% Mizar ND problem: t139_rvsum_1,rvsum_1,1476,30 
fof(dh_c1_101__rvsum_1,definition,
    ( ( m2_subset_1(c1_101__rvsum_1,k1_numbers,k5_numbers)
     => ! [A] : 
          ( m2_finseq_2(A,k1_numbers,k4_finseq_2(c1_101__rvsum_1,k1_numbers))
         => k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,A)) = k7_square_1(k16_rvsum_1(A)) ) )
   => ! [B] : 
        ( m2_subset_1(B,k1_numbers,k5_numbers)
       => ! [C] : 
            ( m2_finseq_2(C,k1_numbers,k4_finseq_2(B,k1_numbers))
           => k16_rvsum_1(k12_rvsum_1(B,C)) = k7_square_1(k16_rvsum_1(C)) ) ) ),
    introduced(definition,[new_symbol(c1_101__rvsum_1),file(rvsum_1,c1_101__rvsum_1)]),
    [interesting(0.8),axiom,file(rvsum_1,c1_101__rvsum_1)]).

fof(dh_c2_101__rvsum_1,definition,
    ( ( m2_finseq_2(c2_101__rvsum_1,k1_numbers,k4_finseq_2(c1_101__rvsum_1,k1_numbers))
     => k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,c2_101__rvsum_1)) = k7_square_1(k16_rvsum_1(c2_101__rvsum_1)) )
   => ! [A] : 
        ( m2_finseq_2(A,k1_numbers,k4_finseq_2(c1_101__rvsum_1,k1_numbers))
       => k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,A)) = k7_square_1(k16_rvsum_1(A)) ) ),
    introduced(definition,[new_symbol(c2_101__rvsum_1),file(rvsum_1,c2_101__rvsum_1)]),
    [interesting(0.8),axiom,file(rvsum_1,c2_101__rvsum_1)]).

fof(dt_k3_funcop_1,axiom,(
    $true ),
    file(funcop_1,k3_funcop_1),
    [interesting(0.9),axiom,file(funcop_1,k3_funcop_1)]).

fof(dt_k5_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k5_relat_1(A,B)) ) ),
    file(relat_1,k5_relat_1),
    [interesting(0.9),axiom,file(relat_1,k5_relat_1)]).

fof(cc1_funct_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(funct_2,cc1_funct_2)]).

fof(fc1_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k5_relat_1(A,B))
        & v1_funct_1(k5_relat_1(A,B)) ) ) ),
    file(funct_1,fc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc1_funct_1)]).

fof(rc3_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]).

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

fof(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    [interesting(0.9),axiom,file(relset_1,m1_relset_1)]).

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    [interesting(0.9),axiom,file(relset_1,m2_relset_1)]).

fof(redefinition_k1_finseqop,definition,(
    ! [A,B,C,D,E,F] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m1_relset_1(D,k2_zfmisc_1(A,B),C)
        & m1_finseq_1(E,A)
        & m1_finseq_1(F,B) )
     => k1_finseqop(A,B,C,D,E,F) = k3_funcop_1(D,E,F) ) ),
    file(finseqop,k1_finseqop),
    [interesting(0.9),axiom,file(finseqop,k1_finseqop)]).

fof(redefinition_k5_finseqop,definition,(
    ! [A,B,C,D] : 
      ( ( m1_finseq_1(C,A)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B) )
     => k5_finseqop(A,B,C,D) = k5_relat_1(C,D) ) ),
    file(finseqop,k5_finseqop),
    [interesting(0.9),axiom,file(finseqop,k5_finseqop)]).

fof(redefinition_m2_relset_1,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    [interesting(0.9),axiom,file(relset_1,m2_relset_1)]).

fof(dt_k1_finseqop,axiom,(
    ! [A,B,C,D,E,F] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k2_zfmisc_1(A,B),C)
        & m1_relset_1(D,k2_zfmisc_1(A,B),C)
        & m1_finseq_1(E,A)
        & m1_finseq_1(F,B) )
     => m2_finseq_1(k1_finseqop(A,B,C,D,E,F),C) ) ),
    file(finseqop,k1_finseqop),
    [interesting(0.9),axiom,file(finseqop,k1_finseqop)]).

fof(dt_k1_rvsum_1,axiom,
    ( v1_funct_1(k1_rvsum_1)
    & v1_funct_2(k1_rvsum_1,k1_numbers,k1_numbers)
    & m2_relset_1(k1_rvsum_1,k1_numbers,k1_numbers) ),
    file(rvsum_1,k1_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k1_rvsum_1)]).

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

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

fof(dt_k5_finseqop,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_finseq_1(C,A)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B) )
     => m2_finseq_1(k5_finseqop(A,B,C,D),B) ) ),
    file(finseqop,k5_finseqop),
    [interesting(0.9),axiom,file(finseqop,k5_finseqop)]).

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    [interesting(0.9),axiom,file(relset_1,m1_relset_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),
    [interesting(0.9),axiom,file(relset_1,m2_relset_1)]).

fof(cc12_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc12_membered),
    [interesting(0.9),axiom,file(membered,cc12_membered)]).

fof(cc13_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc13_membered),
    [interesting(0.9),axiom,file(membered,cc13_membered)]).

fof(cc14_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v4_ordinal2(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc14_membered),
    [interesting(0.9),axiom,file(membered,cc14_membered)]).

fof(cc18_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B) ) ) ) ),
    file(membered,cc18_membered),
    [interesting(0.9),axiom,file(membered,cc18_membered)]).

fof(cc19_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B) ) ) ) ),
    file(membered,cc19_membered),
    [interesting(0.9),axiom,file(membered,cc19_membered)]).

fof(cc1_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => v4_membered(A) ) ),
    file(membered,cc1_membered),
    [interesting(0.9),axiom,file(membered,cc1_membered)]).

fof(cc1_relset_1,theorem,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) ),
    file(relset_1,cc1_relset_1),
    [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]).

fof(cc20_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B)
            & v5_membered(B) ) ) ) ),
    file(membered,cc20_membered),
    [interesting(0.9),axiom,file(membered,cc20_membered)]).

fof(cc2_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) ),
    file(funct_1,cc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]).

fof(cc2_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => v3_membered(A) ) ),
    file(membered,cc2_membered),
    [interesting(0.9),axiom,file(membered,cc2_membered)]).

fof(cc3_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => v2_membered(A) ) ),
    file(membered,cc3_membered),
    [interesting(0.9),axiom,file(membered,cc3_membered)]).

fof(cc5_funct_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(funct_2,cc5_funct_2)]).

fof(cc6_funct_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(funct_2,cc6_funct_2)]).

fof(fc6_membered,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_membered(k1_xboole_0)
    & v2_membered(k1_xboole_0)
    & v3_membered(k1_xboole_0)
    & v4_membered(k1_xboole_0)
    & v5_membered(k1_xboole_0) ),
    file(membered,fc6_membered),
    [interesting(0.9),axiom,file(membered,fc6_membered)]).

fof(rc1_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]).

fof(rc1_funct_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(funct_2,rc1_funct_2)]).

fof(rc1_membered,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_membered(A)
      & v2_membered(A)
      & v3_membered(A)
      & v4_membered(A)
      & v5_membered(A) ) ),
    file(membered,rc1_membered),
    [interesting(0.9),axiom,file(membered,rc1_membered)]).

fof(rc1_nat_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(nat_1,rc1_nat_1),
    [interesting(0.9),axiom,file(nat_1,rc1_nat_1)]).

fof(rc2_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]).

fof(rc2_nat_1,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ~ v1_xboole_0(A)
      & v3_ordinal1(A) ) ),
    file(nat_1,rc2_nat_1),
    [interesting(0.9),axiom,file(nat_1,rc2_nat_1)]).

fof(rc3_nat_1,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k5_numbers)
      & ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(nat_1,rc3_nat_1),
    [interesting(0.9),axiom,file(nat_1,rc3_nat_1)]).

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

fof(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(commutativity_k13_rvsum_1,theorem,(
    ! [A,B] : 
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_finseq_1(B,k1_numbers) )
     => k13_rvsum_1(A,B) = k13_rvsum_1(B,A) ) ),
    file(rvsum_1,k13_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k13_rvsum_1)]).

fof(existence_m1_finseq_1,axiom,(
    ! [A] : 
    ? [B] : m1_finseq_1(B,A) ),
    file(finseq_1,m1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]).

fof(existence_m1_finseq_2,axiom,(
    ! [A] : 
    ? [B] : m1_finseq_2(B,A) ),
    file(finseq_2,m1_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]).

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_finseq_1,axiom,(
    ! [A] : 
    ? [B] : m2_finseq_1(B,A) ),
    file(finseq_1,m2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]).

fof(redefinition_m2_finseq_1,definition,(
    ! [A,B] : 
      ( m2_finseq_1(B,A)
    <=> m1_finseq_1(B,A) ) ),
    file(finseq_1,m2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]).

fof(dt_k11_rvsum_1,axiom,(
    ! [A] : 
      ( m1_finseq_1(A,k1_numbers)
     => m2_finseq_1(k11_rvsum_1(A),k1_numbers) ) ),
    file(rvsum_1,k11_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k11_rvsum_1)]).

fof(dt_k13_rvsum_1,axiom,(
    ! [A,B] : 
      ( ( m1_finseq_1(A,k1_numbers)
        & m1_finseq_1(B,k1_numbers) )
     => m2_finseq_1(k13_rvsum_1(A,B),k1_numbers) ) ),
    file(rvsum_1,k13_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k13_rvsum_1)]).

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_k2_finsop_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(A,A),A)
        & m1_relset_1(C,k2_zfmisc_1(A,A),A) )
     => m1_subset_1(k2_finsop_1(A,B,C),A) ) ),
    file(finsop_1,k2_finsop_1),
    [interesting(0.9),axiom,file(finsop_1,k2_finsop_1)]).

fof(dt_k35_binop_2,axiom,
    ( v1_funct_1(k35_binop_2)
    & v1_funct_2(k35_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
    & m2_relset_1(k35_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ),
    file(binop_2,k35_binop_2),
    [interesting(0.9),axiom,file(binop_2,k35_binop_2)]).

fof(dt_k5_ordinal2,axiom,(
    $true ),
    file(ordinal2,k5_ordinal2),
    [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]).

fof(dt_m1_finseq_1,axiom,(
    ! [A,B] : 
      ( m1_finseq_1(B,A)
     => ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) ) ) ),
    file(finseq_1,m1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]).

fof(dt_m1_finseq_2,axiom,(
    $true ),
    file(finseq_2,m1_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]).

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_finseq_1,axiom,(
    ! [A,B] : 
      ( m2_finseq_1(B,A)
     => ( v1_funct_1(B)
        & v1_finseq_1(B)
        & m2_relset_1(B,k5_numbers,A) ) ) ),
    file(finseq_1,m2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]).

fof(cc10_membered,theorem,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => v1_xcmplx_0(B) ) ) ),
    file(membered,cc10_membered),
    [interesting(0.9),axiom,file(membered,cc10_membered)]).

fof(cc11_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) ),
    file(membered,cc11_membered),
    [interesting(0.9),axiom,file(membered,cc11_membered)]).

fof(cc15_membered,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_membered(A)
        & v2_membered(A)
        & v3_membered(A)
        & v4_membered(A)
        & v5_membered(A) ) ) ),
    file(membered,cc15_membered),
    [interesting(0.9),axiom,file(membered,cc15_membered)]).

fof(cc16_membered,theorem,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_membered(B) ) ) ),
    file(membered,cc16_membered),
    [interesting(0.9),axiom,file(membered,cc16_membered)]).

fof(cc17_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B) ) ) ) ),
    file(membered,cc17_membered),
    [interesting(0.9),axiom,file(membered,cc17_membered)]).

fof(cc1_funct_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]).

fof(cc1_nat_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) ) ) ),
    file(nat_1,cc1_nat_1),
    [interesting(0.9),axiom,file(nat_1,cc1_nat_1)]).

fof(cc2_nat_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) ) ) ),
    file(nat_1,cc2_nat_1),
    [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]).

fof(cc3_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ( v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(nat_1,cc3_nat_1),
    [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]).

fof(cc4_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => v1_membered(A) ) ),
    file(membered,cc4_membered),
    [interesting(0.9),axiom,file(membered,cc4_membered)]).

fof(cc6_membered,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( v1_membered(A)
        & v2_membered(A) ) ) ),
    file(membered,cc6_membered),
    [interesting(0.9),axiom,file(membered,cc6_membered)]).

fof(cc9_membered,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k1_zfmisc_1(k5_numbers))
     => ( v1_membered(A)
        & v2_membered(A)
        & v3_membered(A)
        & v4_membered(A)
        & v5_membered(A) ) ) ),
    file(membered,cc9_membered),
    [interesting(0.9),axiom,file(membered,cc9_membered)]).

fof(fc17_binop_2,theorem,
    ( v1_funct_1(k35_binop_2)
    & v1_funct_2(k35_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
    & v1_binop_1(k35_binop_2,k1_numbers)
    & v2_binop_1(k35_binop_2,k1_numbers)
    & v1_setwiseo(k35_binop_2,k1_numbers) ),
    file(binop_2,fc17_binop_2),
    [interesting(0.9),axiom,file(binop_2,fc17_binop_2)]).

fof(fc1_finseq_2,theorem,(
    ! [A,B] : 
      ( ( v4_ordinal2(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k4_finseq_2(A,B)) ) ),
    file(finseq_2,fc1_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,fc1_finseq_2)]).

fof(fc4_binop_2,theorem,
    ( v1_funct_1(k35_binop_2)
    & v1_funct_2(k35_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers)
    & v1_binop_1(k35_binop_2,k1_numbers)
    & v2_binop_1(k35_binop_2,k1_numbers) ),
    file(binop_2,fc4_binop_2),
    [interesting(0.9),axiom,file(binop_2,fc4_binop_2)]).

fof(fc5_membered,theorem,
    ( v1_membered(k5_ordinal2)
    & v2_membered(k5_ordinal2)
    & v3_membered(k5_ordinal2)
    & v4_membered(k5_ordinal2)
    & v5_membered(k5_ordinal2) ),
    file(membered,fc5_membered),
    [interesting(0.9),axiom,file(membered,fc5_membered)]).

fof(rc1_finseq_2,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_finseq_2(B,A)
      & ~ v1_xboole_0(B) ) ),
    file(finseq_2,rc1_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,rc1_finseq_2)]).

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(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(d8_rvsum_1,definition,(
    ! [A] : 
      ( m2_finseq_1(A,k1_numbers)
     => k11_rvsum_1(A) = k5_finseqop(k1_numbers,k1_numbers,A,k1_rvsum_1) ) ),
    file(rvsum_1,d8_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,d8_rvsum_1)]).

fof(d9_rvsum_1,definition,(
    ! [A] : 
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] : 
          ( m2_finseq_1(B,k1_numbers)
         => k13_rvsum_1(A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k35_binop_2,A,B) ) ) ),
    file(rvsum_1,d9_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,d9_rvsum_1)]).

fof(commutativity_k14_rvsum_1,theorem,(
    ! [A,B,C] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k14_rvsum_1(A,B,C) = k14_rvsum_1(A,C,B) ) ),
    file(rvsum_1,k14_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k14_rvsum_1)]).

fof(existence_m2_finseq_2,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & m1_finseq_2(B,A) )
     => ? [C] : m2_finseq_2(C,A,B) ) ),
    file(finseq_2,m2_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]).

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_k12_rvsum_1,definition,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers)) )
     => k12_rvsum_1(A,B) = k11_rvsum_1(B) ) ),
    file(rvsum_1,k12_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k12_rvsum_1)]).

fof(redefinition_k14_rvsum_1,definition,(
    ! [A,B,C] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => k14_rvsum_1(A,B,C) = k13_rvsum_1(B,C) ) ),
    file(rvsum_1,k14_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k14_rvsum_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_finseq_2,definition,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & m1_finseq_2(B,A) )
     => ! [C] : 
          ( m2_finseq_2(C,A,B)
        <=> m1_subset_1(C,B) ) ) ),
    file(finseq_2,m2_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]).

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_k12_rvsum_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k12_rvsum_1(A,B),k1_numbers,k4_finseq_2(A,k1_numbers)) ) ),
    file(rvsum_1,k12_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k12_rvsum_1)]).

fof(dt_k14_rvsum_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k1_numbers))
        & m1_subset_1(C,k4_finseq_2(A,k1_numbers)) )
     => m2_finseq_2(k14_rvsum_1(A,B,C),k1_numbers,k4_finseq_2(A,k1_numbers)) ) ),
    file(rvsum_1,k14_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k14_rvsum_1)]).

fof(dt_k16_rvsum_1,axiom,(
    ! [A] : 
      ( m1_finseq_1(A,k1_numbers)
     => m1_subset_1(k16_rvsum_1(A),k1_numbers) ) ),
    file(rvsum_1,k16_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,k16_rvsum_1)]).

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

fof(dt_k4_finseq_2,axiom,(
    ! [A,B] : 
      ( v4_ordinal2(A)
     => m1_finseq_2(k4_finseq_2(A,B),B) ) ),
    file(finseq_2,k4_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,k4_finseq_2)]).

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

fof(dt_m2_finseq_2,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(B)
        & m1_finseq_2(B,A) )
     => ! [C] : 
          ( m2_finseq_2(C,A,B)
         => m2_finseq_1(C,A) ) ) ),
    file(finseq_2,m2_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]).

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(dt_c1_101__rvsum_1,assumption,(
    m2_subset_1(c1_101__rvsum_1,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(rvsum_1,c1_101__rvsum_1)]),
    [interesting(0.8),axiom,file(rvsum_1,c1_101__rvsum_1)]).

fof(dt_c2_101__rvsum_1,assumption,(
    m2_finseq_2(c2_101__rvsum_1,k1_numbers,k4_finseq_2(c1_101__rvsum_1,k1_numbers)) ),
    introduced(assumption,[file(rvsum_1,c2_101__rvsum_1)]),
    [interesting(0.8),axiom,file(rvsum_1,c2_101__rvsum_1)]).

fof(fc2_membered,theorem,
    ( ~ v1_xboole_0(k1_numbers)
    & v1_membered(k1_numbers)
    & v2_membered(k1_numbers) ),
    file(membered,fc2_membered),
    [interesting(0.9),axiom,file(membered,fc2_membered)]).

fof(d11_rvsum_1,definition,(
    ! [A] : 
      ( m2_finseq_1(A,k1_numbers)
     => k16_rvsum_1(A) = k2_finsop_1(k1_numbers,A,k35_binop_2) ) ),
    file(rvsum_1,d11_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,d11_rvsum_1)]).

fof(t97_rvsum_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k12_rvsum_1(A,B) = k14_rvsum_1(A,B,B) ) ) ),
    file(rvsum_1,t97_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,t97_rvsum_1)]).

fof(e1_101_1__rvsum_1,plain,(
    k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,c2_101__rvsum_1)) = k16_rvsum_1(k14_rvsum_1(c1_101__rvsum_1,c2_101__rvsum_1,c2_101__rvsum_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_101__rvsum_1,dt_c2_101__rvsum_1])],[dt_k3_funcop_1,dt_k5_relat_1,cc1_funct_2,fc1_funct_1,rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m2_relset_1,redefinition_k1_finseqop,redefinition_k5_finseqop,redefinition_m2_relset_1,dt_k1_finseqop,dt_k1_rvsum_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_finseqop,dt_m1_relset_1,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc5_funct_2,cc6_funct_2,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_nat_1,rc2_funct_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,commutativity_k13_rvsum_1,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k11_rvsum_1,dt_k13_rvsum_1,dt_k1_zfmisc_1,dt_k2_finsop_1,dt_k35_binop_2,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc1_nat_1,cc2_nat_1,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc17_binop_2,fc1_finseq_2,fc4_binop_2,fc5_membered,rc1_finseq_2,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d8_rvsum_1,d9_rvsum_1,commutativity_k14_rvsum_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k12_rvsum_1,redefinition_k14_rvsum_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k12_rvsum_1,dt_k14_rvsum_1,dt_k16_rvsum_1,dt_k1_numbers,dt_k4_finseq_2,dt_k5_numbers,dt_m2_finseq_2,dt_m2_subset_1,dt_c1_101__rvsum_1,dt_c2_101__rvsum_1,fc2_membered,d11_rvsum_1,t97_rvsum_1]),
    [interesting(0.65),file(rvsum_1,e1_101_1__rvsum_1),[file(rvsum_1,e1_101_1__rvsum_1)]]).

fof(fc2_nat_1,theorem,(
    ! [A,B] : 
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => ( v4_ordinal2(k3_xcmplx_0(A,B))
        & v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ),
    file(nat_1,fc2_nat_1),
    [interesting(0.9),axiom,file(nat_1,fc2_nat_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(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(commutativity_k11_binop_2,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k11_binop_2(A,B) = k11_binop_2(B,A) ) ),
    file(binop_2,k11_binop_2),
    [interesting(0.9),axiom,file(binop_2,k11_binop_2)]).

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(redefinition_k11_binop_2,definition,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k11_binop_2(A,B) = k3_xcmplx_0(A,B) ) ),
    file(binop_2,k11_binop_2),
    [interesting(0.9),axiom,file(binop_2,k11_binop_2)]).

fof(dt_k11_binop_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k11_binop_2(A,B),k1_numbers) ) ),
    file(binop_2,k11_binop_2),
    [interesting(0.9),axiom,file(binop_2,k11_binop_2)]).

fof(dt_k3_xcmplx_0,axiom,(
    $true ),
    file(xcmplx_0,k3_xcmplx_0),
    [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]).

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(spc1_boole,theorem,(
    ~ v1_xboole_0(1) ),
    file(boole,spc1_boole),
    [interesting(0.9),axiom,file(boole,spc1_boole)]).

fof(t137_rvsum_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => ! [C] : 
              ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers))
             => k16_rvsum_1(k14_rvsum_1(A,B,C)) = k11_binop_2(k16_rvsum_1(B),k16_rvsum_1(C)) ) ) ) ),
    file(rvsum_1,t137_rvsum_1),
    [interesting(0.9),axiom,file(rvsum_1,t137_rvsum_1)]).

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(e2_101_1__rvsum_1,plain,(
    k16_rvsum_1(k14_rvsum_1(c1_101__rvsum_1,c2_101__rvsum_1,c2_101__rvsum_1)) = k11_binop_2(k16_rvsum_1(c2_101__rvsum_1),k16_rvsum_1(c2_101__rvsum_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_101__rvsum_1,dt_c2_101__rvsum_1])],[dt_k3_funcop_1,cc1_funct_2,rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m2_relset_1,redefinition_k1_finseqop,redefinition_m2_relset_1,dt_k1_finseqop,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc5_funct_2,cc6_funct_2,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_nat_1,rc2_funct_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,commutativity_k13_rvsum_1,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k13_rvsum_1,dt_k1_zfmisc_1,dt_k2_finsop_1,dt_k35_binop_2,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_funct_1,cc1_nat_1,cc2_nat_1,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc17_binop_2,fc1_finseq_2,fc2_nat_1,fc4_binop_2,fc5_membered,rc1_finseq_2,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_boole,t7_boole,t8_boole,d9_rvsum_1,commutativity_k11_binop_2,commutativity_k14_rvsum_1,commutativity_k3_xcmplx_0,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k11_binop_2,redefinition_k14_rvsum_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k11_binop_2,dt_k14_rvsum_1,dt_k16_rvsum_1,dt_k1_numbers,dt_k3_xcmplx_0,dt_k4_finseq_2,dt_k5_numbers,dt_m2_finseq_2,dt_m2_subset_1,dt_c1_101__rvsum_1,dt_c2_101__rvsum_1,fc2_membered,d11_rvsum_1,spc1_numerals,spc1_boole,t137_rvsum_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.65),file(rvsum_1,e2_101_1__rvsum_1),[file(rvsum_1,e2_101_1__rvsum_1)]]).

fof(dt_k5_square_1,axiom,(
    $true ),
    file(square_1,k5_square_1),
    [interesting(0.9),axiom,file(square_1,k5_square_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(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(t1_numerals,theorem,(
    m1_subset_1(0,k5_numbers) ),
    file(numerals,t1_numerals),
    [interesting(0.9),axiom,file(numerals,t1_numerals)]).

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(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(d3_square_1,definition,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k5_square_1(A) = k3_xcmplx_0(A,A) ) ),
    file(square_1,d3_square_1),
    [interesting(0.9),axiom,file(square_1,d3_square_1)]).

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(redefinition_k7_square_1,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => k7_square_1(A) = k5_square_1(A) ) ),
    file(square_1,k7_square_1),
    [interesting(0.9),axiom,file(square_1,k7_square_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(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_square_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k7_square_1(A),k1_numbers) ) ),
    file(square_1,k7_square_1),
    [interesting(0.9),axiom,file(square_1,k7_square_1)]).

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(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(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(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(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(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(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(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(e3_101_1__rvsum_1,plain,(
    k11_binop_2(k16_rvsum_1(c2_101__rvsum_1),k16_rvsum_1(c2_101__rvsum_1)) = k7_square_1(k16_rvsum_1(c2_101__rvsum_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_101__rvsum_1,dt_c2_101__rvsum_1])],[reflexivity_r1_tarski,cc1_funct_2,rc3_funct_1,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_finseq_2,fc2_nat_1,fc5_membered,fc6_membered,rc1_finseq_2,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_nat_1,rc2_funct_1,rc2_nat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_finsop_1,dt_k35_binop_2,dt_k4_finseq_2,dt_k5_numbers,dt_k5_square_1,dt_m1_finseq_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_subset_1,dt_c1_101__rvsum_1,cc15_membered,cc1_funct_1,cc1_nat_1,cc2_nat_1,fc17_binop_2,fc2_membered,fc4_binop_2,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d3_square_1,commutativity_k11_binop_2,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,redefinition_k11_binop_2,redefinition_k7_square_1,dt_k11_binop_2,dt_k16_rvsum_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_k7_square_1,dt_c2_101__rvsum_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,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,d11_rvsum_1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,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]),
    [interesting(0.65),file(rvsum_1,e3_101_1__rvsum_1),[file(rvsum_1,e3_101_1__rvsum_1)]]).

fof(e1_101__rvsum_1,plain,(
    k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,c2_101__rvsum_1)) = k7_square_1(k16_rvsum_1(c2_101__rvsum_1)) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c1_101__rvsum_1,dt_c2_101__rvsum_1])],[e1_101_1__rvsum_1,e2_101_1__rvsum_1,e3_101_1__rvsum_1]),
    [interesting(0.8),file(rvsum_1,e1_101__rvsum_1),[file(rvsum_1,e1_101__rvsum_1)]]).

fof(i3_101__rvsum_1,theorem,(
    $true ),
    introduced(tautology,[file(rvsum_1,i3_101__rvsum_1)]),
    [interesting(0.8),trivial,file(rvsum_1,i3_101__rvsum_1)]).

fof(i2_101__rvsum_1,plain,(
    k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,c2_101__rvsum_1)) = k7_square_1(k16_rvsum_1(c2_101__rvsum_1)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_101__rvsum_1,dt_c2_101__rvsum_1])],[e1_101__rvsum_1,i3_101__rvsum_1]),
    [interesting(0.8),file(rvsum_1,i2_101__rvsum_1),[file(rvsum_1,i2_101__rvsum_1)]]).

fof(i2_101_tmp__rvsum_1,plain,
    ( m2_finseq_2(c2_101__rvsum_1,k1_numbers,k4_finseq_2(c1_101__rvsum_1,k1_numbers))
   => k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,c2_101__rvsum_1)) = k7_square_1(k16_rvsum_1(c2_101__rvsum_1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_101__rvsum_1]),discharge_asm(discharge,[dt_c2_101__rvsum_1])],[dt_c2_101__rvsum_1,i2_101__rvsum_1]),
    [interesting(0.8),i1_101__rvsum_1]).

fof(i1_101__rvsum_1,plain,(
    ! [A] : 
      ( m2_finseq_2(A,k1_numbers,k4_finseq_2(c1_101__rvsum_1,k1_numbers))
     => k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,A)) = k7_square_1(k16_rvsum_1(A)) ) ),
    inference(let,[status(thm),assumptions([dt_c1_101__rvsum_1])],[i2_101_tmp__rvsum_1,dh_c2_101__rvsum_1]),
    [interesting(0.8),file(rvsum_1,i1_101__rvsum_1),[file(rvsum_1,i1_101__rvsum_1)]]).

fof(i1_101_tmp__rvsum_1,plain,
    ( m2_subset_1(c1_101__rvsum_1,k1_numbers,k5_numbers)
   => ! [A] : 
        ( m2_finseq_2(A,k1_numbers,k4_finseq_2(c1_101__rvsum_1,k1_numbers))
       => k16_rvsum_1(k12_rvsum_1(c1_101__rvsum_1,A)) = k7_square_1(k16_rvsum_1(A)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_101__rvsum_1])],[dt_c1_101__rvsum_1,i1_101__rvsum_1]),
    [interesting(1),t139_rvsum_1]).

fof(t139_rvsum_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers))
         => k16_rvsum_1(k12_rvsum_1(A,B)) = k7_square_1(k16_rvsum_1(B)) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_101_tmp__rvsum_1,dh_c1_101__rvsum_1]),
    [interesting(1),file(rvsum_1,t139_rvsum_1),[file(rvsum_1,t139_rvsum_1)]]).