% Mizar ND problem: t9_toprns_1,toprns_1,269,33 
fof(dh_c1_16__toprns_1,definition,
    ( ( m2_subset_1(c1_16__toprns_1,k1_numbers,k5_numbers)
     => ! [A] : 
          ( m1_subset_1(A,k1_numbers)
         => ! [B] : 
              ( ( v1_funct_1(B)
                & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
                & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
             => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(A,c1_16__toprns_1,B)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,B),k18_complex1(A)) ) ) )
   => ! [C] : 
        ( m2_subset_1(C,k1_numbers,k5_numbers)
       => ! [D] : 
            ( m1_subset_1(D,k1_numbers)
           => ! [E] : 
                ( ( v1_funct_1(E)
                  & v1_funct_2(E,k5_numbers,u1_struct_0(k15_euclid(C)))
                  & m2_relset_1(E,k5_numbers,u1_struct_0(k15_euclid(C))) )
               => k6_toprns_1(C,k2_toprns_1(D,C,E)) = k14_seq_1(k6_toprns_1(C,E),k18_complex1(D)) ) ) ) ),
    introduced(definition,[new_symbol(c1_16__toprns_1),file(toprns_1,c1_16__toprns_1)]),
    [interesting(0.8),axiom,file(toprns_1,c1_16__toprns_1)]).

fof(dh_c2_16__toprns_1,definition,
    ( ( m1_subset_1(c2_16__toprns_1,k1_numbers)
     => ! [A] : 
          ( ( v1_funct_1(A)
            & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
            & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
         => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,A)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,A),k18_complex1(c2_16__toprns_1)) ) )
   => ! [B] : 
        ( m1_subset_1(B,k1_numbers)
       => ! [C] : 
            ( ( v1_funct_1(C)
              & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
              & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
           => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(B,c1_16__toprns_1,C)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,C),k18_complex1(B)) ) ) ),
    introduced(definition,[new_symbol(c2_16__toprns_1),file(toprns_1,c2_16__toprns_1)]),
    [interesting(0.8),axiom,file(toprns_1,c2_16__toprns_1)]).

fof(dh_c3_16__toprns_1,definition,
    ( ( ( v1_funct_1(c3_16__toprns_1)
        & v1_funct_2(c3_16__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
        & m2_relset_1(c3_16__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
     => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)) )
   => ! [A] : 
        ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
          & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
       => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,A)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,A),k18_complex1(c2_16__toprns_1)) ) ),
    introduced(definition,[new_symbol(c3_16__toprns_1),file(toprns_1,c3_16__toprns_1)]),
    [interesting(0.8),axiom,file(toprns_1,c3_16__toprns_1)]).

fof(free_g1_pre_topc,definition,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C,D] : 
          ( g1_pre_topc(A,B) = g1_pre_topc(C,D)
         => ( A = C
            & B = D ) ) ) ),
    file(pre_topc,g1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]).

fof(dt_g1_pre_topc,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v1_pre_topc(g1_pre_topc(A,B))
        & l1_pre_topc(g1_pre_topc(A,B)) ) ) ),
    file(pre_topc,g1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]).

fof(dt_u1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ),
    file(pre_topc,u1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,u1_pre_topc)]).

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(abstractness_v1_pre_topc,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ( v1_pre_topc(A)
       => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ),
    file(pre_topc,v1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,v1_pre_topc)]).

fof(existence_l1_pre_topc,axiom,(
    ? [A] : l1_pre_topc(A) ),
    file(pre_topc,l1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]).

fof(existence_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

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_l1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => l1_struct_0(A) ) ),
    file(pre_topc,l1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

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_arytm_3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_ordinal1(B)
            & v2_ordinal1(B)
            & v3_ordinal1(B) ) ) ) ),
    file(arytm_3,cc1_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]).

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

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

fof(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_arytm_3,theorem,(
    ! [A] : 
      ( ( v1_xboole_0(A)
        & v3_ordinal1(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A) ) ) ),
    file(arytm_3,cc2_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]).

fof(cc2_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(cc3_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(xreal_0,cc3_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]).

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

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

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

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

fof(fc1_struct_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ~ v1_xboole_0(u1_struct_0(A)) ) ),
    file(struct_0,fc1_struct_0),
    [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]).

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_arytm_3,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A) ) ),
    file(arytm_3,rc1_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]).

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

fof(rc3_struct_0,theorem,(
    ? [A] : 
      ( l1_struct_0(A)
      & ~ v3_struct_0(A) ) ),
    file(struct_0,rc3_struct_0),
    [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]).

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

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

fof(rc5_struct_0,theorem,(
    ! [A] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B) ) ) ),
    file(struct_0,rc5_struct_0),
    [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]).

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

fof(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(projectivity_k16_complex1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ),
    file(complex1,k16_complex1),
    [interesting(0.9),axiom,file(complex1,k16_complex1)]).

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(dt_k12_seq_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A)
        & v1_xreal_0(B) )
     => ( v1_relat_1(k12_seq_1(A,B))
        & v1_funct_1(k12_seq_1(A,B)) ) ) ),
    file(seq_1,k12_seq_1),
    [interesting(0.9),axiom,file(seq_1,k12_seq_1)]).

fof(dt_k15_euclid,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( v1_pre_topc(k15_euclid(A))
        & v2_pre_topc(k15_euclid(A))
        & l1_pre_topc(k15_euclid(A)) ) ) ),
    file(euclid,k15_euclid),
    [interesting(0.9),axiom,file(euclid,k15_euclid)]).

fof(dt_k16_complex1,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => v1_xcmplx_0(k16_complex1(A)) ) ),
    file(complex1,k16_complex1),
    [interesting(0.9),axiom,file(complex1,k16_complex1)]).

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_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]).

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

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_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    [interesting(0.9),axiom,file(struct_0,u1_struct_0)]).

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

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

fof(cc4_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(cc7_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xboole_0(A)
        & v1_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(xreal_0,cc7_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]).

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

fof(fc2_euclid,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k15_euclid(A))
        & v1_pre_topc(k15_euclid(A))
        & v2_pre_topc(k15_euclid(A)) ) ) ),
    file(euclid,fc2_euclid),
    [interesting(0.9),axiom,file(euclid,fc2_euclid)]).

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_xreal_0,theorem,(
    ? [A] : 
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(xreal_0,rc1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]).

fof(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(projectivity_k18_complex1,theorem,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ),
    file(complex1,k18_complex1),
    [interesting(0.9),axiom,file(complex1,k18_complex1)]).

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_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(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_k14_seq_1,definition,(
    ! [A,B] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_xreal_0(B) )
     => k14_seq_1(A,B) = k12_seq_1(A,B) ) ),
    file(seq_1,k14_seq_1),
    [interesting(0.9),axiom,file(seq_1,k14_seq_1)]).

fof(redefinition_k18_complex1,definition,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => k18_complex1(A) = k16_complex1(A) ) ),
    file(complex1,k18_complex1),
    [interesting(0.9),axiom,file(complex1,k18_complex1)]).

fof(redefinition_k2_seq_1,definition,(
    ! [A,B,C,D] : 
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ),
    file(seq_1,k2_seq_1),
    [interesting(0.9),axiom,file(seq_1,k2_seq_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_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(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_k14_seq_1,axiom,(
    ! [A,B] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_xreal_0(B) )
     => ( v1_funct_1(k14_seq_1(A,B))
        & v1_funct_2(k14_seq_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k14_seq_1(A,B),k5_numbers,k1_numbers) ) ) ),
    file(seq_1,k14_seq_1),
    [interesting(0.9),axiom,file(seq_1,k14_seq_1)]).

fof(dt_k18_complex1,axiom,(
    ! [A] : 
      ( v1_xcmplx_0(A)
     => m1_subset_1(k18_complex1(A),k1_numbers) ) ),
    file(complex1,k18_complex1),
    [interesting(0.9),axiom,file(complex1,k18_complex1)]).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    [interesting(0.9),axiom,file(funct_1,k1_funct_1)]).

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

fof(dt_k2_seq_1,axiom,(
    ! [A,B,C,D] : 
      ( ( v2_membered(B)
        & v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ),
    file(seq_1,k2_seq_1),
    [interesting(0.9),axiom,file(seq_1,k2_seq_1)]).

fof(dt_k2_toprns_1,axiom,(
    ! [A,B,C] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k5_numbers)
        & v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(B)))
        & m1_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(B))) )
     => ( v1_funct_1(k2_toprns_1(A,B,C))
        & v1_funct_2(k2_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(B)))
        & m2_relset_1(k2_toprns_1(A,B,C),k5_numbers,u1_struct_0(k15_euclid(B))) ) ) ),
    file(toprns_1,k2_toprns_1),
    [interesting(0.9),axiom,file(toprns_1,k2_toprns_1)]).

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_k6_toprns_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
        & m1_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
     => ( v1_funct_1(k6_toprns_1(A,B))
        & v1_funct_2(k6_toprns_1(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k6_toprns_1(A,B),k5_numbers,k1_numbers) ) ) ),
    file(toprns_1,k6_toprns_1),
    [interesting(0.9),axiom,file(toprns_1,k6_toprns_1)]).

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_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(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_16__toprns_1,assumption,(
    m2_subset_1(c1_16__toprns_1,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(toprns_1,c1_16__toprns_1)]),
    [interesting(0.8),axiom,file(toprns_1,c1_16__toprns_1)]).

fof(dt_c2_16__toprns_1,assumption,(
    m1_subset_1(c2_16__toprns_1,k1_numbers) ),
    introduced(assumption,[file(toprns_1,c2_16__toprns_1)]),
    [interesting(0.8),axiom,file(toprns_1,c2_16__toprns_1)]).

fof(dt_c3_16__toprns_1,assumption,
    ( v1_funct_1(c3_16__toprns_1)
    & v1_funct_2(c3_16__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
    & m2_relset_1(c3_16__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) ),
    introduced(assumption,[file(toprns_1,c3_16__toprns_1)]),
    [interesting(0.8),axiom,file(toprns_1,c3_16__toprns_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(dh_c1_16_1__toprns_1,definition,
    ( ( m2_subset_1(c1_16_1__toprns_1,k1_numbers,k5_numbers)
     => k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)),c1_16_1__toprns_1) = k2_seq_1(k5_numbers,k1_numbers,k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)),c1_16_1__toprns_1) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)),A) = k2_seq_1(k5_numbers,k1_numbers,k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)),A) ) ),
    introduced(definition,[new_symbol(c1_16_1__toprns_1),file(toprns_1,c1_16_1__toprns_1)]),
    [interesting(0.65),axiom,file(toprns_1,c1_16_1__toprns_1)]).

fof(fc21_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ),
    file(xreal_0,fc21_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,fc21_xreal_0)]).

fof(fc22_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(B,A))
        & v1_xreal_0(k3_xcmplx_0(B,A))
        & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ),
    file(xreal_0,fc22_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,fc22_xreal_0)]).

fof(fc23_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ),
    file(xreal_0,fc23_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,fc23_xreal_0)]).

fof(fc24_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ),
    file(xreal_0,fc24_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,fc24_xreal_0)]).

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

fof(fc4_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ),
    file(xreal_0,fc4_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,fc4_xreal_0)]).

fof(commutativity_k4_real_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_real_1(A,B) = k4_real_1(B,A) ) ),
    file(real_1,k4_real_1),
    [interesting(0.9),axiom,file(real_1,k4_real_1)]).

fof(redefinition_k2_normsp_1,definition,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => k2_normsp_1(A,B,C) = k1_funct_1(B,C) ) ),
    file(normsp_1,k2_normsp_1),
    [interesting(0.9),axiom,file(normsp_1,k2_normsp_1)]).

fof(redefinition_k4_real_1,definition,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ),
    file(real_1,k4_real_1),
    [interesting(0.9),axiom,file(real_1,k4_real_1)]).

fof(dt_k18_euclid,axiom,(
    ! [A,B,C] : 
      ( ( v1_xreal_0(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,u1_struct_0(k15_euclid(B))) )
     => m1_subset_1(k18_euclid(A,B,C),u1_struct_0(k15_euclid(B))) ) ),
    file(euclid,k18_euclid),
    [interesting(0.9),axiom,file(euclid,k18_euclid)]).

fof(dt_k2_normsp_1,axiom,(
    ! [A,B,C] : 
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,u1_struct_0(A))
        & m1_relset_1(B,k5_numbers,u1_struct_0(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k2_normsp_1(A,B,C),u1_struct_0(A)) ) ),
    file(normsp_1,k2_normsp_1),
    [interesting(0.9),axiom,file(normsp_1,k2_normsp_1)]).

fof(dt_k4_real_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ),
    file(real_1,k4_real_1),
    [interesting(0.9),axiom,file(real_1,k4_real_1)]).

fof(dt_k5_toprns_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,u1_struct_0(k15_euclid(A))) )
     => m1_subset_1(k5_toprns_1(A,B),k1_numbers) ) ),
    file(toprns_1,k5_toprns_1),
    [interesting(0.9),axiom,file(toprns_1,k5_toprns_1)]).

fof(dt_c1_16_1__toprns_1,assumption,(
    m2_subset_1(c1_16_1__toprns_1,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(toprns_1,c1_16_1__toprns_1)]),
    [interesting(0.65),axiom,file(toprns_1,c1_16_1__toprns_1)]).

fof(d7_toprns_1,definition,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k6_toprns_1(A,B)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D)) ) ) ) ) ) ),
    file(toprns_1,d7_toprns_1),
    [interesting(0.9),axiom,file(toprns_1,d7_toprns_1)]).

fof(e1_16_1_1__toprns_1,plain,(
    k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)),c1_16_1__toprns_1) = k5_toprns_1(c1_16__toprns_1,k2_normsp_1(k15_euclid(c1_16__toprns_1),k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1),c1_16_1__toprns_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,t1_subset,t4_subset,t5_subset,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc1_struct_0,fc2_euclid,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k2_seq_1,dt_k2_toprns_1,dt_k5_numbers,dt_k5_toprns_1,dt_k6_toprns_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1,fc2_membered,d7_toprns_1]),
    [interesting(0.5),file(toprns_1,e1_16_1_1__toprns_1),[file(toprns_1,e1_16_1_1__toprns_1)]]).

fof(d3_toprns_1,definition,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(B)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(B))) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(B)))
                    & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(B))) )
                 => ( D = k2_toprns_1(A,B,C)
                  <=> ! [E] : 
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k2_normsp_1(k15_euclid(B),D,E) = k18_euclid(A,B,k2_normsp_1(k15_euclid(B),C,E)) ) ) ) ) ) ) ),
    file(toprns_1,d3_toprns_1),
    [interesting(0.9),axiom,file(toprns_1,d3_toprns_1)]).

fof(e2_16_1_1__toprns_1,plain,(
    k5_toprns_1(c1_16__toprns_1,k2_normsp_1(k15_euclid(c1_16__toprns_1),k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1),c1_16_1__toprns_1)) = k5_toprns_1(c1_16__toprns_1,k18_euclid(c2_16__toprns_1,c1_16__toprns_1,k2_normsp_1(k15_euclid(c1_16__toprns_1),c3_16__toprns_1,c1_16_1__toprns_1))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k15_euclid,dt_k18_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k2_toprns_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1,fc2_euclid,fc2_membered,d3_toprns_1]),
    [interesting(0.5),file(toprns_1,e2_16_1_1__toprns_1),[file(toprns_1,e2_16_1_1__toprns_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(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(t8_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => k4_real_1(k18_complex1(B),k5_toprns_1(A,C)) = k5_toprns_1(A,k18_euclid(B,A,C)) ) ) ) ),
    file(toprns_1,t8_toprns_1),
    [interesting(0.9),axiom,file(toprns_1,t8_toprns_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(e3_16_1_1__toprns_1,plain,(
    k5_toprns_1(c1_16__toprns_1,k18_euclid(c2_16__toprns_1,c1_16__toprns_1,k2_normsp_1(k15_euclid(c1_16__toprns_1),c3_16__toprns_1,c1_16_1__toprns_1))) = k4_real_1(k18_complex1(c2_16__toprns_1),k5_toprns_1(c1_16__toprns_1,k2_normsp_1(k15_euclid(c1_16__toprns_1),c3_16__toprns_1,c1_16_1__toprns_1))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,projectivity_k16_complex1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc23_xreal_0,fc4_xreal_0,fc5_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k4_real_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k18_complex1,dt_k18_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1,fc2_euclid,fc2_membered,spc1_numerals,spc1_boole,t8_toprns_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.5),file(toprns_1,e3_16_1_1__toprns_1),[file(toprns_1,e3_16_1_1__toprns_1)]]).

fof(e4_16_1_1__toprns_1,plain,(
    k4_real_1(k18_complex1(c2_16__toprns_1),k5_toprns_1(c1_16__toprns_1,k2_normsp_1(k15_euclid(c1_16__toprns_1),c3_16__toprns_1,c1_16_1__toprns_1))) = k4_real_1(k18_complex1(c2_16__toprns_1),k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),c1_16_1__toprns_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc4_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,t1_subset,t4_subset,t5_subset,projectivity_k16_complex1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc1_struct_0,fc2_euclid,fc5_membered,rc3_struct_0,rc5_struct_0,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k2_seq_1,redefinition_k4_real_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k15_euclid,dt_k18_complex1,dt_k1_numbers,dt_k2_normsp_1,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k5_numbers,dt_k5_toprns_1,dt_k6_toprns_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1,fc2_membered,spc1_numerals,spc1_boole,d7_toprns_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.5),file(toprns_1,e4_16_1_1__toprns_1),[file(toprns_1,e4_16_1_1__toprns_1)]]).

fof(t13_seq_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( B = k14_seq_1(C,A)
              <=> ! [D] : 
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,B,D) = k3_xcmplx_0(A,k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ),
    file(seq_1,t13_seq_1),
    [interesting(0.9),axiom,file(seq_1,t13_seq_1)]).

fof(e5_16_1_1__toprns_1,plain,(
    k4_real_1(k18_complex1(c2_16__toprns_1),k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),c1_16_1__toprns_1)) = k2_seq_1(k5_numbers,k1_numbers,k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)),c1_16_1__toprns_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k1_xboole_0,dt_l1_pre_topc,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc2_xreal_0,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,projectivity_k16_complex1,existence_m1_relset_1,existence_m1_subset_1,dt_k12_seq_1,dt_k15_euclid,dt_k16_complex1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc23_xreal_0,fc2_euclid,fc5_membered,rc1_xreal_0,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k14_seq_1,redefinition_k18_complex1,redefinition_k2_seq_1,redefinition_k4_real_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k14_seq_1,dt_k18_complex1,dt_k1_numbers,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k5_numbers,dt_k6_toprns_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1,cc2_xreal_0,fc2_membered,fc4_xreal_0,spc1_numerals,spc1_boole,t13_seq_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]),
    [interesting(0.5),file(toprns_1,e5_16_1_1__toprns_1),[file(toprns_1,e5_16_1_1__toprns_1)]]).

fof(e1_16_1__toprns_1,plain,(
    k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)),c1_16_1__toprns_1) = k2_seq_1(k5_numbers,k1_numbers,k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)),c1_16_1__toprns_1) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,rc1_arytm_3,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,projectivity_k16_complex1,commutativity_k3_xcmplx_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k12_seq_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k3_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc2_euclid,fc4_xreal_0,fc5_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,projectivity_k18_complex1,commutativity_k4_real_1,redefinition_k14_seq_1,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k2_seq_1,redefinition_k4_real_1,redefinition_k5_numbers,dt_k14_seq_1,dt_k15_euclid,dt_k18_complex1,dt_k18_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k2_seq_1,dt_k2_toprns_1,dt_k4_real_1,dt_k5_numbers,dt_k5_toprns_1,dt_k6_toprns_1,dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1,fc2_membered,e1_16_1_1__toprns_1,e2_16_1_1__toprns_1,e3_16_1_1__toprns_1,e4_16_1_1__toprns_1,e5_16_1_1__toprns_1]),
    [interesting(0.65),file(toprns_1,e1_16_1__toprns_1),[file(toprns_1,e1_16_1__toprns_1)]]).

fof(i2_16_1__toprns_1,theorem,(
    $true ),
    introduced(tautology,[file(toprns_1,i2_16_1__toprns_1)]),
    [interesting(0.65),trivial,file(toprns_1,i2_16_1__toprns_1)]).

fof(i1_16_1__toprns_1,plain,(
    k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)),c1_16_1__toprns_1) = k2_seq_1(k5_numbers,k1_numbers,k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)),c1_16_1__toprns_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c1_16_1__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[e1_16_1__toprns_1,i2_16_1__toprns_1]),
    [interesting(0.65),file(toprns_1,i1_16_1__toprns_1),[file(toprns_1,i1_16_1__toprns_1)]]).

fof(i1_16_1_tmp__toprns_1,plain,
    ( m2_subset_1(c1_16_1__toprns_1,k1_numbers,k5_numbers)
   => k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)),c1_16_1__toprns_1) = k2_seq_1(k5_numbers,k1_numbers,k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)),c1_16_1__toprns_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1]),discharge_asm(discharge,[dt_c1_16_1__toprns_1])],[dt_c1_16_1__toprns_1,i1_16_1__toprns_1]),
    [interesting(0.8),e1_16__toprns_1]).

fof(e1_16__toprns_1,plain,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_seq_1(k5_numbers,k1_numbers,k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)),A) = k2_seq_1(k5_numbers,k1_numbers,k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)),A) ) ),
    inference(let,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[i1_16_1_tmp__toprns_1,dh_c1_16_1__toprns_1]),
    [interesting(0.8),file(toprns_1,e1_16__toprns_1),[file(toprns_1,e1_16__toprns_1)]]).

fof(t113_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) )
         => ( ! [E] : 
                ( m1_subset_1(E,A)
               => k1_funct_1(C,E) = k1_funct_1(D,E) )
           => C = D ) ) ) ),
    file(funct_2,t113_funct_2),
    [interesting(0.9),axiom,file(funct_2,t113_funct_2)]).

fof(e2_16__toprns_1,plain,(
    k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k1_xboole_0,dt_l1_pre_topc,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_struct_0,fc6_membered,rc1_arytm_3,rc1_membered,rc2_xreal_0,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,projectivity_k16_complex1,existence_m1_relset_1,dt_k12_seq_1,dt_k15_euclid,dt_k16_complex1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc2_euclid,fc5_membered,rc1_xreal_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k14_seq_1,redefinition_k18_complex1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k14_seq_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_toprns_1,dt_k5_numbers,dt_k6_toprns_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_16__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1,fc2_membered,e1_16__toprns_1,t113_funct_2]),
    [interesting(0.8),file(toprns_1,e2_16__toprns_1),[file(toprns_1,e2_16__toprns_1)]]).

fof(i4_16__toprns_1,theorem,(
    $true ),
    introduced(tautology,[file(toprns_1,i4_16__toprns_1)]),
    [interesting(0.8),trivial,file(toprns_1,i4_16__toprns_1)]).

fof(i3_16__toprns_1,plain,(
    k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c2_16__toprns_1,dt_c3_16__toprns_1])],[e2_16__toprns_1,i4_16__toprns_1]),
    [interesting(0.8),file(toprns_1,i3_16__toprns_1),[file(toprns_1,i3_16__toprns_1)]]).

fof(i3_16_tmp__toprns_1,plain,
    ( ( v1_funct_1(c3_16__toprns_1)
      & v1_funct_2(c3_16__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
      & m2_relset_1(c3_16__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
   => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,c3_16__toprns_1)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,c3_16__toprns_1),k18_complex1(c2_16__toprns_1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c2_16__toprns_1]),discharge_asm(discharge,[dt_c3_16__toprns_1])],[dt_c3_16__toprns_1,i3_16__toprns_1]),
    [interesting(0.8),i2_16__toprns_1]).

fof(i2_16__toprns_1,plain,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
        & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
     => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,A)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,A),k18_complex1(c2_16__toprns_1)) ) ),
    inference(let,[status(thm),assumptions([dt_c1_16__toprns_1,dt_c2_16__toprns_1])],[i3_16_tmp__toprns_1,dh_c3_16__toprns_1]),
    [interesting(0.8),file(toprns_1,i2_16__toprns_1),[file(toprns_1,i2_16__toprns_1)]]).

fof(i2_16_tmp__toprns_1,plain,
    ( m1_subset_1(c2_16__toprns_1,k1_numbers)
   => ! [A] : 
        ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
          & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
       => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(c2_16__toprns_1,c1_16__toprns_1,A)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,A),k18_complex1(c2_16__toprns_1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_16__toprns_1]),discharge_asm(discharge,[dt_c2_16__toprns_1])],[dt_c2_16__toprns_1,i2_16__toprns_1]),
    [interesting(0.8),i1_16__toprns_1]).

fof(i1_16__toprns_1,plain,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
            & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
         => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(A,c1_16__toprns_1,B)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,B),k18_complex1(A)) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_16__toprns_1])],[i2_16_tmp__toprns_1,dh_c2_16__toprns_1]),
    [interesting(0.8),file(toprns_1,i1_16__toprns_1),[file(toprns_1,i1_16__toprns_1)]]).

fof(i1_16_tmp__toprns_1,plain,
    ( m2_subset_1(c1_16__toprns_1,k1_numbers,k5_numbers)
   => ! [A] : 
        ( m1_subset_1(A,k1_numbers)
       => ! [B] : 
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1)))
              & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(c1_16__toprns_1))) )
           => k6_toprns_1(c1_16__toprns_1,k2_toprns_1(A,c1_16__toprns_1,B)) = k14_seq_1(k6_toprns_1(c1_16__toprns_1,B),k18_complex1(A)) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_16__toprns_1])],[dt_c1_16__toprns_1,i1_16__toprns_1]),
    [interesting(1),t9_toprns_1]).

fof(t9_toprns_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,k5_numbers,u1_struct_0(k15_euclid(A))) )
             => k6_toprns_1(A,k2_toprns_1(B,A,C)) = k14_seq_1(k6_toprns_1(A,C),k18_complex1(B)) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_16_tmp__toprns_1,dh_c1_16__toprns_1]),
    [interesting(1),file(toprns_1,t9_toprns_1),[file(toprns_1,t9_toprns_1)]]).