% Mizar ND problem: t6_int_2,int_2,93,24 
fof(dh_c1_5__int_2,definition,
    ( ( m2_subset_1(c1_5__int_2,k1_numbers,k5_numbers)
     => ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => k2_nat_1(c1_5__int_2,A) = k2_nat_1(k5_nat_1(c1_5__int_2,A),k6_nat_1(c1_5__int_2,A)) ) )
   => ! [B] : 
        ( m2_subset_1(B,k1_numbers,k5_numbers)
       => ! [C] : 
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => k2_nat_1(B,C) = k2_nat_1(k5_nat_1(B,C),k6_nat_1(B,C)) ) ) ),
    introduced(definition,[new_symbol(c1_5__int_2),file(int_2,c1_5__int_2)]),
    [interesting(0.8),axiom,file(int_2,c1_5__int_2)]).

fof(dh_c2_5__int_2,definition,
    ( ( m2_subset_1(c2_5__int_2,k1_numbers,k5_numbers)
     => k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => k2_nat_1(c1_5__int_2,A) = k2_nat_1(k5_nat_1(c1_5__int_2,A),k6_nat_1(c1_5__int_2,A)) ) ),
    introduced(definition,[new_symbol(c2_5__int_2),file(int_2,c2_5__int_2)]),
    [interesting(0.8),axiom,file(int_2,c2_5__int_2)]).

fof(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

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(t1_subset,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    [interesting(0.9),axiom,file(subset,t1_subset)]).

fof(t4_subset,theorem,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    [interesting(0.9),axiom,file(subset,t4_subset)]).

fof(t5_subset,theorem,(
    ! [A,B,C] : 
      ~ ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C))
        & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    [interesting(0.9),axiom,file(subset,t5_subset)]).

fof(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_int_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => v1_int_1(A) ) ),
    file(int_1,cc3_int_1),
    [interesting(0.9),axiom,file(int_1,cc3_int_1)]).

fof(cc4_int_1,theorem,(
    ! [A] : 
      ( v1_int_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(int_1,cc4_int_1),
    [interesting(0.9),axiom,file(int_1,cc4_int_1)]).

fof(fc2_int_1,theorem,(
    ! [A,B] : 
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & v1_int_1(k3_xcmplx_0(A,B)) ) ) ),
    file(int_1,fc2_int_1),
    [interesting(0.9),axiom,file(int_1,fc2_int_1)]).

fof(fc7_int_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(B,A))
        & v1_xreal_0(k3_xcmplx_0(B,A))
        & v1_int_1(k3_xcmplx_0(B,A)) ) ) ),
    file(int_1,fc7_int_1),
    [interesting(0.9),axiom,file(int_1,fc7_int_1)]).

fof(rc1_int_1,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k1_numbers)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v1_int_1(A) ) ),
    file(int_1,rc1_int_1),
    [interesting(0.9),axiom,file(int_1,rc1_int_1)]).

fof(rc2_int_1,theorem,(
    ? [A] : v1_int_1(A) ),
    file(int_1,rc2_int_1),
    [interesting(0.9),axiom,file(int_1,rc2_int_1)]).

fof(spc7_arithm,theorem,(
    ! [A,B,C] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B)
        & v1_xcmplx_0(C) )
     => k3_xcmplx_0(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(A,k3_xcmplx_0(B,C)) ) ),
    file(arithm,spc6_arithm),
    [interesting(0.9),axiom,file(arithm,spc6_arithm)]).

fof(t2_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k3_xcmplx_0(A,0) = 0 ) ),
    file(arithm,t2_arithm),
    [interesting(0.9),axiom,file(arithm,t2_arithm)]).

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(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(commutativity_k3_xcmplx_0,theorem,(
    ! [A,B] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ),
    file(xcmplx_0,k3_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(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_k3_xcmplx_0,axiom,(
    $true ),
    file(xcmplx_0,k3_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]).

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(cc2_int_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_int_1(A) ) ) ),
    file(int_1,cc2_int_1),
    [interesting(0.9),axiom,file(int_1,cc2_int_1)]).

fof(rqRealMult__k3_xcmplx_0__r0_r0_r0,theorem,(
    k3_xcmplx_0(0,0) = 0 ),
    file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0),
    [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0)]).

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

fof(commutativity_k2_nat_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k2_nat_1(A,B) = k2_nat_1(B,A) ) ),
    file(nat_1,k2_nat_1),
    [interesting(0.9),axiom,file(nat_1,k2_nat_1)]).

fof(commutativity_k5_nat_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k5_nat_1(A,B) = k5_nat_1(B,A) ) ),
    file(nat_1,k5_nat_1),
    [interesting(0.9),axiom,file(nat_1,k5_nat_1)]).

fof(idempotence_k5_nat_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k5_nat_1(A,A) = A ) ),
    file(nat_1,k5_nat_1),
    [interesting(0.9),axiom,file(nat_1,k5_nat_1)]).

fof(commutativity_k6_nat_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k6_nat_1(A,B) = k6_nat_1(B,A) ) ),
    file(nat_1,k6_nat_1),
    [interesting(0.9),axiom,file(nat_1,k6_nat_1)]).

fof(idempotence_k6_nat_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k6_nat_1(A,A) = A ) ),
    file(nat_1,k6_nat_1),
    [interesting(0.9),axiom,file(nat_1,k6_nat_1)]).

fof(redefinition_k2_nat_1,definition,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) ),
    file(nat_1,k2_nat_1),
    [interesting(0.9),axiom,file(nat_1,k2_nat_1)]).

fof(dt_k2_nat_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) ),
    file(nat_1,k2_nat_1),
    [interesting(0.9),axiom,file(nat_1,k2_nat_1)]).

fof(dt_k5_nat_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k5_nat_1(A,B),k1_numbers,k5_numbers) ) ),
    file(nat_1,k5_nat_1),
    [interesting(0.9),axiom,file(nat_1,k5_nat_1)]).

fof(dt_k6_nat_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k6_nat_1(A,B),k1_numbers,k5_numbers) ) ),
    file(nat_1,k6_nat_1),
    [interesting(0.9),axiom,file(nat_1,k6_nat_1)]).

fof(dt_c1_5__int_2,assumption,(
    m2_subset_1(c1_5__int_2,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(int_2,c1_5__int_2)]),
    [interesting(0.8),axiom,file(int_2,c1_5__int_2)]).

fof(dt_c2_5__int_2,assumption,(
    m2_subset_1(c2_5__int_2,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(int_2,c2_5__int_2)]),
    [interesting(0.8),axiom,file(int_2,c2_5__int_2)]).

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(e1_5_2__int_2,assumption,
    ( c1_5__int_2 = 0
    | c2_5__int_2 = 0 ),
    introduced(assumption,[file(int_2,e1_5_2__int_2)]),
    [interesting(0.65),axiom,file(int_2,e1_5_2__int_2)]).

fof(t3_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k3_xcmplx_0(1,A) = A ) ),
    file(arithm,t3_arithm),
    [interesting(0.9),axiom,file(arithm,t3_arithm)]).

fof(rqRealMult__k3_xcmplx_0__r0_r1_r0,theorem,(
    k3_xcmplx_0(0,1) = 0 ),
    file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0),
    [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0)]).

fof(rqRealMult__k3_xcmplx_0__r1_r0_r0,theorem,(
    k3_xcmplx_0(1,0) = 0 ),
    file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0),
    [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0)]).

fof(spc1_numerals,theorem,
    ( v2_xreal_0(1)
    & m2_subset_1(1,k1_numbers,k5_numbers)
    & m1_subset_1(1,k5_numbers)
    & m1_subset_1(1,k1_numbers) ),
    file(numerals,spc1_numerals),
    [interesting(0.9),axiom,file(numerals,spc1_numerals)]).

fof(rqRealMult__k3_xcmplx_0__r1_r1_r1,theorem,(
    k3_xcmplx_0(1,1) = 1 ),
    file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1),
    [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1)]).

fof(e1_5_2_1__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(0,k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_2__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,spc7_arithm,t1_numerals,t2_arithm,t3_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k6_nat_1,idempotence_k6_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k6_nat_1,dt_c1_5__int_2,dt_c2_5__int_2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc0_numerals,spc1_numerals,e1_5_2__int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.5),file(int_2,e1_5_2_1__int_2),[file(int_2,e1_5_2_1__int_2)]]).

fof(t4_int_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( A = 0
              | B = 0 )
          <=> k5_nat_1(A,B) = 0 ) ) ) ),
    file(int_2,t4_int_2),
    [interesting(0.9),axiom,file(int_2,t4_int_2)]).

fof(e2_5_2_1__int_2,plain,(
    k2_nat_1(0,k6_nat_1(c1_5__int_2,c2_5__int_2)) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_2__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc3_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,spc7_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,commutativity_k6_nat_1,idempotence_k6_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_k6_nat_1,dt_m2_subset_1,dt_c1_5__int_2,dt_c2_5__int_2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc0_numerals,spc1_numerals,e1_5_2__int_2,t4_int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.5),file(int_2,e2_5_2_1__int_2),[file(int_2,e2_5_2_1__int_2)]]).

fof(e2_5_2__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_2__int_2])],[e1_5_2_1__int_2,e2_5_2_1__int_2]),
    [interesting(0.65),file(int_2,e2_5_2__int_2),[file(int_2,e2_5_2__int_2)]]).

fof(e3_5_2__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_2__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc1_int_1,rc2_int_1,spc7_arithm,t2_subset,t3_subset,commutativity_k3_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k3_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc2_int_1,commutativity_k2_nat_1,commutativity_k5_nat_1,idempotence_k5_nat_1,commutativity_k6_nat_1,idempotence_k6_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k5_nat_1,dt_k6_nat_1,dt_c1_5__int_2,dt_c2_5__int_2,e2_5_2__int_2]),
    [interesting(0.65),file(int_2,e3_5_2__int_2),[file(int_2,e3_5_2__int_2)]]).

fof(i2_5_2__int_2,theorem,(
    $true ),
    introduced(tautology,[file(int_2,i2_5_2__int_2)]),
    [interesting(0.65),trivial,file(int_2,i2_5_2__int_2)]).

fof(i1_5_2__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_2__int_2])],[e3_5_2__int_2,i2_5_2__int_2]),
    [interesting(0.65),file(int_2,i1_5_2__int_2),[file(int_2,i1_5_2__int_2)]]).

fof(e2_5__int_2,plain,
    ( ( c1_5__int_2 = 0
      | c2_5__int_2 = 0 )
   => k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2]),discharge_asm(discharge,[e1_5_2__int_2])],[e1_5_2__int_2,i1_5_2__int_2]),
    [interesting(0.8),file(int_2,e2_5__int_2),[file(int_2,e2_5__int_2)]]).

fof(e1_5_1__int_2,assumption,
    ( c1_5__int_2 != 0
    & c2_5__int_2 != 0 ),
    introduced(assumption,[file(int_2,e1_5_1__int_2)]),
    [interesting(0.65),axiom,file(int_2,e1_5_1__int_2)]).

fof(fc3_int_1,theorem,(
    ! [A] : 
      ( v1_int_1(A)
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A))
        & v1_int_1(k4_xcmplx_0(A)) ) ) ),
    file(int_1,fc3_int_1),
    [interesting(0.9),axiom,file(int_1,fc3_int_1)]).

fof(fc5_int_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A))
        & ~ v2_xreal_0(k4_xcmplx_0(A))
        & v1_int_1(k4_xcmplx_0(A)) ) ) ),
    file(int_1,fc5_int_1),
    [interesting(0.9),axiom,file(int_1,fc5_int_1)]).

fof(spc2_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k3_xcmplx_0(A,k4_xcmplx_0(1)) = k4_xcmplx_0(A) ) ),
    file(arithm,spc2_arithm),
    [interesting(0.9),axiom,file(arithm,spc2_arithm)]).

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(reflexivity_r1_nat_1,theorem,(
    ! [A,B] : 
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => r1_nat_1(A,A) ) ),
    file(nat_1,r1_nat_1),
    [interesting(0.9),axiom,file(nat_1,r1_nat_1)]).

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(dh_c1_5_1__int_2,definition,
    ( ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),A) )
   => ( m2_subset_1(c1_5_1__int_2,k1_numbers,k5_numbers)
      & k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),c1_5_1__int_2) ) ),
    introduced(definition,[new_symbol(c1_5_1__int_2),file(int_2,c1_5_1__int_2)]),
    [interesting(0.65),axiom,file(int_2,c1_5_1__int_2)]).

fof(t56_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ! [C] : 
              ( v4_ordinal2(C)
             => ( r1_nat_1(A,B)
               => r1_nat_1(A,k3_xcmplx_0(B,C)) ) ) ) ) ),
    file(nat_1,t56_nat_1),
    [interesting(0.9),axiom,file(nat_1,t56_nat_1)]).

fof(e3_5_1__int_2,plain,
    ( r1_nat_1(c2_5__int_2,c2_5__int_2)
   => r1_nat_1(c2_5__int_2,k2_nat_1(c2_5__int_2,c1_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,rc1_int_1,t2_subset,t3_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,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t3_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_c1_5__int_2,dt_c2_5__int_2,cc3_int_1,spc1_numerals,t56_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.65),file(int_2,e3_5_1__int_2),[file(int_2,e3_5_1__int_2)]]).

fof(e2_5_1__int_2,plain,
    ( r1_nat_1(c1_5__int_2,c1_5__int_2)
   => r1_nat_1(c1_5__int_2,k2_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,rc1_int_1,t2_subset,t3_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,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t3_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_c1_5__int_2,dt_c2_5__int_2,cc3_int_1,spc1_numerals,t56_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.65),file(int_2,e2_5_1__int_2),[file(int_2,e2_5_1__int_2)]]).

fof(d4_nat_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k5_nat_1(A,B)
              <=> ( r1_nat_1(A,C)
                  & r1_nat_1(B,C)
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_nat_1(A,D)
                          & r1_nat_1(B,D) )
                       => r1_nat_1(C,D) ) ) ) ) ) ) ) ),
    file(nat_1,d4_nat_1),
    [interesting(0.9),axiom,file(nat_1,d4_nat_1)]).

fof(e4_5_1__int_2,plain,(
    r1_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k2_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc4_int_1,fc2_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc3_int_1,spc7_arithm,t2_subset,t3_arithm,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c2_5__int_2,spc1_numerals,e3_5_1__int_2,e2_5_1__int_2,d4_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.65),file(int_2,e4_5_1__int_2),[file(int_2,e4_5_1__int_2)]]).

fof(d3_nat_1,definition,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r1_nat_1(A,B)
          <=> ? [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & B = k3_xcmplx_0(A,C) ) ) ) ) ),
    file(nat_1,d3_nat_1),
    [interesting(0.9),axiom,file(nat_1,d3_nat_1)]).

fof(e5_5_1__int_2,plain,(
    ? [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
      & k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t2_subset,t3_arithm,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c2_5__int_2,cc3_int_1,spc1_numerals,e4_5_1__int_2,d3_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.65),file(int_2,e5_5_1__int_2),[file(int_2,e5_5_1__int_2)]]).

fof(dt_c1_5_1__int_2,plain,(
    m2_subset_1(c1_5_1__int_2,k1_numbers,k5_numbers) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[dh_c1_5_1__int_2,e5_5_1__int_2]),
    [interesting(0.65),file(int_2,c1_5_1__int_2),[file(int_2,c1_5_1__int_2)]]).

fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,(
    k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ),
    file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1),
    [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]).

fof(dh_c1_5_1_1__int_2,definition,
    ( ( m2_subset_1(c1_5_1_1__int_2,k1_numbers,k5_numbers)
     => ( ( r1_nat_1(c1_5_1_1__int_2,c1_5__int_2)
          & r1_nat_1(c1_5_1_1__int_2,c2_5__int_2) )
       => r1_nat_1(c1_5_1_1__int_2,c1_5_1__int_2) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_nat_1(A,c1_5__int_2)
            & r1_nat_1(A,c2_5__int_2) )
         => r1_nat_1(A,c1_5_1__int_2) ) ) ),
    introduced(definition,[new_symbol(c1_5_1_1__int_2),file(int_2,c1_5_1_1__int_2)]),
    [interesting(0.5),axiom,file(int_2,c1_5_1_1__int_2)]).

fof(e1_5_1_1__int_2,assumption,
    ( r1_nat_1(c1_5_1_1__int_2,c1_5__int_2)
    & r1_nat_1(c1_5_1_1__int_2,c2_5__int_2) ),
    introduced(assumption,[file(int_2,e1_5_1_1__int_2)]),
    [interesting(0.5),axiom,file(int_2,e1_5_1_1__int_2)]).

fof(dt_c1_5_1_1__int_2,assumption,(
    m2_subset_1(c1_5_1_1__int_2,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(int_2,c1_5_1_1__int_2)]),
    [interesting(0.5),axiom,file(int_2,c1_5_1_1__int_2)]).

fof(dh_c4_5_1_1__int_2,definition,
    ( ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),A) )
   => ( m2_subset_1(c4_5_1_1__int_2,k1_numbers,k5_numbers)
      & k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),c4_5_1_1__int_2) ) ),
    introduced(definition,[new_symbol(c4_5_1_1__int_2),file(int_2,c4_5_1_1__int_2)]),
    [interesting(0.5),axiom,file(int_2,c4_5_1_1__int_2)]).

fof(dh_c2_5_1_1__int_2,definition,
    ( ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & c1_5__int_2 = k2_nat_1(c1_5_1_1__int_2,A) )
   => ( m2_subset_1(c2_5_1_1__int_2,k1_numbers,k5_numbers)
      & c1_5__int_2 = k2_nat_1(c1_5_1_1__int_2,c2_5_1_1__int_2) ) ),
    introduced(definition,[new_symbol(c2_5_1_1__int_2),file(int_2,c2_5_1_1__int_2)]),
    [interesting(0.5),axiom,file(int_2,c2_5_1_1__int_2)]).

fof(e2_5_1_1__int_2,plain,(
    ? [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
      & c1_5__int_2 = k2_nat_1(c1_5_1_1__int_2,A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t2_subset,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,cc3_int_1,e1_5_1_1__int_2,d3_nat_1]),
    [interesting(0.5),file(int_2,e2_5_1_1__int_2),[file(int_2,e2_5_1_1__int_2)]]).

fof(dt_c2_5_1_1__int_2,plain,(
    m2_subset_1(c2_5_1_1__int_2,k1_numbers,k5_numbers) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[dh_c2_5_1_1__int_2,e2_5_1_1__int_2]),
    [interesting(0.5),file(int_2,c2_5_1_1__int_2),[file(int_2,c2_5_1_1__int_2)]]).

fof(dh_c3_5_1_1__int_2,definition,
    ( ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & c2_5__int_2 = k2_nat_1(c1_5_1_1__int_2,A) )
   => ( m2_subset_1(c3_5_1_1__int_2,k1_numbers,k5_numbers)
      & c2_5__int_2 = k2_nat_1(c1_5_1_1__int_2,c3_5_1_1__int_2) ) ),
    introduced(definition,[new_symbol(c3_5_1_1__int_2),file(int_2,c3_5_1_1__int_2)]),
    [interesting(0.5),axiom,file(int_2,c3_5_1_1__int_2)]).

fof(e4_5_1_1__int_2,plain,(
    ? [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
      & c2_5__int_2 = k2_nat_1(c1_5_1_1__int_2,A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t2_subset,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,cc3_int_1,e1_5_1_1__int_2,d3_nat_1]),
    [interesting(0.5),file(int_2,e4_5_1_1__int_2),[file(int_2,e4_5_1_1__int_2)]]).

fof(dt_c3_5_1_1__int_2,plain,(
    m2_subset_1(c3_5_1_1__int_2,k1_numbers,k5_numbers) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[dh_c3_5_1_1__int_2,e4_5_1_1__int_2]),
    [interesting(0.5),file(int_2,c3_5_1_1__int_2),[file(int_2,c3_5_1_1__int_2)]]).

fof(fc4_int_1,theorem,(
    ! [A,B] : 
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_int_1(k6_xcmplx_0(A,B)) ) ) ),
    file(int_1,fc4_int_1),
    [interesting(0.9),axiom,file(int_1,fc4_int_1)]).

fof(fc8_int_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & v1_int_1(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(B,A))
        & v1_xreal_0(k6_xcmplx_0(B,A))
        & v1_int_1(k6_xcmplx_0(B,A)) ) ) ),
    file(int_1,fc8_int_1),
    [interesting(0.9),axiom,file(int_1,fc8_int_1)]).

fof(fc9_int_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v1_int_1(k6_xcmplx_0(A,B)) ) ) ),
    file(int_1,fc9_int_1),
    [interesting(0.9),axiom,file(int_1,fc9_int_1)]).

fof(spc4_arithm,theorem,(
    ! [A,B,C] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B)
        & v1_xcmplx_0(C) )
     => k3_xcmplx_0(A,k7_xcmplx_0(B,C)) = k7_xcmplx_0(k3_xcmplx_0(A,B),C) ) ),
    file(arithm,spc4_arithm),
    [interesting(0.9),axiom,file(arithm,spc4_arithm)]).

fof(spc9_arithm,theorem,(
    ! [A,B] : 
      ( ( v1_xcmplx_0(A)
        & v1_xcmplx_0(B) )
     => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ),
    file(arithm,spc9_arithm),
    [interesting(0.9),axiom,file(arithm,spc9_arithm)]).

fof(t4_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k6_xcmplx_0(A,0) = A ) ),
    file(arithm,t4_arithm),
    [interesting(0.9),axiom,file(arithm,t4_arithm)]).

fof(t5_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k7_xcmplx_0(0,A) = 0 ) ),
    file(arithm,t5_arithm),
    [interesting(0.9),axiom,file(arithm,t5_arithm)]).

fof(t6_arithm,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k7_xcmplx_0(A,1) = A ) ),
    file(arithm,t6_arithm),
    [interesting(0.9),axiom,file(arithm,t6_arithm)]).

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_k7_xcmplx_0,axiom,(
    $true ),
    file(xcmplx_0,k7_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k7_xcmplx_0)]).

fof(rqRealDiff__k6_xcmplx_0__r0_r0_r0,theorem,(
    k6_xcmplx_0(0,0) = 0 ),
    file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0),
    [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0)]).

fof(rqRealDiff__k6_xcmplx_0__r0_r1_rm1,theorem,(
    k6_xcmplx_0(0,1) = k4_xcmplx_0(1) ),
    file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1),
    [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1)]).

fof(rqRealDiff__k6_xcmplx_0__r0_rm1_r1,theorem,(
    k6_xcmplx_0(0,k4_xcmplx_0(1)) = 1 ),
    file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1),
    [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1)]).

fof(rqRealDiff__k6_xcmplx_0__r1_r0_r1,theorem,(
    k6_xcmplx_0(1,0) = 1 ),
    file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1),
    [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1)]).

fof(rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,theorem,(
    k6_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ),
    file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1),
    [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1)]).

fof(rqRealDiv__k7_xcmplx_0__r0_r1_r0,theorem,(
    k7_xcmplx_0(0,1) = 0 ),
    file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0),
    [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0)]).

fof(rqRealDiv__k7_xcmplx_0__r1_r1_r1,theorem,(
    k7_xcmplx_0(1,1) = 1 ),
    file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1),
    [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1)]).

fof(rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,theorem,(
    k7_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(1) ),
    file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1),
    [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1)]).

fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,(
    k4_xcmplx_0(0) = 0 ),
    file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0),
    [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]).

fof(e3_5_1_1__int_2,plain,(
    c1_5__int_2 = k2_nat_1(c1_5_1_1__int_2,c2_5_1_1__int_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[dh_c2_5_1_1__int_2,e2_5_1_1__int_2]),
    [interesting(0.5),file(int_2,e3_5_1_1__int_2),[file(int_2,e3_5_1_1__int_2)]]).

fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,(
    k4_xcmplx_0(1) = k4_xcmplx_0(1) ),
    file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1),
    [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]).

fof(e7_5_1_1__int_2,plain,(
    k2_nat_1(c1_5__int_2,c3_5_1_1__int_2) = k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,spc2_arithm,spc7_arithm,t3_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5_1_1__int_2,dt_c3_5_1_1__int_2,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,e3_5_1_1__int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]),
    [interesting(0.5),file(int_2,e7_5_1_1__int_2),[file(int_2,e7_5_1_1__int_2)]]).

fof(e5_5_1_1__int_2,plain,(
    c2_5__int_2 = k2_nat_1(c1_5_1_1__int_2,c3_5_1_1__int_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[dh_c3_5_1_1__int_2,e4_5_1_1__int_2]),
    [interesting(0.5),file(int_2,e5_5_1_1__int_2),[file(int_2,e5_5_1_1__int_2)]]).

fof(e6_5_1_1__int_2,plain,(
    k2_nat_1(c2_5__int_2,c2_5_1_1__int_2) = k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,spc2_arithm,spc7_arithm,t3_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c2_5_1_1__int_2,dt_c3_5_1_1__int_2,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,e5_5_1_1__int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]),
    [interesting(0.5),file(int_2,e6_5_1_1__int_2),[file(int_2,e6_5_1_1__int_2)]]).

fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,(
    k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ),
    file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0),
    [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]).

fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,(
    k6_xcmplx_0(1,1) = 0 ),
    file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0),
    [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]).

fof(rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,theorem,(
    k7_xcmplx_0(1,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ),
    file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1),
    [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1)]).

fof(e8_5_1_1__int_2,plain,
    ( r1_nat_1(c1_5__int_2,k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2))
    & r1_nat_1(c2_5__int_2,k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc5_int_1,fc7_int_1,fc8_int_1,fc9_int_1,rc2_int_1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_arithm,t5_arithm,t6_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c2_5_1_1__int_2,dt_c3_5_1_1__int_2,cc3_int_1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,spc0_numerals,spc1_numerals,e7_5_1_1__int_2,e6_5_1_1__int_2,d3_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1]),
    [interesting(0.5),file(int_2,e8_5_1_1__int_2),[file(int_2,e8_5_1_1__int_2)]]).

fof(e9_5_1_1__int_2,plain,(
    r1_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc4_int_1,fc2_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc3_int_1,spc7_arithm,t2_subset,t3_arithm,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c2_5_1_1__int_2,dt_c3_5_1_1__int_2,spc1_numerals,e8_5_1_1__int_2,d4_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.5),file(int_2,e9_5_1_1__int_2),[file(int_2,e9_5_1_1__int_2)]]).

fof(e10_5_1_1__int_2,plain,(
    ? [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
      & k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t2_subset,t3_arithm,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c2_5_1_1__int_2,dt_c3_5_1_1__int_2,cc3_int_1,spc1_numerals,e9_5_1_1__int_2,d3_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.5),file(int_2,e10_5_1_1__int_2),[file(int_2,e10_5_1_1__int_2)]]).

fof(dt_c4_5_1_1__int_2,plain,(
    m2_subset_1(c4_5_1_1__int_2,k1_numbers,k5_numbers) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[dh_c4_5_1_1__int_2,e10_5_1_1__int_2]),
    [interesting(0.5),file(int_2,c4_5_1_1__int_2),[file(int_2,c4_5_1_1__int_2)]]).

fof(e13_5_1_1__int_2,plain,(
    k5_nat_1(c1_5__int_2,c2_5__int_2) != 0 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc3_int_1,cc4_int_1,rc1_int_1,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,t1_numerals,t2_subset,t3_subset,commutativity_k5_nat_1,idempotence_k5_nat_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c2_5__int_2,spc0_numerals,e1_5_1__int_2,t4_int_2]),
    [interesting(0.5),file(int_2,e13_5_1_1__int_2),[file(int_2,e13_5_1_1__int_2)]]).

fof(e6_5_1__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),c1_5_1__int_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[dh_c1_5_1__int_2,e5_5_1__int_2]),
    [interesting(0.65),file(int_2,e6_5_1__int_2),[file(int_2,e6_5_1__int_2)]]).

fof(e1_5_1_1_1__int_2,plain,(
    k2_nat_1(c1_5_1__int_2,k5_nat_1(c1_5__int_2,c2_5__int_2)) = k2_nat_1(k2_nat_1(c2_5_1_1__int_2,k2_nat_1(c3_5_1_1__int_2,c1_5_1_1__int_2)),c1_5_1_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc7_int_1,fc8_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,fc9_int_1,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t3_arithm,t4_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_nat_1,dt_k6_xcmplx_0,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c2_5_1_1__int_2,dt_c3_5_1_1__int_2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,spc0_numerals,spc1_numerals,e6_5_1__int_2,e3_5_1_1__int_2,e5_5_1_1__int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1]),
    [interesting(0.35),file(int_2,e1_5_1_1_1__int_2),[file(int_2,e1_5_1_1_1__int_2)]]).

fof(e11_5_1_1__int_2,plain,(
    k2_nat_1(k2_nat_1(c2_5_1_1__int_2,c3_5_1_1__int_2),c1_5_1_1__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),c4_5_1_1__int_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[dh_c4_5_1_1__int_2,e10_5_1_1__int_2]),
    [interesting(0.5),file(int_2,e11_5_1_1__int_2),[file(int_2,e11_5_1_1__int_2)]]).

fof(e2_5_1_1_1__int_2,plain,(
    k2_nat_1(k2_nat_1(c2_5_1_1__int_2,k2_nat_1(c3_5_1_1__int_2,c1_5_1_1__int_2)),c1_5_1_1__int_2) = k2_nat_1(k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),c4_5_1_1__int_2),c1_5_1_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,spc2_arithm,spc7_arithm,t3_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_nat_1,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c2_5_1_1__int_2,dt_c3_5_1_1__int_2,dt_c4_5_1_1__int_2,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,e11_5_1_1__int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]),
    [interesting(0.35),file(int_2,e2_5_1_1_1__int_2),[file(int_2,e2_5_1_1_1__int_2)]]).

fof(e3_5_1_1_1__int_2,plain,(
    k2_nat_1(k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),c4_5_1_1__int_2),c1_5_1_1__int_2) = k2_nat_1(k2_nat_1(c4_5_1_1__int_2,c1_5_1_1__int_2),k5_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,spc7_arithm,t3_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c4_5_1_1__int_2,spc1_numerals,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.35),file(int_2,e3_5_1_1_1__int_2),[file(int_2,e3_5_1_1_1__int_2)]]).

fof(e12_5_1_1__int_2,plain,(
    k2_nat_1(c1_5_1__int_2,k5_nat_1(c1_5__int_2,c2_5__int_2)) = k2_nat_1(k2_nat_1(c4_5_1_1__int_2,c1_5_1_1__int_2),k5_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[e1_5_1_1_1__int_2,e2_5_1_1_1__int_2,e3_5_1_1_1__int_2]),
    [interesting(0.5),file(int_2,e12_5_1_1__int_2),[file(int_2,e12_5_1_1__int_2)]]).

fof(t5_xcmplx_1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => ! [B] : 
          ( v1_xcmplx_0(B)
         => ! [C] : 
              ( v1_xcmplx_0(C)
             => ( k3_xcmplx_0(B,A) = k3_xcmplx_0(C,A)
               => ( A = 0
                  | B = C ) ) ) ) ) ),
    file(xcmplx_1,t5_xcmplx_1),
    [interesting(0.9),axiom,file(xcmplx_1,t5_xcmplx_1)]).

fof(e14_5_1_1__int_2,plain,(
    c1_5_1__int_2 = k2_nat_1(c1_5_1_1__int_2,c4_5_1_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([e1_5_1__int_2,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc7_int_1,fc8_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,fc9_int_1,t1_numerals,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_nat_1,dt_k6_xcmplx_0,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,dt_c4_5_1_1__int_2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc2_arithm,spc7_arithm,spc9_arithm,t2_arithm,t3_arithm,t4_arithm,spc0_numerals,spc1_numerals,e13_5_1_1__int_2,e12_5_1_1__int_2,t5_xcmplx_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1]),
    [interesting(0.5),file(int_2,e14_5_1_1__int_2),[file(int_2,e14_5_1_1__int_2)]]).

fof(e15_5_1_1__int_2,plain,(
    r1_nat_1(c1_5_1_1__int_2,c1_5_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([e1_5_1__int_2,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc5_int_1,fc7_int_1,rc2_int_1,spc2_arithm,spc7_arithm,t2_subset,t3_arithm,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_m2_subset_1,dt_c1_5_1__int_2,dt_c1_5_1_1__int_2,dt_c4_5_1_1__int_2,cc3_int_1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,e14_5_1_1__int_2,d3_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]),
    [interesting(0.5),file(int_2,e15_5_1_1__int_2),[file(int_2,e15_5_1_1__int_2)]]).

fof(i3_5_1_1__int_2,theorem,(
    $true ),
    introduced(tautology,[file(int_2,i3_5_1_1__int_2)]),
    [interesting(0.5),trivial,file(int_2,i3_5_1_1__int_2)]).

fof(i2_5_1_1__int_2,plain,(
    r1_nat_1(c1_5_1_1__int_2,c1_5_1__int_2) ),
    inference(conclusion,[status(thm),assumptions([e1_5_1__int_2,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2,e1_5_1_1__int_2])],[e15_5_1_1__int_2,i3_5_1_1__int_2]),
    [interesting(0.5),file(int_2,i2_5_1_1__int_2),[file(int_2,i2_5_1_1__int_2)]]).

fof(i1_5_1_1__int_2,plain,
    ( ( r1_nat_1(c1_5_1_1__int_2,c1_5__int_2)
      & r1_nat_1(c1_5_1_1__int_2,c2_5__int_2) )
   => r1_nat_1(c1_5_1_1__int_2,c1_5_1__int_2) ),
    inference(discharge_asm,[status(thm),assumptions([e1_5_1__int_2,dt_c1_5__int_2,dt_c1_5_1_1__int_2,dt_c2_5__int_2]),discharge_asm(discharge,[e1_5_1_1__int_2])],[e1_5_1_1__int_2,i2_5_1_1__int_2]),
    [interesting(0.5),file(int_2,i1_5_1_1__int_2),[file(int_2,i1_5_1_1__int_2)]]).

fof(i1_5_1_1_tmp__int_2,plain,
    ( m2_subset_1(c1_5_1_1__int_2,k1_numbers,k5_numbers)
   => ( ( r1_nat_1(c1_5_1_1__int_2,c1_5__int_2)
        & r1_nat_1(c1_5_1_1__int_2,c2_5__int_2) )
     => r1_nat_1(c1_5_1_1__int_2,c1_5_1__int_2) ) ),
    inference(discharge_asm,[status(thm),assumptions([e1_5_1__int_2,dt_c1_5__int_2,dt_c2_5__int_2]),discharge_asm(discharge,[dt_c1_5_1_1__int_2])],[dt_c1_5_1_1__int_2,i1_5_1_1__int_2]),
    [interesting(0.65),e17_5_1__int_2]).

fof(e17_5_1__int_2,plain,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ( r1_nat_1(A,c1_5__int_2)
          & r1_nat_1(A,c2_5__int_2) )
       => r1_nat_1(A,c1_5_1__int_2) ) ) ),
    inference(let,[status(thm),assumptions([e1_5_1__int_2,dt_c1_5__int_2,dt_c2_5__int_2])],[i1_5_1_1_tmp__int_2,dh_c1_5_1_1__int_2]),
    [interesting(0.65),file(int_2,e17_5_1__int_2),[file(int_2,e17_5_1__int_2)]]).

fof(dh_c2_5_1__int_2,definition,
    ( ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c1_5__int_2,A) )
   => ( m2_subset_1(c2_5_1__int_2,k1_numbers,k5_numbers)
      & k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c1_5__int_2,c2_5_1__int_2) ) ),
    introduced(definition,[new_symbol(c2_5_1__int_2),file(int_2,c2_5_1__int_2)]),
    [interesting(0.65),axiom,file(int_2,c2_5_1__int_2)]).

fof(e7_5_1__int_2,plain,
    ( r1_nat_1(c1_5__int_2,k5_nat_1(c1_5__int_2,c2_5__int_2))
    & r1_nat_1(c2_5__int_2,k5_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc4_int_1,rc1_int_1,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc3_int_1,t2_subset,t3_subset,commutativity_k5_nat_1,idempotence_k5_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c2_5__int_2,d4_nat_1]),
    [interesting(0.65),file(int_2,e7_5_1__int_2),[file(int_2,e7_5_1__int_2)]]).

fof(e8_5_1__int_2,plain,(
    ? [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
      & k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c1_5__int_2,A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t2_subset,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c2_5__int_2,cc3_int_1,e7_5_1__int_2,d3_nat_1]),
    [interesting(0.65),file(int_2,e8_5_1__int_2),[file(int_2,e8_5_1__int_2)]]).

fof(dt_c2_5_1__int_2,plain,(
    m2_subset_1(c2_5_1__int_2,k1_numbers,k5_numbers) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[dh_c2_5_1__int_2,e8_5_1__int_2]),
    [interesting(0.65),file(int_2,c2_5_1__int_2),[file(int_2,c2_5_1__int_2)]]).

fof(dh_c3_5_1__int_2,definition,
    ( ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c2_5__int_2,A) )
   => ( m2_subset_1(c3_5_1__int_2,k1_numbers,k5_numbers)
      & k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c2_5__int_2,c3_5_1__int_2) ) ),
    introduced(definition,[new_symbol(c3_5_1__int_2),file(int_2,c3_5_1__int_2)]),
    [interesting(0.65),axiom,file(int_2,c3_5_1__int_2)]).

fof(e10_5_1__int_2,plain,(
    ? [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
      & k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c2_5__int_2,A) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc2_int_1,spc7_arithm,t2_subset,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__int_2,dt_c2_5__int_2,cc3_int_1,e7_5_1__int_2,d3_nat_1]),
    [interesting(0.65),file(int_2,e10_5_1__int_2),[file(int_2,e10_5_1__int_2)]]).

fof(dt_c3_5_1__int_2,plain,(
    m2_subset_1(c3_5_1__int_2,k1_numbers,k5_numbers) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[dh_c3_5_1__int_2,e10_5_1__int_2]),
    [interesting(0.65),file(int_2,c3_5_1__int_2),[file(int_2,c3_5_1__int_2)]]).

fof(e11_5_1__int_2,plain,(
    k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c2_5__int_2,c3_5_1__int_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[dh_c3_5_1__int_2,e10_5_1__int_2]),
    [interesting(0.65),file(int_2,e11_5_1__int_2),[file(int_2,e11_5_1__int_2)]]).

fof(e14_5_1__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c2_5__int_2,k2_nat_1(c1_5_1__int_2,c3_5_1__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc7_int_1,fc8_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,fc9_int_1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t3_arithm,t4_arithm,t5_arithm,t6_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_nat_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c2_5__int_2,dt_c3_5_1__int_2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,spc0_numerals,spc1_numerals,e6_5_1__int_2,e11_5_1__int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0]),
    [interesting(0.65),file(int_2,e14_5_1__int_2),[file(int_2,e14_5_1__int_2)]]).

fof(e15_5_1__int_2,plain,(
    c1_5__int_2 = k2_nat_1(c1_5_1__int_2,c3_5_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc7_int_1,fc8_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,fc9_int_1,t1_numerals,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c2_5__int_2,dt_c3_5_1__int_2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc2_arithm,spc7_arithm,spc9_arithm,t2_arithm,t3_arithm,t4_arithm,spc0_numerals,spc1_numerals,e14_5_1__int_2,e1_5_1__int_2,t5_xcmplx_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1]),
    [interesting(0.65),file(int_2,e15_5_1__int_2),[file(int_2,e15_5_1__int_2)]]).

fof(e9_5_1__int_2,plain,(
    k5_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c1_5__int_2,c2_5_1__int_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[dh_c2_5_1__int_2,e8_5_1__int_2]),
    [interesting(0.65),file(int_2,e9_5_1__int_2),[file(int_2,e9_5_1__int_2)]]).

fof(e12_5_1__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(c1_5__int_2,k2_nat_1(c1_5_1__int_2,c2_5_1__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc7_int_1,fc8_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,fc9_int_1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t3_arithm,t4_arithm,t5_arithm,t6_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_nat_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c2_5__int_2,dt_c2_5_1__int_2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,spc0_numerals,spc1_numerals,e6_5_1__int_2,e9_5_1__int_2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0]),
    [interesting(0.65),file(int_2,e12_5_1__int_2),[file(int_2,e12_5_1__int_2)]]).

fof(e13_5_1__int_2,plain,(
    c2_5__int_2 = k2_nat_1(c1_5_1__int_2,c2_5_1__int_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc7_int_1,fc8_int_1,rc1_int_1,rc2_int_1,t2_subset,t3_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,cc2_int_1,fc5_int_1,fc9_int_1,t1_numerals,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,redefinition_k2_nat_1,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c2_5__int_2,dt_c2_5_1__int_2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc2_arithm,spc7_arithm,spc9_arithm,t2_arithm,t3_arithm,t4_arithm,spc0_numerals,spc1_numerals,e12_5_1__int_2,e1_5_1__int_2,t5_xcmplx_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1]),
    [interesting(0.65),file(int_2,e13_5_1__int_2),[file(int_2,e13_5_1__int_2)]]).

fof(e16_5_1__int_2,plain,
    ( r1_nat_1(c1_5_1__int_2,c1_5__int_2)
    & r1_nat_1(c1_5_1__int_2,c2_5__int_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,rc1_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc4_int_1,fc2_int_1,fc3_int_1,fc4_int_1,fc5_int_1,fc7_int_1,fc8_int_1,fc9_int_1,rc2_int_1,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_arithm,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c2_5__int_2,dt_c2_5_1__int_2,dt_c3_5_1__int_2,cc3_int_1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,spc0_numerals,spc1_numerals,e15_5_1__int_2,e13_5_1__int_2,d3_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1]),
    [interesting(0.65),file(int_2,e16_5_1__int_2),[file(int_2,e16_5_1__int_2)]]).

fof(d5_nat_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( C = k6_nat_1(A,B)
              <=> ( r1_nat_1(C,A)
                  & r1_nat_1(C,B)
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_nat_1(D,A)
                          & r1_nat_1(D,B) )
                       => r1_nat_1(D,C) ) ) ) ) ) ) ) ),
    file(nat_1,d5_nat_1),
    [interesting(0.9),axiom,file(nat_1,d5_nat_1)]).

fof(e18_5_1__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_1__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc4_int_1,fc2_int_1,fc3_int_1,fc7_int_1,rc1_int_1,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc2_int_1,cc3_int_1,fc5_int_1,spc2_arithm,spc7_arithm,t2_subset,t3_arithm,t3_subset,commutativity_k2_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,commutativity_k5_nat_1,idempotence_k5_nat_1,commutativity_k6_nat_1,idempotence_k6_nat_1,reflexivity_r1_nat_1,existence_m2_subset_1,redefinition_k2_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_nat_1,dt_k5_numbers,dt_k6_nat_1,dt_m2_subset_1,dt_c1_5__int_2,dt_c1_5_1__int_2,dt_c2_5__int_2,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,e17_5_1__int_2,e6_5_1__int_2,e16_5_1__int_2,d5_nat_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]),
    [interesting(0.65),file(int_2,e18_5_1__int_2),[file(int_2,e18_5_1__int_2)]]).

fof(i2_5_1__int_2,theorem,(
    $true ),
    introduced(tautology,[file(int_2,i2_5_1__int_2)]),
    [interesting(0.65),trivial,file(int_2,i2_5_1__int_2)]).

fof(i1_5_1__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2,e1_5_1__int_2])],[e18_5_1__int_2,i2_5_1__int_2]),
    [interesting(0.65),file(int_2,i1_5_1__int_2),[file(int_2,i1_5_1__int_2)]]).

fof(e1_5__int_2,plain,(
    ~ ( c1_5__int_2 != 0
      & c2_5__int_2 != 0
      & k2_nat_1(c1_5__int_2,c2_5__int_2) != k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2]),discharge_asm(discharge,[e1_5_1__int_2])],[e1_5_1__int_2,i1_5_1__int_2]),
    [interesting(0.8),file(int_2,e1_5__int_2),[file(int_2,e1_5__int_2)]]).

fof(e3_5__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,dt_k1_zfmisc_1,dt_k5_ordinal2,cc3_int_1,cc4_int_1,fc2_int_1,fc7_int_1,rc1_int_1,rc2_int_1,spc7_arithm,t2_arithm,t2_subset,t3_subset,commutativity_k3_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k3_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc2_int_1,rqRealMult__k3_xcmplx_0__r0_r0_r0,t1_numerals,commutativity_k2_nat_1,commutativity_k5_nat_1,idempotence_k5_nat_1,commutativity_k6_nat_1,idempotence_k6_nat_1,redefinition_k2_nat_1,dt_k2_nat_1,dt_k5_nat_1,dt_k6_nat_1,dt_c1_5__int_2,dt_c2_5__int_2,spc0_numerals,e2_5__int_2,e1_5__int_2]),
    [interesting(0.8),file(int_2,e3_5__int_2),[file(int_2,e3_5__int_2)]]).

fof(i3_5__int_2,theorem,(
    $true ),
    introduced(tautology,[file(int_2,i3_5__int_2)]),
    [interesting(0.8),trivial,file(int_2,i3_5__int_2)]).

fof(i2_5__int_2,plain,(
    k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__int_2,dt_c2_5__int_2])],[e3_5__int_2,i3_5__int_2]),
    [interesting(0.8),file(int_2,i2_5__int_2),[file(int_2,i2_5__int_2)]]).

fof(i2_5_tmp__int_2,plain,
    ( m2_subset_1(c2_5__int_2,k1_numbers,k5_numbers)
   => k2_nat_1(c1_5__int_2,c2_5__int_2) = k2_nat_1(k5_nat_1(c1_5__int_2,c2_5__int_2),k6_nat_1(c1_5__int_2,c2_5__int_2)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__int_2]),discharge_asm(discharge,[dt_c2_5__int_2])],[dt_c2_5__int_2,i2_5__int_2]),
    [interesting(0.8),i1_5__int_2]).

fof(i1_5__int_2,plain,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_nat_1(c1_5__int_2,A) = k2_nat_1(k5_nat_1(c1_5__int_2,A),k6_nat_1(c1_5__int_2,A)) ) ),
    inference(let,[status(thm),assumptions([dt_c1_5__int_2])],[i2_5_tmp__int_2,dh_c2_5__int_2]),
    [interesting(0.8),file(int_2,i1_5__int_2),[file(int_2,i1_5__int_2)]]).

fof(i1_5_tmp__int_2,plain,
    ( m2_subset_1(c1_5__int_2,k1_numbers,k5_numbers)
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => k2_nat_1(c1_5__int_2,A) = k2_nat_1(k5_nat_1(c1_5__int_2,A),k6_nat_1(c1_5__int_2,A)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5__int_2])],[dt_c1_5__int_2,i1_5__int_2]),
    [interesting(1),t6_int_2]).

fof(t6_int_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_nat_1(A,B) = k2_nat_1(k5_nat_1(A,B),k6_nat_1(A,B)) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_5_tmp__int_2,dh_c1_5__int_2]),
    [interesting(1),file(int_2,t6_int_2),[file(int_2,t6_int_2)]]).