% Mizar ND problem: t49_rfunct_3,rfunct_3,1633,65 
fof(dh_c1_68__rfunct_3,definition,
    ( ( ~ v1_xboole_0(c1_68__rfunct_3)
     => ! [A] : 
          ( ( v1_funct_1(A)
            & m2_relset_1(A,c1_68__rfunct_3,k1_numbers) )
         => ( ! [B] : 
                ( m1_subset_1(B,c1_68__rfunct_3)
               => ( r2_hidden(B,k4_relset_1(c1_68__rfunct_3,k1_numbers,A))
                 => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,A,B)) ) )
           => k19_rfunct_3(c1_68__rfunct_3,A) = A ) ) )
   => ! [C] : 
        ( ~ v1_xboole_0(C)
       => ! [D] : 
            ( ( v1_funct_1(D)
              & m2_relset_1(D,C,k1_numbers) )
           => ( ! [E] : 
                  ( m1_subset_1(E,C)
                 => ( r2_hidden(E,k4_relset_1(C,k1_numbers,D))
                   => r1_xreal_0(0,k2_seq_1(C,k1_numbers,D,E)) ) )
             => k19_rfunct_3(C,D) = D ) ) ) ),
    introduced(definition,[new_symbol(c1_68__rfunct_3),file(rfunct_3,c1_68__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c1_68__rfunct_3)]).

fof(dh_c2_68__rfunct_3,definition,
    ( ( ( v1_funct_1(c2_68__rfunct_3)
        & m2_relset_1(c2_68__rfunct_3,c1_68__rfunct_3,k1_numbers) )
     => ( ! [A] : 
            ( m1_subset_1(A,c1_68__rfunct_3)
           => ( r2_hidden(A,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
             => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,A)) ) )
       => k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3) = c2_68__rfunct_3 ) )
   => ! [B] : 
        ( ( v1_funct_1(B)
          & m2_relset_1(B,c1_68__rfunct_3,k1_numbers) )
       => ( ! [C] : 
              ( m1_subset_1(C,c1_68__rfunct_3)
             => ( r2_hidden(C,k4_relset_1(c1_68__rfunct_3,k1_numbers,B))
               => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,B,C)) ) )
         => k19_rfunct_3(c1_68__rfunct_3,B) = B ) ) ),
    introduced(definition,[new_symbol(c2_68__rfunct_3),file(rfunct_3,c2_68__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c2_68__rfunct_3)]).

fof(e1_68__rfunct_3,assumption,(
    ! [A] : 
      ( m1_subset_1(A,c1_68__rfunct_3)
     => ( r2_hidden(A,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
       => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,A)) ) ) ),
    introduced(assumption,[file(rfunct_3,e1_68__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,e1_68__rfunct_3)]).

fof(cc1_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A) ) ) ),
    file(finseq_1,cc1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]).

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(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(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(rc1_finseq_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]).

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(rc1_partfun1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A) ) ),
    file(partfun1,rc1_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc1_partfun1)]).

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_finseq_1,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc3_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]).

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(rc6_finseq_1,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A) ) ),
    file(finseq_1,rc6_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]).

fof(rc8_finseq_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc8_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc8_finseq_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(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_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_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => v4_membered(A) ) ),
    file(membered,cc1_membered),
    [interesting(0.9),axiom,file(membered,cc1_membered)]).

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_finset_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_finset_1(B) ) ) ),
    file(finset_1,cc2_finset_1),
    [interesting(0.9),axiom,file(finset_1,cc2_finset_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(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_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => v2_membered(A) ) ),
    file(membered,cc3_membered),
    [interesting(0.9),axiom,file(membered,cc3_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(fc14_finset_1,theorem,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_zfmisc_1(A,B)) ) ),
    file(finset_1,fc14_finset_1),
    [interesting(0.9),axiom,file(finset_1,fc14_finset_1)]).

fof(fc17_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A) )
     => v1_finset_1(k1_relat_1(A)) ) ),
    file(finseq_1,fc17_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,fc17_finseq_1)]).

fof(fc1_rfunct_3,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_finset_1(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_relat_1(k19_rfunct_3(A,B))
        & v1_funct_1(k19_rfunct_3(A,B))
        & v1_finset_1(k19_rfunct_3(A,B))
        & v1_seq_1(k19_rfunct_3(A,B)) ) ) ),
    file(rfunct_3,fc1_rfunct_3),
    [interesting(0.9),axiom,file(rfunct_3,fc1_rfunct_3)]).

fof(fc1_seq_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_seq_1(A) )
     => ( v1_xcmplx_0(k1_funct_1(A,B))
        & v1_xreal_0(k1_funct_1(A,B)) ) ) ),
    file(seq_1,fc1_seq_1),
    [interesting(0.9),axiom,file(seq_1,fc1_seq_1)]).

fof(fc2_finseq_1,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_finset_1(k1_xboole_0)
    & v1_finseq_1(k1_xboole_0) ),
    file(finseq_1,fc2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]).

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_finset_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) ),
    file(finset_1,rc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]).

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_seq_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_seq_1(A) ) ),
    file(seq_1,rc1_seq_1),
    [interesting(0.9),axiom,file(seq_1,rc1_seq_1)]).

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(rc2_rfunct_3,theorem,(
    ! [A,B] : 
    ? [C] : 
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C)
      & v1_finset_1(C) ) ),
    file(rfunct_3,rc2_rfunct_3),
    [interesting(0.9),axiom,file(rfunct_3,rc2_rfunct_3)]).

fof(rc3_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc3_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]).

fof(rc4_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc4_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc4_finset_1)]).

fof(rc7_finseq_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A) ) ),
    file(finseq_1,rc7_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]).

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_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    [interesting(0.9),axiom,file(relat_1,k1_relat_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_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]).

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    [interesting(0.9),axiom,file(relset_1,m1_relset_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_finset_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) ),
    file(finset_1,cc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]).

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(cc1_seq_1,theorem,(
    ! [A,B] : 
      ( v2_membered(B)
     => ! [C] : 
          ( m1_relset_1(C,A,B)
         => ( v1_funct_1(C)
           => ( v1_funct_1(C)
              & v1_seq_1(C) ) ) ) ) ),
    file(seq_1,cc1_seq_1),
    [interesting(0.9),axiom,file(seq_1,cc1_seq_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(rc2_partfun1,theorem,(
    ! [A,B] : 
    ? [C] : 
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C) ) ),
    file(partfun1,rc2_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc2_partfun1)]).

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(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(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(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(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_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(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_k4_relset_1,definition,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => k4_relset_1(A,B,C) = k1_relat_1(C) ) ),
    file(relset_1,k4_relset_1),
    [interesting(0.9),axiom,file(relset_1,k4_relset_1)]).

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_k19_rfunct_3,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & m1_relset_1(B,A,k1_numbers) )
     => ( v1_funct_1(k19_rfunct_3(A,B))
        & m2_relset_1(k19_rfunct_3(A,B),A,k1_numbers) ) ) ),
    file(rfunct_3,k19_rfunct_3),
    [interesting(0.9),axiom,file(rfunct_3,k19_rfunct_3)]).

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_k4_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(relset_1,k4_relset_1),
    [interesting(0.9),axiom,file(relset_1,k4_relset_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_c1_68__rfunct_3,assumption,(
    ~ v1_xboole_0(c1_68__rfunct_3) ),
    introduced(assumption,[file(rfunct_3,c1_68__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c1_68__rfunct_3)]).

fof(dt_c2_68__rfunct_3,assumption,
    ( v1_funct_1(c2_68__rfunct_3)
    & m2_relset_1(c2_68__rfunct_3,c1_68__rfunct_3,k1_numbers) ),
    introduced(assumption,[file(rfunct_3,c2_68__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c2_68__rfunct_3)]).

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(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(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(dh_c1_68_1__rfunct_3,definition,
    ( ( m1_subset_1(c1_68_1__rfunct_3,c1_68__rfunct_3)
     => ( r2_hidden(c1_68_1__rfunct_3,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
       => k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),c1_68_1__rfunct_3) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3) ) )
   => ! [A] : 
        ( m1_subset_1(A,c1_68__rfunct_3)
       => ( r2_hidden(A,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
         => k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),A) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,A) ) ) ),
    introduced(definition,[new_symbol(c1_68_1__rfunct_3),file(rfunct_3,c1_68_1__rfunct_3)]),
    [interesting(0.65),axiom,file(rfunct_3,c1_68_1__rfunct_3)]).

fof(e1_68_1__rfunct_3,assumption,(
    r2_hidden(c1_68_1__rfunct_3,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3)) ),
    introduced(assumption,[file(rfunct_3,e1_68_1__rfunct_3)]),
    [interesting(0.65),axiom,file(rfunct_3,e1_68_1__rfunct_3)]).

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

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(rc2_finset_1,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v2_funct_1(B)
      & v1_ordinal1(B)
      & v2_ordinal1(B)
      & v3_ordinal1(B)
      & v4_ordinal2(B)
      & v1_finset_1(B) ) ),
    file(finset_1,rc2_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc2_finset_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(existence_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m2_subset_1(C,A,B) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(redefinition_k5_numbers,definition,(
    k5_numbers = k5_ordinal2 ),
    file(numbers,k5_numbers),
    [interesting(0.9),axiom,file(numbers,k5_numbers)]).

fof(redefinition_m2_subset_1,definition,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(dt_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_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
         => m1_subset_1(C,A) ) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(cc1_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(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(rc2_finseq_1,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_relset_1(B,k5_numbers,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v1_finset_1(B)
      & v1_finseq_1(B) ) ),
    file(finseq_1,rc2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc2_finseq_1)]).

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

fof(commutativity_k2_square_1,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => k2_square_1(A,B) = k2_square_1(B,A) ) ),
    file(square_1,k2_square_1),
    [interesting(0.9),axiom,file(square_1,k2_square_1)]).

fof(idempotence_k2_square_1,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => k2_square_1(A,A) = A ) ),
    file(square_1,k2_square_1),
    [interesting(0.9),axiom,file(square_1,k2_square_1)]).

fof(dt_k2_square_1,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => v1_xreal_0(k2_square_1(A,B)) ) ),
    file(square_1,k2_square_1),
    [interesting(0.9),axiom,file(square_1,k2_square_1)]).

fof(spc0_boole,theorem,(
    v1_xboole_0(0) ),
    file(boole,spc0_boole),
    [interesting(0.9),axiom,file(boole,spc0_boole)]).

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(dt_k2_rfunct_3,axiom,(
    ! [A] : 
      ( v1_xreal_0(A)
     => m1_subset_1(k2_rfunct_3(A),k1_numbers) ) ),
    file(rfunct_3,k2_rfunct_3),
    [interesting(0.9),axiom,file(rfunct_3,k2_rfunct_3)]).

fof(dt_c1_68_1__rfunct_3,assumption,(
    m1_subset_1(c1_68_1__rfunct_3,c1_68__rfunct_3) ),
    introduced(assumption,[file(rfunct_3,c1_68_1__rfunct_3)]),
    [interesting(0.65),axiom,file(rfunct_3,c1_68_1__rfunct_3)]).

fof(d1_rfunct_3,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => k2_rfunct_3(A) = k2_square_1(A,0) ) ),
    file(rfunct_3,d1_rfunct_3),
    [interesting(0.9),axiom,file(rfunct_3,d1_rfunct_3)]).

fof(d10_rfunct_3,definition,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] : 
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,k1_numbers) )
             => ( C = k19_rfunct_3(A,B)
              <=> ( k4_relset_1(A,k1_numbers,C) = k4_relset_1(A,k1_numbers,B)
                  & ! [D] : 
                      ( m1_subset_1(D,A)
                     => ( r2_hidden(D,k4_relset_1(A,k1_numbers,C))
                       => k2_seq_1(A,k1_numbers,C,D) = k2_rfunct_3(k2_seq_1(A,k1_numbers,B,D)) ) ) ) ) ) ) ) ),
    file(rfunct_3,d10_rfunct_3),
    [interesting(0.9),axiom,file(rfunct_3,d10_rfunct_3)]).

fof(e2_68__rfunct_3,plain,(
    k4_relset_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3)) = k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c2_68__rfunct_3])],[dt_k5_ordinal2,fc5_membered,rc2_finset_1,rc2_nat_1,rc3_nat_1,reflexivity_r1_tarski,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k5_numbers,dt_m2_subset_1,cc1_finseq_1,cc1_nat_1,cc1_xreal_0,cc2_nat_1,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,cc9_membered,fc17_finseq_1,fc1_seq_1,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_partfun1,rc2_rfunct_3,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_numerals,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_square_1,dt_k2_zfmisc_1,dt_m1_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,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc4_membered,cc6_membered,cc7_xreal_0,fc14_finset_1,fc1_rfunct_3,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc4_finset_1,spc0_boole,t3_subset,t4_subset,t5_subset,spc0_numerals,spc0_boole,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_rfunct_3,dt_k2_seq_1,dt_k4_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_68__rfunct_3,dt_c2_68__rfunct_3,cc15_membered,cc1_finset_1,fc2_membered,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole,d1_rfunct_3,d10_rfunct_3]),
    [interesting(0.8),file(rfunct_3,e2_68__rfunct_3),[file(rfunct_3,e2_68__rfunct_3)]]).

fof(e1_68_1_1__rfunct_3,plain,(
    k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),c1_68_1__rfunct_3) = k2_rfunct_3(k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_68_1__rfunct_3,dt_c1_68__rfunct_3,dt_c2_68__rfunct_3,e1_68_1__rfunct_3])],[dt_k5_ordinal2,fc5_membered,rc2_finset_1,rc2_nat_1,rc3_nat_1,reflexivity_r1_tarski,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k5_numbers,dt_m2_subset_1,cc1_finseq_1,cc1_nat_1,cc1_xreal_0,cc2_nat_1,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,cc9_membered,fc17_finseq_1,fc1_seq_1,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_partfun1,rc2_rfunct_3,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_numerals,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_square_1,dt_k2_zfmisc_1,dt_m1_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,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc4_membered,cc6_membered,cc7_xreal_0,fc14_finset_1,fc1_rfunct_3,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc4_finset_1,spc0_boole,t3_subset,t4_subset,t5_subset,spc0_numerals,spc0_boole,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_rfunct_3,dt_k2_seq_1,dt_k4_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,cc15_membered,cc1_finset_1,fc2_membered,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole,d1_rfunct_3,e2_68__rfunct_3,e1_68_1__rfunct_3,d10_rfunct_3]),
    [interesting(0.5),file(rfunct_3,e1_68_1_1__rfunct_3),[file(rfunct_3,e1_68_1_1__rfunct_3)]]).

fof(commutativity_k4_square_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_square_1(A,B) = k4_square_1(B,A) ) ),
    file(square_1,k4_square_1),
    [interesting(0.9),axiom,file(square_1,k4_square_1)]).

fof(idempotence_k4_square_1,theorem,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_square_1(A,A) = A ) ),
    file(square_1,k4_square_1),
    [interesting(0.9),axiom,file(square_1,k4_square_1)]).

fof(redefinition_k4_square_1,definition,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_square_1(A,B) = k2_square_1(A,B) ) ),
    file(square_1,k4_square_1),
    [interesting(0.9),axiom,file(square_1,k4_square_1)]).

fof(dt_k4_square_1,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k4_square_1(A,B),k1_numbers) ) ),
    file(square_1,k4_square_1),
    [interesting(0.9),axiom,file(square_1,k4_square_1)]).

fof(e2_68_1_1__rfunct_3,plain,(
    k2_rfunct_3(k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3)) = k4_square_1(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_rfunct_3,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k2_square_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k4_square_1,idempotence_k4_square_1,redefinition_k2_seq_1,redefinition_k4_square_1,dt_k1_numbers,dt_k2_rfunct_3,dt_k2_seq_1,dt_k4_square_1,dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,fc2_membered,d1_rfunct_3,spc0_numerals,spc0_boole]),
    [interesting(0.5),file(rfunct_3,e2_68_1_1__rfunct_3),[file(rfunct_3,e2_68_1_1__rfunct_3)]]).

fof(t2_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(A,B)
              & v3_xreal_0(B) )
           => v3_xreal_0(A) ) ) ) ),
    file(real,t2_real),
    [interesting(0.9),axiom,file(real,t2_real)]).

fof(t3_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( r1_xreal_0(A,B)
              & ~ v3_xreal_0(A)
              & v3_xreal_0(B) ) ) ) ),
    file(real,t3_real),
    [interesting(0.9),axiom,file(real,t3_real)]).

fof(t5_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ( v1_xboole_0(B)
              | v3_xreal_0(A)
              | v2_xreal_0(B) ) ) ) ) ),
    file(real,t5_real),
    [interesting(0.9),axiom,file(real,t5_real)]).

fof(t6_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => ( v1_xboole_0(A)
              | v2_xreal_0(B)
              | v3_xreal_0(A) ) ) ) ) ),
    file(real,t6_real),
    [interesting(0.9),axiom,file(real,t6_real)]).

fof(t7_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,B)
              & ~ v2_xreal_0(A)
              & ~ v3_xreal_0(B) ) ) ) ),
    file(real,t7_real),
    [interesting(0.9),axiom,file(real,t7_real)]).

fof(t8_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,B)
              & ~ v3_xreal_0(B)
              & ~ v2_xreal_0(A) ) ) ) ),
    file(real,t8_real),
    [interesting(0.9),axiom,file(real,t8_real)]).

fof(t1_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(A,B)
              & v2_xreal_0(A) )
           => v2_xreal_0(B) ) ) ) ),
    file(real,t1_real),
    [interesting(0.9),axiom,file(real,t1_real)]).

fof(t4_real,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( r1_xreal_0(A,B)
              & ~ v2_xreal_0(B)
              & v2_xreal_0(A) ) ) ) ),
    file(real,t4_real),
    [interesting(0.9),axiom,file(real,t4_real)]).

fof(reflexivity_r1_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => r1_xreal_0(A,A) ) ),
    file(xreal_0,r1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]).

fof(connectedness_r1_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( r1_xreal_0(A,B)
        | r1_xreal_0(B,A) ) ) ),
    file(xreal_0,r1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]).

fof(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,(
    r1_xreal_0(0,0) ),
    file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0),
    [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]).

fof(e2_68_1__rfunct_3,plain,(
    r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,e1_68_1__rfunct_3,e1_68__rfunct_3])],[cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc17_finseq_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_rfunct_3,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,dt_k1_numbers,dt_k2_seq_1,dt_k4_relset_1,dt_m1_subset_1,dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_subset,t7_boole,spc0_numerals,spc0_boole,e1_68_1__rfunct_3,e1_68__rfunct_3]),
    [interesting(0.65),file(rfunct_3,e2_68_1__rfunct_3),[file(rfunct_3,e2_68_1__rfunct_3)]]).

fof(d2_square_1,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(B,A)
             => k2_square_1(A,B) = A )
            & ( ~ r1_xreal_0(B,A)
             => k2_square_1(A,B) = B ) ) ) ) ),
    file(square_1,d2_square_1),
    [interesting(0.9),axiom,file(square_1,d2_square_1)]).

fof(e3_68_1_1__rfunct_3,plain,(
    k4_square_1(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3)) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,e1_68_1__rfunct_3,e1_68__rfunct_3])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc2_nat_1,rc2_partfun1,rc2_rfunct_3,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,rc1_xreal_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k2_square_1,idempotence_k2_square_1,commutativity_k4_square_1,idempotence_k4_square_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_seq_1,redefinition_k4_square_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_square_1,dt_k4_square_1,dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,cc2_xreal_0,fc2_membered,spc0_numerals,spc0_boole,e2_68_1__rfunct_3,d2_square_1,rqLessOrEqual__r1_xreal_0__r0_r0]),
    [interesting(0.5),file(rfunct_3,e3_68_1_1__rfunct_3),[file(rfunct_3,e3_68_1_1__rfunct_3)]]).

fof(e3_68_1__rfunct_3,plain,(
    k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),c1_68_1__rfunct_3) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,e1_68_1__rfunct_3,e1_68__rfunct_3])],[e1_68_1_1__rfunct_3,e2_68_1_1__rfunct_3,e3_68_1_1__rfunct_3]),
    [interesting(0.65),file(rfunct_3,e3_68_1__rfunct_3),[file(rfunct_3,e3_68_1__rfunct_3)]]).

fof(i3_68_1__rfunct_3,theorem,(
    $true ),
    introduced(tautology,[file(rfunct_3,i3_68_1__rfunct_3)]),
    [interesting(0.65),trivial,file(rfunct_3,i3_68_1__rfunct_3)]).

fof(i2_68_1__rfunct_3,plain,(
    k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),c1_68_1__rfunct_3) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,e1_68_1__rfunct_3,e1_68__rfunct_3])],[e3_68_1__rfunct_3,i3_68_1__rfunct_3]),
    [interesting(0.65),file(rfunct_3,i2_68_1__rfunct_3),[file(rfunct_3,i2_68_1__rfunct_3)]]).

fof(i1_68_1__rfunct_3,plain,
    ( r2_hidden(c1_68_1__rfunct_3,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
   => k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),c1_68_1__rfunct_3) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c1_68_1__rfunct_3,dt_c2_68__rfunct_3,e1_68__rfunct_3]),discharge_asm(discharge,[e1_68_1__rfunct_3])],[e1_68_1__rfunct_3,i2_68_1__rfunct_3]),
    [interesting(0.65),file(rfunct_3,i1_68_1__rfunct_3),[file(rfunct_3,i1_68_1__rfunct_3)]]).

fof(i1_68_1_tmp__rfunct_3,plain,
    ( m1_subset_1(c1_68_1__rfunct_3,c1_68__rfunct_3)
   => ( r2_hidden(c1_68_1__rfunct_3,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
     => k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),c1_68_1__rfunct_3) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,c1_68_1__rfunct_3) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c2_68__rfunct_3,e1_68__rfunct_3]),discharge_asm(discharge,[dt_c1_68_1__rfunct_3])],[dt_c1_68_1__rfunct_3,i1_68_1__rfunct_3]),
    [interesting(0.8),e3_68__rfunct_3]).

fof(e3_68__rfunct_3,plain,(
    ! [A] : 
      ( m1_subset_1(A,c1_68__rfunct_3)
     => ( r2_hidden(A,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
       => k2_seq_1(c1_68__rfunct_3,k1_numbers,k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3),A) = k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,A) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c2_68__rfunct_3,e1_68__rfunct_3])],[i1_68_1_tmp__rfunct_3,dh_c1_68_1__rfunct_3]),
    [interesting(0.8),file(rfunct_3,e3_68__rfunct_3),[file(rfunct_3,e3_68__rfunct_3)]]).

fof(t34_partfun1,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] : 
          ( ( v1_funct_1(D)
            & m2_relset_1(D,A,B) )
         => ( ( k4_relset_1(A,B,C) = k4_relset_1(A,B,D)
              & ! [E] : 
                  ( m1_subset_1(E,A)
                 => ( r2_hidden(E,k4_relset_1(A,B,C))
                   => k1_funct_1(C,E) = k1_funct_1(D,E) ) ) )
           => C = D ) ) ) ),
    file(partfun1,t34_partfun1),
    [interesting(0.9),axiom,file(partfun1,t34_partfun1)]).

fof(e4_68__rfunct_3,plain,(
    k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3) = c2_68__rfunct_3 ),
    inference(mizar_by,[status(thm),assumptions([e1_68__rfunct_3,dt_c1_68__rfunct_3,dt_c2_68__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc14_finset_1,fc17_finseq_1,fc1_rfunct_3,fc1_seq_1,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,existence_m1_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_relset_1,cc1_seq_1,cc4_membered,cc6_membered,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k19_rfunct_3,dt_k1_funct_1,dt_k1_numbers,dt_k2_seq_1,dt_k4_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_68__rfunct_3,dt_c2_68__rfunct_3,fc2_membered,t1_subset,t7_boole,e3_68__rfunct_3,e2_68__rfunct_3,t34_partfun1]),
    [interesting(0.8),file(rfunct_3,e4_68__rfunct_3),[file(rfunct_3,e4_68__rfunct_3)]]).

fof(i4_68__rfunct_3,theorem,(
    $true ),
    introduced(tautology,[file(rfunct_3,i4_68__rfunct_3)]),
    [interesting(0.8),trivial,file(rfunct_3,i4_68__rfunct_3)]).

fof(i3_68__rfunct_3,plain,(
    k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3) = c2_68__rfunct_3 ),
    inference(conclusion,[status(thm),assumptions([e1_68__rfunct_3,dt_c1_68__rfunct_3,dt_c2_68__rfunct_3])],[e4_68__rfunct_3,i4_68__rfunct_3]),
    [interesting(0.8),file(rfunct_3,i3_68__rfunct_3),[file(rfunct_3,i3_68__rfunct_3)]]).

fof(i2_68__rfunct_3,plain,
    ( ! [A] : 
        ( m1_subset_1(A,c1_68__rfunct_3)
       => ( r2_hidden(A,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
         => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,A)) ) )
   => k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3) = c2_68__rfunct_3 ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_68__rfunct_3,dt_c2_68__rfunct_3]),discharge_asm(discharge,[e1_68__rfunct_3])],[e1_68__rfunct_3,i3_68__rfunct_3]),
    [interesting(0.8),file(rfunct_3,i2_68__rfunct_3),[file(rfunct_3,i2_68__rfunct_3)]]).

fof(i2_68_tmp__rfunct_3,plain,
    ( ( v1_funct_1(c2_68__rfunct_3)
      & m2_relset_1(c2_68__rfunct_3,c1_68__rfunct_3,k1_numbers) )
   => ( ! [A] : 
          ( m1_subset_1(A,c1_68__rfunct_3)
         => ( r2_hidden(A,k4_relset_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3))
           => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,c2_68__rfunct_3,A)) ) )
     => k19_rfunct_3(c1_68__rfunct_3,c2_68__rfunct_3) = c2_68__rfunct_3 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_68__rfunct_3]),discharge_asm(discharge,[dt_c2_68__rfunct_3])],[dt_c2_68__rfunct_3,i2_68__rfunct_3]),
    [interesting(0.8),i1_68__rfunct_3]).

fof(i1_68__rfunct_3,plain,(
    ! [A] : 
      ( ( v1_funct_1(A)
        & m2_relset_1(A,c1_68__rfunct_3,k1_numbers) )
     => ( ! [B] : 
            ( m1_subset_1(B,c1_68__rfunct_3)
           => ( r2_hidden(B,k4_relset_1(c1_68__rfunct_3,k1_numbers,A))
             => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,A,B)) ) )
       => k19_rfunct_3(c1_68__rfunct_3,A) = A ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_68__rfunct_3])],[i2_68_tmp__rfunct_3,dh_c2_68__rfunct_3]),
    [interesting(0.8),file(rfunct_3,i1_68__rfunct_3),[file(rfunct_3,i1_68__rfunct_3)]]).

fof(i1_68_tmp__rfunct_3,plain,
    ( ~ v1_xboole_0(c1_68__rfunct_3)
   => ! [A] : 
        ( ( v1_funct_1(A)
          & m2_relset_1(A,c1_68__rfunct_3,k1_numbers) )
       => ( ! [B] : 
              ( m1_subset_1(B,c1_68__rfunct_3)
             => ( r2_hidden(B,k4_relset_1(c1_68__rfunct_3,k1_numbers,A))
               => r1_xreal_0(0,k2_seq_1(c1_68__rfunct_3,k1_numbers,A,B)) ) )
         => k19_rfunct_3(c1_68__rfunct_3,A) = A ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_68__rfunct_3])],[dt_c1_68__rfunct_3,i1_68__rfunct_3]),
    [interesting(1),t49_rfunct_3]).

fof(t49_rfunct_3,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ( v1_funct_1(B)
            & m2_relset_1(B,A,k1_numbers) )
         => ( ! [C] : 
                ( m1_subset_1(C,A)
               => ( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
                 => r1_xreal_0(0,k2_seq_1(A,k1_numbers,B,C)) ) )
           => k19_rfunct_3(A,B) = B ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_68_tmp__rfunct_3,dh_c1_68__rfunct_3]),
    [interesting(1),file(rfunct_3,t49_rfunct_3),[file(rfunct_3,t49_rfunct_3)]]).