) ),
    file(arytm_3,k13_arytm_3),
    []).

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

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

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

fof(dt_k2_numbers,axiom,(
    $true ),
    file(numbers,k2_numbers),
    []).

fof(dt_k2_tarski,axiom,(
    $true ),
    file(tarski,k2_tarski),
    []).

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

fof(dt_k4_funct_4,axiom,(
    $true ),
    file(funct_4,k4_funct_4),
    []).

fof(dt_k4_ordinal2,axiom,
    ( v3_ordinal1(k4_ordinal2)
    & ~ v1_xboole_0(k4_ordinal2) ),
    file(ordinal2,k4_ordinal2),
    []).

fof(dt_k5_arytm_0,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k5_arytm_0(A,B),k2_numbers) ) ),
    file(arytm_0,k5_arytm_0),
    []).

fof(dt_k5_funct_4,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => ( v1_funct_1(k5_funct_4(A,B,C,D,E))
        & v1_funct_2(k5_funct_4(A,B,C,D,E),k2_tarski(B,C),A)
        & m2_relset_1(k5_funct_4(A,B,C,D,E),k2_tarski(B,C),A) ) ) ),
    file(funct_4,k5_funct_4),
    []).

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

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    []).

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

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    []).

fof(fc1_arytm_3,axiom,
    ( ~ v1_xboole_0(k4_ordinal2)
    & v1_ordinal1(k4_ordinal2)
    & v2_ordinal1(k4_ordinal2)
    & v3_ordinal1(k4_ordinal2)
    & v4_ordinal2(k4_ordinal2) ),
    file(arytm_3,fc1_arytm_3),
    []).

fof(fc1_funct_4,axiom,(
    ! [A,B,C,D] : 
      ( v1_relat_1(k4_funct_4(A,B,C,D))
      & v1_funct_1(k4_funct_4(A,B,C,D)) ) ),
    file(funct_4,fc1_funct_4),
    []).

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

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

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc2_numbers,axiom,(
    ~ v1_xboole_0(k2_numbers) ),
    file(numbers,fc2_numbers),
    []).

fof(fc3_subset_1,axiom,(
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ),
    file(subset_1,fc3_subset_1),
    []).

fof(fc4_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    []).

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

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

fof(rc1_funct_2,axiom,(
    ! [A,B] : 
    ? [C] : 
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C)
      & v1_funct_2(C,A,B) ) ),
    file(funct_2,rc1_funct_2),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_arytm_3,axiom,(
    ? [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),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(rc3_arytm_3,axiom,(
    ? [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),
    []).

fof(redefinition_k13_arytm_3,axiom,(
    k13_arytm_3 = k4_ordinal2 ),
    file(arytm_3,k13_arytm_3),
    []).

fof(redefinition_k5_funct_4,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => k5_funct_4(A,B,C,D,E) = k4_funct_4(B,C,D,E) ) ),
    file(funct_4,k5_funct_4),
    []).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    []).

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

fof(spc0_boole,axiom,(
    v1_xboole_0(0) ),
    file(boole,spc0_boole),
    []).

fof(t10_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ~ r2_hidden(k5_funct_4(k1_numbers,0,k13_arytm_3,A,B),k1_numbers) ) ) ),
    file(arytm_0,t10_arytm_0),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

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

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

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

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).