% Mizar ND problem: t2_graph_2,graph_2,70,20 
fof(dh_c1_3__graph_2,definition,
    ( ( ( v1_relat_1(c1_3__graph_2)
        & v1_funct_1(c1_3__graph_2)
        & v1_finseq_1(c1_3__graph_2) )
     => ! [A] : 
          ( ( v1_relat_1(A)
            & v1_funct_1(A)
            & v1_finseq_1(A) )
         => ! [B] : 
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( c1_3__graph_2 = k7_relat_1(A,k2_finseq_1(B))
               => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(A))
                  & ! [C] : 
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(1,C)
                          & r1_xreal_0(C,k3_finseq_1(c1_3__graph_2)) )
                       => k1_funct_1(A,C) = k1_funct_1(c1_3__graph_2,C) ) ) ) ) ) ) )
   => ! [D] : 
        ( ( v1_relat_1(D)
          & v1_funct_1(D)
          & v1_finseq_1(D) )
       => ! [E] : 
            ( ( v1_relat_1(E)
              & v1_funct_1(E)
              & v1_finseq_1(E) )
           => ! [F] : 
                ( m2_subset_1(F,k1_numbers,k5_numbers)
               => ( D = k7_relat_1(E,k2_finseq_1(F))
                 => ( r1_xreal_0(k3_finseq_1(D),k3_finseq_1(E))
                    & ! [G] : 
                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(1,G)
                            & r1_xreal_0(G,k3_finseq_1(D)) )
                         => k1_funct_1(E,G) = k1_funct_1(D,G) ) ) ) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_3__graph_2),file(graph_2,c1_3__graph_2)]),
    [interesting(0.8),axiom,file(graph_2,c1_3__graph_2)]).

fof(dh_c2_3__graph_2,definition,
    ( ( ( v1_relat_1(c2_3__graph_2)
        & v1_funct_1(c2_3__graph_2)
        & v1_finseq_1(c2_3__graph_2) )
     => ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(A))
           => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
              & ! [B] : 
                  ( m2_subset_1(B,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(1,B)
                      & r1_xreal_0(B,k3_finseq_1(c1_3__graph_2)) )
                   => k1_funct_1(c2_3__graph_2,B) = k1_funct_1(c1_3__graph_2,B) ) ) ) ) ) )
   => ! [C] : 
        ( ( v1_relat_1(C)
          & v1_funct_1(C)
          & v1_finseq_1(C) )
       => ! [D] : 
            ( m2_subset_1(D,k1_numbers,k5_numbers)
           => ( c1_3__graph_2 = k7_relat_1(C,k2_finseq_1(D))
             => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(C))
                & ! [E] : 
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                   => ( ( r1_xreal_0(1,E)
                        & r1_xreal_0(E,k3_finseq_1(c1_3__graph_2)) )
                     => k1_funct_1(C,E) = k1_funct_1(c1_3__graph_2,E) ) ) ) ) ) ) ),
    introduced(definition,[new_symbol(c2_3__graph_2),file(graph_2,c2_3__graph_2)]),
    [interesting(0.8),axiom,file(graph_2,c2_3__graph_2)]).

fof(dh_c3_3__graph_2,definition,
    ( ( m2_subset_1(c3_3__graph_2,k1_numbers,k5_numbers)
     => ( c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(c3_3__graph_2))
       => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
          & ! [A] : 
              ( m2_subset_1(A,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(1,A)
                  & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
               => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ) ) )
   => ! [B] : 
        ( m2_subset_1(B,k1_numbers,k5_numbers)
       => ( c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(B))
         => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
            & ! [C] : 
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ( r1_xreal_0(1,C)
                    & r1_xreal_0(C,k3_finseq_1(c1_3__graph_2)) )
                 => k1_funct_1(c2_3__graph_2,C) = k1_funct_1(c1_3__graph_2,C) ) ) ) ) ) ),
    introduced(definition,[new_symbol(c3_3__graph_2),file(graph_2,c3_3__graph_2)]),
    [interesting(0.8),axiom,file(graph_2,c3_3__graph_2)]).

fof(e1_3__graph_2,assumption,(
    c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(c3_3__graph_2)) ),
    introduced(assumption,[file(graph_2,e1_3__graph_2)]),
    [interesting(0.8),axiom,file(graph_2,e1_3__graph_2)]).

fof(e1_3_1_1__graph_2,assumption,(
    r1_xreal_0(c3_3__graph_2,k3_finseq_1(c2_3__graph_2)) ),
    introduced(assumption,[file(graph_2,e1_3_1_1__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,e1_3_1_1__graph_2)]).

fof(rc4_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]).

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

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(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(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_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]).

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

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

fof(cc1_card_1,theorem,(
    ! [A] : 
      ( v1_card_1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(card_1,cc1_card_1),
    [interesting(0.9),axiom,file(card_1,cc1_card_1)]).

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_funct_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]).

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

fof(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(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(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(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_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(rc1_card_1,theorem,(
    ? [A] : v1_card_1(A) ),
    file(card_1,rc1_card_1),
    [interesting(0.9),axiom,file(card_1,rc1_card_1)]).

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_nat_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(nat_1,rc1_nat_1),
    [interesting(0.9),axiom,file(nat_1,rc1_nat_1)]).

fof(rc2_card_1,theorem,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v1_finset_1(A)
      & v1_card_1(A) ) ),
    file(card_1,rc2_card_1),
    [interesting(0.9),axiom,file(card_1,rc2_card_1)]).

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

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

fof(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_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(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(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_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(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(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(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(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(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(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(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_boole,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    [interesting(0.9),axiom,file(boole,t6_boole)]).

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

fof(existence_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m2_subset_1(C,A,B) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

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

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

fof(dt_k1_card_1,axiom,(
    ! [A] : v1_card_1(k1_card_1(A)) ),
    file(card_1,k1_card_1),
    [interesting(0.9),axiom,file(card_1,k1_card_1)]).

fof(dt_k1_finseq_1,axiom,(
    $true ),
    file(finseq_1,k1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k1_finseq_1)]).

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

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_k5_numbers,axiom,(
    m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ),
    file(numbers,k5_numbers),
    [interesting(0.9),axiom,file(numbers,k5_numbers)]).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(dt_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
         => m1_subset_1(C,A) ) ) ),
    file(subset_1,m2_subset_1),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(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_card_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_card_1(A) ) ) ),
    file(card_1,cc2_card_1),
    [interesting(0.9),axiom,file(card_1,cc2_card_1)]).

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_int_1,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_int_1(A) ) ) ),
    file(int_1,cc2_int_1),
    [interesting(0.9),axiom,file(int_1,cc2_int_1)]).

fof(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(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_card_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_finset_1(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_card_1(A) ) ) ),
    file(card_1,cc3_card_1),
    [interesting(0.9),axiom,file(card_1,cc3_card_1)]).

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

fof(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(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_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => v1_finset_1(k1_finseq_1(A)) ) ),
    file(finseq_1,fc1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,fc1_finseq_1)]).

fof(fc2_card_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ( v1_ordinal1(k1_card_1(A))
        & v2_ordinal1(k1_card_1(A))
        & v3_ordinal1(k1_card_1(A))
        & v1_finset_1(k1_card_1(A))
        & v1_card_1(k1_card_1(A)) ) ) ),
    file(card_1,fc2_card_1),
    [interesting(0.9),axiom,file(card_1,fc2_card_1)]).

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

fof(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_int_1,theorem,(
    ? [A] : v1_int_1(A) ),
    file(int_1,rc2_int_1),
    [interesting(0.9),axiom,file(int_1,rc2_int_1)]).

fof(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(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(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(redefinition_k2_finseq_1,definition,(
    ! [A] : 
      ( v4_ordinal2(A)
     => k2_finseq_1(A) = k1_finseq_1(A) ) ),
    file(finseq_1,k2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]).

fof(redefinition_k3_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => k3_finseq_1(A) = k1_card_1(A) ) ),
    file(finseq_1,k3_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]).

fof(redefinition_k4_finseq_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => k4_finseq_1(A) = k1_relat_1(A) ) ),
    file(finseq_1,k4_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k4_finseq_1)]).

fof(dt_k2_finseq_1,axiom,(
    ! [A] : 
      ( v4_ordinal2(A)
     => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ),
    file(finseq_1,k2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]).

fof(dt_k3_finseq_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ),
    file(finseq_1,k3_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]).

fof(dt_k4_finseq_1,axiom,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ),
    file(finseq_1,k4_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,k4_finseq_1)]).

fof(dt_k7_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(A)
     => v1_relat_1(k7_relat_1(A,B)) ) ),
    file(relat_1,k7_relat_1),
    [interesting(0.9),axiom,file(relat_1,k7_relat_1)]).

fof(dt_c1_3__graph_2,assumption,
    ( v1_relat_1(c1_3__graph_2)
    & v1_funct_1(c1_3__graph_2)
    & v1_finseq_1(c1_3__graph_2) ),
    introduced(assumption,[file(graph_2,c1_3__graph_2)]),
    [interesting(0.8),axiom,file(graph_2,c1_3__graph_2)]).

fof(dt_c2_3__graph_2,assumption,
    ( v1_relat_1(c2_3__graph_2)
    & v1_funct_1(c2_3__graph_2)
    & v1_finseq_1(c2_3__graph_2) ),
    introduced(assumption,[file(graph_2,c2_3__graph_2)]),
    [interesting(0.8),axiom,file(graph_2,c2_3__graph_2)]).

fof(dt_c3_3__graph_2,assumption,(
    m2_subset_1(c3_3__graph_2,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(graph_2,c3_3__graph_2)]),
    [interesting(0.8),axiom,file(graph_2,c3_3__graph_2)]).

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

fof(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(fc4_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B)) ) ) ),
    file(funct_1,fc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc4_funct_1)]).

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_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]).

fof(t21_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( r1_xreal_0(A,k3_finseq_1(B))
                  & C = k7_relat_1(B,k2_finseq_1(A)) )
               => ( k3_finseq_1(C) = A
                  & k4_finseq_1(C) = k2_finseq_1(A) ) ) ) ) ) ),
    file(finseq_1,t21_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,t21_finseq_1)]).

fof(e3_3_1_1__graph_2,plain,(
    r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2,e1_3_1_1__graph_2])],[rc4_funct_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc6_finseq_1,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,dt_k5_ordinal2,cc1_card_1,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_ordinal2,rc1_card_1,rc1_finset_1,rc1_nat_1,rc2_card_1,rc2_funct_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,t1_real,t2_subset,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc4_int_1,cc5_xreal_0,fc17_finseq_1,fc1_finseq_1,fc2_card_1,rc1_int_1,rc1_xreal_0,rc2_int_1,t2_real,t3_real,t3_subset,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k4_finseq_1,dt_k2_finseq_1,dt_k3_finseq_1,dt_k4_finseq_1,dt_k7_relat_1,dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,cc1_finseq_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc4_funct_1,rc1_finseq_1,rc1_funct_1,e1_3__graph_2,e1_3_1_1__graph_2,t21_finseq_1]),
    [interesting(0.5),file(graph_2,e3_3_1_1__graph_2),[file(graph_2,e3_3_1_1__graph_2)]]).

fof(dh_c1_3_1_1__graph_2,definition,
    ( ( m2_subset_1(c1_3_1_1__graph_2,k1_numbers,k5_numbers)
     => ( ( r1_xreal_0(1,c1_3_1_1__graph_2)
          & r1_xreal_0(c1_3_1_1__graph_2,k3_finseq_1(c1_3__graph_2)) )
       => k1_funct_1(c2_3__graph_2,c1_3_1_1__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_1__graph_2) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(1,A)
            & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
         => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ),
    introduced(definition,[new_symbol(c1_3_1_1__graph_2),file(graph_2,c1_3_1_1__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,c1_3_1_1__graph_2)]).

fof(e4_3_1_1__graph_2,assumption,(
    r1_xreal_0(1,c1_3_1_1__graph_2) ),
    introduced(assumption,[file(graph_2,e4_3_1_1__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,e4_3_1_1__graph_2)]).

fof(e5_3_1_1__graph_2,assumption,(
    r1_xreal_0(c1_3_1_1__graph_2,k3_finseq_1(c1_3__graph_2)) ),
    introduced(assumption,[file(graph_2,e5_3_1_1__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,e5_3_1_1__graph_2)]).

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_c1_3_1_1__graph_2,assumption,(
    m2_subset_1(c1_3_1_1__graph_2,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(graph_2,c1_3_1_1__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,c1_3_1_1__graph_2)]).

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(e2_3_1_1__graph_2,plain,(
    k3_finseq_1(c1_3__graph_2) = c3_3__graph_2 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e1_3__graph_2])],[rc4_funct_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc6_finseq_1,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,dt_k5_ordinal2,cc1_card_1,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_ordinal2,rc1_card_1,rc1_finset_1,rc1_nat_1,rc2_card_1,rc2_funct_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,t1_real,t2_subset,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc4_int_1,cc5_xreal_0,fc17_finseq_1,fc1_finseq_1,fc2_card_1,rc1_int_1,rc1_xreal_0,rc2_int_1,t2_real,t3_real,t3_subset,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k4_finseq_1,dt_k2_finseq_1,dt_k3_finseq_1,dt_k4_finseq_1,dt_k7_relat_1,dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,cc1_finseq_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc4_funct_1,rc1_finseq_1,rc1_funct_1,e1_3_1_1__graph_2,e1_3__graph_2,t21_finseq_1]),
    [interesting(0.5),file(graph_2,e2_3_1_1__graph_2),[file(graph_2,e2_3_1_1__graph_2)]]).

fof(t3_finseq_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( r2_hidden(A,k2_finseq_1(B))
          <=> ( r1_xreal_0(1,A)
              & r1_xreal_0(A,B) ) ) ) ) ),
    file(finseq_1,t3_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,t3_finseq_1)]).

fof(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,(
    r1_xreal_0(1,1) ),
    file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1),
    [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]).

fof(e6_3_1_1__graph_2,plain,(
    r2_hidden(c1_3_1_1__graph_2,k2_finseq_1(c3_3__graph_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3_1_1__graph_2,dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e1_3__graph_2,e4_3_1_1__graph_2,e5_3_1_1__graph_2])],[rc4_funct_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_card_1,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_nat_1,rc4_finset_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_finseq_1,rc1_finseq_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_real,t2_real,t2_subset,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k2_finseq_1,redefinition_k3_finseq_1,dt_k2_finseq_1,dt_k3_finseq_1,dt_c1_3__graph_2,dt_c1_3_1_1__graph_2,dt_c3_3__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,t1_subset,t7_boole,spc1_numerals,spc1_boole,e2_3_1_1__graph_2,e4_3_1_1__graph_2,e5_3_1_1__graph_2,t3_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1]),
    [interesting(0.5),file(graph_2,e6_3_1_1__graph_2),[file(graph_2,e6_3_1_1__graph_2)]]).

fof(t72_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,A)
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ),
    file(funct_1,t72_funct_1),
    [interesting(0.9),axiom,file(funct_1,t72_funct_1)]).

fof(e7_3_1_1__graph_2,plain,(
    k1_funct_1(c2_3__graph_2,c1_3_1_1__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_1__graph_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3_1_1__graph_2,dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e4_3_1_1__graph_2,e5_3_1_1__graph_2,e1_3__graph_2])],[cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,cc2_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_nat_1,rc4_finset_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_card_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_int_1,cc3_nat_1,fc1_finseq_1,rc1_finseq_1,rc2_funct_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k2_finseq_1,dt_k1_funct_1,dt_k2_finseq_1,dt_k7_relat_1,dt_c1_3__graph_2,dt_c1_3_1_1__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,fc4_funct_1,rc1_funct_1,t1_subset,t7_boole,e6_3_1_1__graph_2,e1_3__graph_2,t72_funct_1]),
    [interesting(0.5),file(graph_2,e7_3_1_1__graph_2),[file(graph_2,e7_3_1_1__graph_2)]]).

fof(i5_3_1_1__graph_2,theorem,(
    $true ),
    introduced(tautology,[file(graph_2,i5_3_1_1__graph_2)]),
    [interesting(0.5),trivial,file(graph_2,i5_3_1_1__graph_2)]).

fof(i4_3_1_1__graph_2,plain,(
    k1_funct_1(c2_3__graph_2,c1_3_1_1__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_1__graph_2) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3_1_1__graph_2,dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e4_3_1_1__graph_2,e5_3_1_1__graph_2,e1_3__graph_2])],[e7_3_1_1__graph_2,i5_3_1_1__graph_2]),
    [interesting(0.5),file(graph_2,i4_3_1_1__graph_2),[file(graph_2,i4_3_1_1__graph_2)]]).

fof(i3_3_1_1__graph_2,plain,
    ( ( r1_xreal_0(1,c1_3_1_1__graph_2)
      & r1_xreal_0(c1_3_1_1__graph_2,k3_finseq_1(c1_3__graph_2)) )
   => k1_funct_1(c2_3__graph_2,c1_3_1_1__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_1__graph_2) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3_1_1__graph_2,dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e1_3__graph_2]),discharge_asm(discharge,[e4_3_1_1__graph_2,e5_3_1_1__graph_2])],[e4_3_1_1__graph_2,e5_3_1_1__graph_2,i4_3_1_1__graph_2]),
    [interesting(0.5),file(graph_2,i3_3_1_1__graph_2),[file(graph_2,i3_3_1_1__graph_2)]]).

fof(i3_3_1_1_tmp__graph_2,plain,
    ( m2_subset_1(c1_3_1_1__graph_2,k1_numbers,k5_numbers)
   => ( ( r1_xreal_0(1,c1_3_1_1__graph_2)
        & r1_xreal_0(c1_3_1_1__graph_2,k3_finseq_1(c1_3__graph_2)) )
     => k1_funct_1(c2_3__graph_2,c1_3_1_1__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_1__graph_2) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e1_3__graph_2]),discharge_asm(discharge,[dt_c1_3_1_1__graph_2])],[dt_c1_3_1_1__graph_2,i3_3_1_1__graph_2]),
    [interesting(0.5),i2_3_1_1__graph_2]).

fof(i2_3_1_1__graph_2,plain,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ( r1_xreal_0(1,A)
          & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
       => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e1_3__graph_2])],[i3_3_1_1_tmp__graph_2,dh_c1_3_1_1__graph_2]),
    [interesting(0.5),file(graph_2,i2_3_1_1__graph_2),[file(graph_2,i2_3_1_1__graph_2)]]).

fof(i1_3_1_1__graph_2,plain,
    ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
    & ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(1,A)
            & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
         => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_1__graph_2,e1_3__graph_2])],[e3_3_1_1__graph_2,i2_3_1_1__graph_2]),
    [interesting(0.5),file(graph_2,i1_3_1_1__graph_2),[file(graph_2,i1_3_1_1__graph_2)]]).

fof(i1_3_1__graph_2,plain,
    ( r1_xreal_0(c3_3__graph_2,k3_finseq_1(c2_3__graph_2))
   => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(1,A)
              & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
           => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2]),discharge_asm(discharge,[e1_3_1_1__graph_2])],[e1_3_1_1__graph_2,i1_3_1_1__graph_2]),
    [interesting(0.65),file(graph_2,i1_3_1__graph_2),[file(graph_2,i1_3_1__graph_2)]]).

fof(e1_3_1_2__graph_2,assumption,(
    r1_xreal_0(k3_finseq_1(c2_3__graph_2),c3_3__graph_2) ),
    introduced(assumption,[file(graph_2,e1_3_1_2__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,e1_3_1_2__graph_2)]).

fof(t23_finseq_2,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( B = k7_relat_1(C,k2_finseq_1(A))
                  & r1_xreal_0(k3_finseq_1(C),A) )
               => C = B ) ) ) ) ),
    file(finseq_2,t23_finseq_2),
    [interesting(0.9),axiom,file(finseq_2,t23_finseq_2)]).

fof(e2_3_1_2__graph_2,plain,(
    r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3_1_2__graph_2,e1_3__graph_2])],[rc4_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc2_finseq_1,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_int_1,cc3_nat_1,cc7_xreal_0,fc1_finseq_1,fc1_ordinal2,fc2_card_1,rc1_finset_1,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_finseq_1,dt_k3_finseq_1,dt_k5_numbers,dt_k7_relat_1,dt_m2_subset_1,dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,cc1_finseq_1,fc4_funct_1,rc1_finseq_1,rc1_funct_1,e1_3_1_2__graph_2,e1_3__graph_2,t23_finseq_2]),
    [interesting(0.5),file(graph_2,e2_3_1_2__graph_2),[file(graph_2,e2_3_1_2__graph_2)]]).

fof(dh_c1_3_1_2__graph_2,definition,
    ( ( m2_subset_1(c1_3_1_2__graph_2,k1_numbers,k5_numbers)
     => ( ( r1_xreal_0(1,c1_3_1_2__graph_2)
          & r1_xreal_0(c1_3_1_2__graph_2,k3_finseq_1(c1_3__graph_2)) )
       => k1_funct_1(c2_3__graph_2,c1_3_1_2__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_2__graph_2) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(1,A)
            & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
         => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ),
    introduced(definition,[new_symbol(c1_3_1_2__graph_2),file(graph_2,c1_3_1_2__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,c1_3_1_2__graph_2)]).

fof(e3_3_1_2__graph_2,assumption,
    ( r1_xreal_0(1,c1_3_1_2__graph_2)
    & r1_xreal_0(c1_3_1_2__graph_2,k3_finseq_1(c1_3__graph_2)) ),
    introduced(assumption,[file(graph_2,e3_3_1_2__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,e3_3_1_2__graph_2)]).

fof(dt_c1_3_1_2__graph_2,assumption,(
    m2_subset_1(c1_3_1_2__graph_2,k1_numbers,k5_numbers) ),
    introduced(assumption,[file(graph_2,c1_3_1_2__graph_2)]),
    [interesting(0.5),axiom,file(graph_2,c1_3_1_2__graph_2)]).

fof(e4_3_1_2__graph_2,plain,(
    k1_funct_1(c2_3__graph_2,c1_3_1_2__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_2__graph_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__graph_2,dt_c1_3_1_2__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2,e1_3_1_2__graph_2])],[rc4_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc2_finseq_1,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_int_1,cc3_nat_1,cc7_xreal_0,fc1_finseq_1,fc1_ordinal2,fc2_card_1,rc1_finset_1,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_finseq_1,dt_k3_finseq_1,dt_k5_numbers,dt_k7_relat_1,dt_m2_subset_1,dt_c1_3__graph_2,dt_c1_3_1_2__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,cc1_finseq_1,fc4_funct_1,rc1_finseq_1,rc1_funct_1,e1_3__graph_2,e1_3_1_2__graph_2,t23_finseq_2]),
    [interesting(0.5),file(graph_2,e4_3_1_2__graph_2),[file(graph_2,e4_3_1_2__graph_2)]]).

fof(i5_3_1_2__graph_2,theorem,(
    $true ),
    introduced(tautology,[file(graph_2,i5_3_1_2__graph_2)]),
    [interesting(0.5),trivial,file(graph_2,i5_3_1_2__graph_2)]).

fof(i4_3_1_2__graph_2,plain,(
    k1_funct_1(c2_3__graph_2,c1_3_1_2__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_2__graph_2) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3__graph_2,dt_c1_3_1_2__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2,e1_3_1_2__graph_2])],[e4_3_1_2__graph_2,i5_3_1_2__graph_2]),
    [interesting(0.5),file(graph_2,i4_3_1_2__graph_2),[file(graph_2,i4_3_1_2__graph_2)]]).

fof(i3_3_1_2__graph_2,plain,
    ( ( r1_xreal_0(1,c1_3_1_2__graph_2)
      & r1_xreal_0(c1_3_1_2__graph_2,k3_finseq_1(c1_3__graph_2)) )
   => k1_funct_1(c2_3__graph_2,c1_3_1_2__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_2__graph_2) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2,dt_c1_3_1_2__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2,e1_3_1_2__graph_2]),discharge_asm(discharge,[e3_3_1_2__graph_2])],[e3_3_1_2__graph_2,i4_3_1_2__graph_2]),
    [interesting(0.5),file(graph_2,i3_3_1_2__graph_2),[file(graph_2,i3_3_1_2__graph_2)]]).

fof(i3_3_1_2_tmp__graph_2,plain,
    ( m2_subset_1(c1_3_1_2__graph_2,k1_numbers,k5_numbers)
   => ( ( r1_xreal_0(1,c1_3_1_2__graph_2)
        & r1_xreal_0(c1_3_1_2__graph_2,k3_finseq_1(c1_3__graph_2)) )
     => k1_funct_1(c2_3__graph_2,c1_3_1_2__graph_2) = k1_funct_1(c1_3__graph_2,c1_3_1_2__graph_2) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2,e1_3_1_2__graph_2]),discharge_asm(discharge,[dt_c1_3_1_2__graph_2])],[dt_c1_3_1_2__graph_2,i3_3_1_2__graph_2]),
    [interesting(0.5),i2_3_1_2__graph_2]).

fof(i2_3_1_2__graph_2,plain,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ( r1_xreal_0(1,A)
          & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
       => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2,e1_3_1_2__graph_2])],[i3_3_1_2_tmp__graph_2,dh_c1_3_1_2__graph_2]),
    [interesting(0.5),file(graph_2,i2_3_1_2__graph_2),[file(graph_2,i2_3_1_2__graph_2)]]).

fof(i1_3_1_2__graph_2,plain,
    ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
    & ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(1,A)
            & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
         => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2,e1_3_1_2__graph_2])],[e2_3_1_2__graph_2,i2_3_1_2__graph_2]),
    [interesting(0.5),file(graph_2,i1_3_1_2__graph_2),[file(graph_2,i1_3_1_2__graph_2)]]).

fof(i2_3_1__graph_2,plain,
    ( r1_xreal_0(k3_finseq_1(c2_3__graph_2),c3_3__graph_2)
   => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(1,A)
              & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
           => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2,e1_3__graph_2]),discharge_asm(discharge,[e1_3_1_2__graph_2])],[e1_3_1_2__graph_2,i1_3_1_2__graph_2]),
    [interesting(0.65),file(graph_2,i2_3_1__graph_2),[file(graph_2,i2_3_1__graph_2)]]).

fof(e1_3_1__graph_2,plain,
    ( r1_xreal_0(c3_3__graph_2,k3_finseq_1(c2_3__graph_2))
    | r1_xreal_0(k3_finseq_1(c2_3__graph_2),c3_3__graph_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_3__graph_2,dt_c3_3__graph_2])],[rc4_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc2_finseq_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_card_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc7_xreal_0,fc1_ordinal2,fc2_card_1,rc1_card_1,rc1_finset_1,rc1_xreal_0,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_finseq_1,cc2_xreal_0,rc1_finseq_1,rc1_funct_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k3_finseq_1,dt_k3_finseq_1,dt_c2_3__graph_2,dt_c3_3__graph_2]),
    [interesting(0.65),file(graph_2,e1_3_1__graph_2),[file(graph_2,e1_3_1__graph_2)]]).

fof(i4_3__graph_2,plain,
    ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
    & ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(1,A)
            & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
         => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ),
    inference(percases,[status(thm),assumptions([dt_c1_3__graph_2,e1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2])],[i1_3_1__graph_2,i2_3_1__graph_2,e1_3_1__graph_2]),
    [interesting(0.8),file(graph_2,i4_3__graph_2),[file(graph_2,i4_3__graph_2)]]).

fof(i3_3__graph_2,plain,
    ( c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(c3_3__graph_2))
   => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(1,A)
              & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
           => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2,dt_c3_3__graph_2]),discharge_asm(discharge,[e1_3__graph_2])],[e1_3__graph_2,i4_3__graph_2]),
    [interesting(0.8),file(graph_2,i3_3__graph_2),[file(graph_2,i3_3__graph_2)]]).

fof(i3_3_tmp__graph_2,plain,
    ( m2_subset_1(c3_3__graph_2,k1_numbers,k5_numbers)
   => ( c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(c3_3__graph_2))
     => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
        & ! [A] : 
            ( m2_subset_1(A,k1_numbers,k5_numbers)
           => ( ( r1_xreal_0(1,A)
                & r1_xreal_0(A,k3_finseq_1(c1_3__graph_2)) )
             => k1_funct_1(c2_3__graph_2,A) = k1_funct_1(c1_3__graph_2,A) ) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2]),discharge_asm(discharge,[dt_c3_3__graph_2])],[dt_c3_3__graph_2,i3_3__graph_2]),
    [interesting(0.8),i2_3__graph_2]).

fof(i2_3__graph_2,plain,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(A))
       => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
          & ! [B] : 
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(1,B)
                  & r1_xreal_0(B,k3_finseq_1(c1_3__graph_2)) )
               => k1_funct_1(c2_3__graph_2,B) = k1_funct_1(c1_3__graph_2,B) ) ) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__graph_2,dt_c2_3__graph_2])],[i3_3_tmp__graph_2,dh_c3_3__graph_2]),
    [interesting(0.8),file(graph_2,i2_3__graph_2),[file(graph_2,i2_3__graph_2)]]).

fof(i2_3_tmp__graph_2,plain,
    ( ( v1_relat_1(c2_3__graph_2)
      & v1_funct_1(c2_3__graph_2)
      & v1_finseq_1(c2_3__graph_2) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( c1_3__graph_2 = k7_relat_1(c2_3__graph_2,k2_finseq_1(A))
         => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(c2_3__graph_2))
            & ! [B] : 
                ( m2_subset_1(B,k1_numbers,k5_numbers)
               => ( ( r1_xreal_0(1,B)
                    & r1_xreal_0(B,k3_finseq_1(c1_3__graph_2)) )
                 => k1_funct_1(c2_3__graph_2,B) = k1_funct_1(c1_3__graph_2,B) ) ) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__graph_2]),discharge_asm(discharge,[dt_c2_3__graph_2])],[dt_c2_3__graph_2,i2_3__graph_2]),
    [interesting(0.8),i1_3__graph_2]).

fof(i1_3__graph_2,plain,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( c1_3__graph_2 = k7_relat_1(A,k2_finseq_1(B))
           => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(A))
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(1,C)
                      & r1_xreal_0(C,k3_finseq_1(c1_3__graph_2)) )
                   => k1_funct_1(A,C) = k1_funct_1(c1_3__graph_2,C) ) ) ) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__graph_2])],[i2_3_tmp__graph_2,dh_c2_3__graph_2]),
    [interesting(0.8),file(graph_2,i1_3__graph_2),[file(graph_2,i1_3__graph_2)]]).

fof(i1_3_tmp__graph_2,plain,
    ( ( v1_relat_1(c1_3__graph_2)
      & v1_funct_1(c1_3__graph_2)
      & v1_finseq_1(c1_3__graph_2) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A) )
       => ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( c1_3__graph_2 = k7_relat_1(A,k2_finseq_1(B))
             => ( r1_xreal_0(k3_finseq_1(c1_3__graph_2),k3_finseq_1(A))
                & ! [C] : 
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                   => ( ( r1_xreal_0(1,C)
                        & r1_xreal_0(C,k3_finseq_1(c1_3__graph_2)) )
                     => k1_funct_1(A,C) = k1_funct_1(c1_3__graph_2,C) ) ) ) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_3__graph_2])],[dt_c1_3__graph_2,i1_3__graph_2]),
    [interesting(1),t2_graph_2]).

fof(t2_graph_2,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] : 
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( A = k7_relat_1(B,k2_finseq_1(C))
               => ( r1_xreal_0(k3_finseq_1(A),k3_finseq_1(B))
                  & ! [D] : 
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(1,D)
                          & r1_xreal_0(D,k3_finseq_1(A)) )
                       => k1_funct_1(B,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_3_tmp__graph_2,dh_c1_3__graph_2]),
    [interesting(1),file(graph_2,t2_graph_2),[file(graph_2,t2_graph_2)]]).