SET007 Axioms: SET007+152.ax


%------------------------------------------------------------------------------
% File     : SET007+152 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Koenig's Lemma
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : trees_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  104 (   6 unit)
%            Number of atoms       :  665 (  60 equality)
%            Maximal formula depth :   22 (   8 average)
%            Number of connectives :  675 ( 114 ~  ;   2  |; 317  &)
%                                         (  21 <=>; 221 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   1 propositional; 0-3 arity)
%            Number of functors    :   70 (  24 constant; 0-3 arity)
%            Number of variables   :  269 (   0 singleton; 233 !;  36 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_trees_2,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_trees_1(A)
      & v1_trees_2(A) ) )).

fof(fc1_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_trees_2(A)
        & m1_subset_1(B,A) )
     => v1_finset_1(k1_trees_2(A,B)) ) )).

fof(rc2_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ? [B] :
          ( m1_trees_2(B,A)
          & ~ v1_xboole_0(B) ) ) )).

fof(rc3_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ? [B] :
          ( m1_trees_2(B,A)
          & v2_trees_2(B,A) ) ) )).

fof(cc1_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_2(B,A)
         => ( v2_trees_2(B,A)
           => ~ v1_xboole_0(B) ) ) ) )).

fof(rc4_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ? [B] :
          ( m1_trees_2(B,A)
          & v1_finset_1(B) ) ) )).

fof(rc5_trees_2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v3_trees_2(A) ) )).

fof(fc2_trees_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( ~ v1_xboole_0(k1_relat_1(A))
        & v1_trees_1(k1_relat_1(A)) ) ) )).

fof(rc6_trees_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m3_trees_2(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v3_trees_2(B) ) ) )).

fof(fc3_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( v1_relat_1(k2_funcop_1(A,B))
        & v1_funct_1(k2_funcop_1(A,B))
        & v3_trees_2(k2_funcop_1(A,B)) ) ) )).

fof(fc4_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A)
        & m1_subset_1(B,A) )
     => v1_finset_1(k1_trees_2(A,B)) ) )).

fof(rc7_trees_2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v3_trees_2(A) ) )).

fof(rc8_trees_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m3_trees_2(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_finset_1(B)
          & v3_trees_2(B) ) ) )).

fof(fc5_trees_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_finset_1(A) )
     => v1_finset_1(k1_relat_1(A)) ) )).

fof(rc9_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ? [C] :
          ( m1_relset_1(C,A,B)
          & v1_relat_1(C)
          & ~ v1_xboole_0(C) ) ) )).

fof(t1_trees_2,axiom,(
    ! [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] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( r1_tarski(A,C)
                  & r1_tarski(B,C) )
               => r3_xboole_0(A,B) ) ) ) ) )).

fof(t2_trees_2,axiom,(
    ! [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] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( r2_xboole_0(A,C)
                  & r1_tarski(B,C) )
               => r3_xboole_0(A,B) ) ) ) ) )).

fof(t3_trees_2,axiom,(
    $true )).

fof(t4_trees_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ~ ( k3_finseq_1(B) = k1_nat_1(A,np__1)
              & ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v1_finseq_1(C) )
                 => ! [D] :
                      ~ ( B = k7_finseq_1(C,k9_finseq_1(D))
                        & k3_finseq_1(C) = A ) ) ) ) ) )).

fof(t5_trees_2,axiom,(
    $true )).

fof(t6_trees_2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k1_trees_1(k7_finseq_1(B,k9_finseq_1(A))) = k2_xboole_0(k1_trees_1(B),k1_tarski(B)) ) )).

fof(t7_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_trees_1(B) )
         => ( ! [C] :
                ( m2_finseq_1(C,k5_numbers)
               => ( r2_hidden(C,A)
                <=> r2_hidden(C,B) ) )
           => A = B ) ) ) )).

fof(d1_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_trees_1(B) )
         => ( A = B
          <=> ! [C] :
                ( m2_finseq_1(C,k5_numbers)
               => ( r2_hidden(C,A)
                <=> r2_hidden(C,B) ) ) ) ) ) )).

fof(t8_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ( r2_hidden(B,A)
           => A = k5_trees_1(A,B,k4_trees_1(A,B)) ) ) ) )).

fof(t9_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_trees_1(B) )
         => ! [C] :
              ( m2_finseq_1(C,k5_numbers)
             => ! [D] :
                  ( m2_finseq_1(D,k5_numbers)
                 => ( ( r2_hidden(C,A)
                      & r2_hidden(D,A) )
                   => ( r1_tarski(C,D)
                      | r2_hidden(D,k5_trees_1(A,C,B)) ) ) ) ) ) ) )).

fof(t10_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_trees_1(B) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v1_trees_1(C) )
             => ! [D] :
                  ( m2_finseq_1(D,k5_numbers)
                 => ! [E] :
                      ( m2_finseq_1(E,k5_numbers)
                     => ( ( r2_hidden(D,A)
                          & r2_hidden(E,A) )
                       => ( r3_xboole_0(D,E)
                          | k5_trees_1(k5_trees_1(A,D,B),E,C) = k5_trees_1(k5_trees_1(A,E,C),D,B) ) ) ) ) ) ) ) )).

fof(d2_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( v1_trees_2(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ! [C] :
                ( m1_trees_1(C,A)
               => ~ r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,B)),A) ) ) ) ) )).

fof(d3_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( m1_trees_2(B,A)
          <=> ! [C] :
                ( m2_finseq_1(C,k5_numbers)
               => ! [D] :
                    ( m2_finseq_1(D,k5_numbers)
                   => ( ( r2_hidden(C,B)
                        & r2_hidden(D,B) )
                     => r3_xboole_0(C,D) ) ) ) ) ) ) )).

fof(t11_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_trees_2(B,A)
         => m4_trees_1(B,A) ) ) )).

fof(t12_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => m4_trees_1(k1_trees_2(A,B),A) ) ) )).

fof(t13_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m4_trees_1(B,A)
         => ! [C] :
              ( m1_trees_2(C,A)
             => ? [D] :
                  ( m1_trees_1(D,A)
                  & r1_tarski(k3_xboole_0(B,C),k1_tarski(D)) ) ) ) ) )).

fof(t14_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)),k1_trees_2(A,B))
              <=> r2_hidden(k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C)),A) ) ) ) ) )).

fof(t15_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => ( B = k1_xboole_0
           => k2_trees_2(A,np__1) = k1_trees_2(A,B) ) ) ) )).

fof(t18_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_trees_2(B,A)
         => ? [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
              & B = k2_trees_2(A,C) ) ) ) )).

fof(t19_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ~ ( v1_trees_2(A)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ? [C] :
                  ( m1_trees_1(C,A)
                  & ! [D] :
                      ( v1_finset_1(D)
                     => ~ ( D = k1_trees_2(A,C)
                          & r1_xreal_0(k4_card_1(D),B) ) ) ) ) ) ) )).

fof(t20_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => ( v1_trees_2(A)
           => v1_finset_1(k1_trees_2(A,B)) ) ) ) )).

fof(t21_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => m1_trees_2(k1_xboole_0,A) ) )).

fof(t22_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => m1_trees_2(k1_tarski(k1_xboole_0),A) ) )).

fof(d7_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_2(B,A)
         => ( v2_trees_2(B,A)
          <=> ( ! [C] :
                  ( m2_finseq_1(C,k5_numbers)
                 => ( r2_hidden(C,B)
                   => r1_tarski(k1_trees_1(C),B) ) )
              & ! [C] :
                  ( m2_finseq_1(C,k5_numbers)
                 => ~ ( r2_hidden(C,A)
                      & ! [D] :
                          ( m2_finseq_1(D,k5_numbers)
                         => ( r2_hidden(D,B)
                           => r2_xboole_0(D,C) ) ) ) ) ) ) ) ) )).

fof(t23_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(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) )
             => ! [D] :
                  ( m1_trees_2(D,A)
                 => ~ ( r2_hidden(B,D)
                      & r2_hidden(C,D)
                      & ~ r2_hidden(B,k1_trees_1(C))
                      & ~ r1_tarski(C,B) ) ) ) ) ) )).

fof(t24_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(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) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D) )
                 => ! [E] :
                      ( m1_trees_2(E,A)
                     => ~ ( r2_hidden(B,E)
                          & r2_hidden(C,E)
                          & r1_tarski(D,C)
                          & ~ r2_hidden(B,k1_trees_1(D))
                          & ~ r1_tarski(D,B) ) ) ) ) ) ) )).

fof(t25_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_trees_2(C,A) )
             => ~ ( ~ r1_xreal_0(k4_card_1(C),B)
                  & ! [D] :
                      ( m2_finseq_1(D,k5_numbers)
                     => ~ ( r2_hidden(D,C)
                          & r1_xreal_0(B,k3_finseq_1(D)) ) ) ) ) ) ) )).

fof(t27_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k5_numbers)
             => ! [D] :
                  ( ( v2_trees_2(D,A)
                    & m1_trees_2(D,A) )
                 => ( ( r1_tarski(B,C)
                      & r2_hidden(C,D) )
                   => r2_hidden(B,D) ) ) ) ) ) )).

fof(t28_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( ( v2_trees_2(B,A)
            & m1_trees_2(B,A) )
         => r2_hidden(k1_xboole_0,B) ) ) )).

fof(t29_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k5_numbers)
             => ! [D] :
                  ( m1_trees_2(D,A)
                 => ( ( r2_hidden(B,D)
                      & r2_hidden(C,D)
                      & r1_xreal_0(k3_finseq_1(B),k3_finseq_1(C)) )
                   => r1_tarski(B,C) ) ) ) ) ) )).

fof(t30_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_2(B,A)
         => ? [C] :
              ( v2_trees_2(C,A)
              & m1_trees_2(C,A)
              & r1_tarski(B,C) ) ) ) )).

fof(t31_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ~ ( ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ? [C] :
                  ( v1_finset_1(C)
                  & m1_trees_2(C,A)
                  & k4_card_1(C) = B ) )
          & ! [B] :
              ( m1_trees_1(B,A)
             => v1_finset_1(k1_trees_2(A,B)) )
          & ! [B] :
              ( m1_trees_2(B,A)
             => v1_finset_1(B) ) ) ) )).

fof(t32_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_trees_2(A) )
     => ~ ( ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ? [C] :
                  ( v1_finset_1(C)
                  & m1_trees_2(C,A)
                  & k4_card_1(C) = B ) )
          & ! [B] :
              ( m1_trees_2(B,A)
             => v1_finset_1(B) ) ) ) )).

fof(d8_trees_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v3_trees_2(A)
      <=> ( ~ v1_xboole_0(k1_relat_1(A))
          & v1_trees_1(k1_relat_1(A)) ) ) ) )).

fof(d9_trees_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( m3_trees_2(B,A)
      <=> r1_tarski(k2_relat_1(B),A) ) ) )).

fof(t33_trees_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v3_trees_2(B) )
         => ( ( k1_relat_1(A) = k1_relat_1(B)
              & ! [C] :
                  ( m2_finseq_1(C,k5_numbers)
                 => ( r2_hidden(C,k1_relat_1(A))
                   => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
           => A = B ) ) ) )).

fof(d10_trees_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => k4_trees_2(A) = k9_relat_1(A,k3_trees_1(k1_relat_1(A))) ) )).

fof(d11_trees_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v3_trees_2(C) )
             => ( C = k5_trees_2(A,B)
              <=> ( k1_relat_1(C) = k4_trees_1(k1_relat_1(A),B)
                  & ! [D] :
                      ( m2_finseq_1(D,k5_numbers)
                     => ( r2_hidden(D,k4_trees_1(k1_relat_1(A),B))
                       => k1_funct_1(C,D) = k1_funct_1(A,k8_finseq_1(k5_numbers,B,D)) ) ) ) ) ) ) ) )).

fof(t34_trees_2,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v3_trees_2(B) )
         => ( r2_hidden(A,k1_relat_1(B))
           => r1_tarski(k2_relat_1(k5_trees_2(B,A)),k2_relat_1(B)) ) ) ) )).

fof(d12_trees_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v3_trees_2(C) )
             => ( r2_hidden(B,k1_relat_1(A))
               => ! [D] :
                    ( ( v1_relat_1(D)
                      & v1_funct_1(D)
                      & v3_trees_2(D) )
                   => ( D = k8_trees_2(A,B,C)
                    <=> ( k1_relat_1(D) = k5_trees_1(k1_relat_1(A),B,k1_relat_1(C))
                        & ! [E] :
                            ( m2_finseq_1(E,k5_numbers)
                           => ~ ( r2_hidden(E,k5_trees_1(k1_relat_1(A),B,k1_relat_1(C)))
                                & ~ ( ~ r1_tarski(B,E)
                                    & k1_funct_1(D,E) = k1_funct_1(A,E) )
                                & ! [F] :
                                    ( m2_finseq_1(F,k5_numbers)
                                   => ~ ( r2_hidden(F,k1_relat_1(C))
                                        & E = k8_finseq_1(k5_numbers,B,F)
                                        & k1_funct_1(D,E) = k1_funct_1(C,F) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t35_trees_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ! [B] :
            ( r2_hidden(B,A)
           => ( ~ v1_xboole_0(B)
              & v1_trees_1(B) ) )
       => ( ~ v1_xboole_0(k3_tarski(A))
          & v1_trees_1(k3_tarski(A)) ) ) ) )).

fof(t36_trees_2,axiom,(
    ! [A] :
      ( ( ! [B] :
            ( r2_hidden(B,A)
           => ( v1_relat_1(B)
              & v1_funct_1(B) ) )
        & v6_ordinal1(A) )
     => ( v1_relat_1(k3_tarski(A))
        & v1_funct_1(k3_tarski(A)) ) ) )).

fof(t37_trees_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ( ! [B] :
              ( r2_hidden(B,A)
             => ( v1_relat_1(B)
                & v1_funct_1(B)
                & v3_trees_2(B) ) )
          & v6_ordinal1(A) )
       => ( v1_relat_1(k3_tarski(A))
          & v1_funct_1(k3_tarski(A))
          & v3_trees_2(k3_tarski(A)) ) ) ) )).

fof(t38_trees_2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( ( ! [C] :
                  ( r2_hidden(C,A)
                 => ( v1_funct_1(C)
                    & v3_trees_2(C)
                    & m3_trees_2(C,B) ) )
              & v6_ordinal1(A) )
           => ( v1_funct_1(k3_tarski(A))
              & v3_trees_2(k3_tarski(A))
              & m3_trees_2(k3_tarski(A),B) ) ) ) ) )).

fof(d13_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => k10_trees_2(A,B) = k4_card_1(k1_trees_2(A,B)) ) ) )).

fof(s1_trees_2,axiom,
    ( ( p1_s1_trees_2(k1_xboole_0)
      & ! [A] :
          ( m1_trees_1(A,f1_s1_trees_2)
         => ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( ( p1_s1_trees_2(A)
                  & r2_hidden(k8_finseq_1(k5_numbers,A,k12_finseq_1(k5_numbers,B)),f1_s1_trees_2) )
               => p1_s1_trees_2(k8_finseq_1(k5_numbers,A,k12_finseq_1(k5_numbers,B))) ) ) ) )
   => ! [A] :
        ( m1_trees_1(A,f1_s1_trees_2)
       => p1_s1_trees_2(A) ) )).

fof(s4_trees_2,axiom,
    ( ! [A] :
        ~ ( r2_hidden(A,f1_s4_trees_2)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ p1_s4_trees_2(A,B) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s4_trees_2
        & ! [B] :
            ~ ( r2_hidden(B,f1_s4_trees_2)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( k1_funct_1(A,B) = C
                      & p1_s4_trees_2(B,C)
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ( p1_s4_trees_2(B,D)
                           => r1_xreal_0(C,D) ) ) ) ) ) ) )).

fof(s5_trees_2,axiom,
    ( ( r2_hidden(f2_s5_trees_2,f1_s5_trees_2)
      & p1_s5_trees_2(f2_s5_trees_2)
      & ! [A] :
          ~ ( r2_hidden(A,f1_s5_trees_2)
            & p1_s5_trees_2(A)
            & ! [B] :
                ~ ( r2_hidden(B,f1_s5_trees_2)
                  & p2_s5_trees_2(A,B)
                  & p1_s5_trees_2(B) ) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = k5_numbers
        & r1_tarski(k2_relat_1(A),f1_s5_trees_2)
        & k1_funct_1(A,np__0) = f2_s5_trees_2
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( p2_s5_trees_2(k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(B,np__1)))
              & p1_s5_trees_2(k1_funct_1(A,B)) ) ) ) )).

fof(s6_trees_2,axiom,
    ( ! [A] :
        ( m2_finseq_1(A,k5_numbers)
       => ~ ( r2_hidden(A,f1_s6_trees_2)
            & ! [B] : ~ p1_s6_trees_2(A,B) ) )
   => ? [A] :
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & k1_relat_1(A) = f1_s6_trees_2
        & ! [B] :
            ( m2_finseq_1(B,k5_numbers)
           => ( r2_hidden(B,f1_s6_trees_2)
             => p1_s6_trees_2(B,k1_funct_1(A,B)) ) ) ) )).

fof(s7_trees_2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v3_trees_2(A)
      & k1_relat_1(A) = f1_s7_trees_2
      & ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ( r2_hidden(B,f1_s7_trees_2)
           => k1_funct_1(A,B) = f2_s7_trees_2(B) ) ) ) )).

fof(dt_m1_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_2(B,A)
         => m1_subset_1(B,k1_zfmisc_1(A)) ) ) )).

fof(existence_m1_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ? [B] : m1_trees_2(B,A) ) )).

fof(dt_m2_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_trees_2(B,A)
         => m1_subset_1(B,k1_zfmisc_1(A)) ) ) )).

fof(existence_m2_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ? [B] : m2_trees_2(B,A) ) )).

fof(dt_m3_trees_2,axiom,(
    ! [A,B] :
      ( m3_trees_2(B,A)
     => v1_relat_1(B) ) )).

fof(existence_m3_trees_2,axiom,(
    ! [A] :
    ? [B] : m3_trees_2(B,A) )).

fof(dt_k1_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k1_trees_2(A,B),k1_zfmisc_1(A)) ) )).

fof(dt_k2_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & m1_subset_1(B,k5_numbers) )
     => m2_trees_2(k2_trees_2(A,B),A) ) )).

fof(dt_k3_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A)
        & m1_subset_1(C,k1_relat_1(B)) )
     => m1_subset_1(k3_trees_2(A,B,C),A) ) )).

fof(redefinition_k3_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A)
        & m1_subset_1(C,k1_relat_1(B)) )
     => k3_trees_2(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k4_trees_2,axiom,(
    $true )).

fof(dt_k5_trees_2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & m1_finseq_1(B,k5_numbers) )
     => ( v1_relat_1(k5_trees_2(A,B))
        & v1_funct_1(k5_trees_2(A,B))
        & v3_trees_2(k5_trees_2(A,B)) ) ) )).

fof(dt_k6_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A) )
     => m1_subset_1(k6_trees_2(A,B),k1_zfmisc_1(A)) ) )).

fof(redefinition_k6_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A) )
     => k6_trees_2(A,B) = k4_trees_2(B) ) )).

fof(dt_k7_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A)
        & m1_subset_1(C,k1_relat_1(B)) )
     => ( v1_funct_1(k7_trees_2(A,B,C))
        & v3_trees_2(k7_trees_2(A,B,C))
        & m3_trees_2(k7_trees_2(A,B,C),A) ) ) )).

fof(redefinition_k7_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A)
        & m1_subset_1(C,k1_relat_1(B)) )
     => k7_trees_2(A,B,C) = k5_trees_2(B,C) ) )).

fof(dt_k8_trees_2,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & m1_finseq_1(B,k5_numbers)
        & v1_relat_1(C)
        & v1_funct_1(C)
        & v3_trees_2(C) )
     => ( v1_relat_1(k8_trees_2(A,B,C))
        & v1_funct_1(k8_trees_2(A,B,C))
        & v3_trees_2(k8_trees_2(A,B,C)) ) ) )).

fof(dt_k9_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_trees_1(B)
        & m1_subset_1(C,A) )
     => ( v1_funct_1(k9_trees_2(A,B,C))
        & v3_trees_2(k9_trees_2(A,B,C))
        & m3_trees_2(k9_trees_2(A,B,C),A) ) ) )).

fof(redefinition_k9_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_trees_1(B)
        & m1_subset_1(C,A) )
     => k9_trees_2(A,B,C) = k2_funcop_1(B,C) ) )).

fof(dt_k10_trees_2,axiom,(
    $true )).

fof(d4_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( m2_trees_2(B,A)
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & B = a_2_0_trees_2(A,C) ) ) ) ) )).

fof(d5_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => k1_trees_2(A,B) = a_2_1_trees_2(A,B) ) ) )).

fof(d6_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_trees_2(A,B) = a_2_0_trees_2(A,B) ) ) )).

fof(t16_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => A = k3_tarski(a_1_0_trees_2(A)) ) )).

fof(t17_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => A = k3_tarski(a_1_1_trees_2(A)) ) )).

fof(t26_trees_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_2(B,A)
         => m1_trees_2(a_2_2_trees_2(A,B),A) ) ) )).

fof(s2_trees_2,axiom,(
    r1_tarski(k1_card_1(a_0_0_trees_2),k1_card_1(f2_s2_trees_2)) )).

fof(s3_trees_2,axiom,
    ( f3_s3_trees_2 = a_0_1_trees_2
   => r1_xreal_0(k4_card_1(f3_s3_trees_2),k4_card_1(f2_s3_trees_2)) )).

fof(s8_trees_2,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s8_trees_2)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ( r1_xreal_0(B,C)
                    & r2_hidden(C,f3_s8_trees_2(A)) )
                 => r2_hidden(B,f3_s8_trees_2(A)) ) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v3_trees_2(A)
        & m3_trees_2(A,f1_s8_trees_2)
        & k1_funct_1(A,k1_xboole_0) = f2_s8_trees_2
        & ! [B] :
            ( m1_trees_1(B,k1_relat_1(A))
           => ( k1_trees_2(k1_relat_1(A),B) = a_2_3_trees_2(A,B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,f3_s8_trees_2(k3_trees_2(f1_s8_trees_2,A,B)))
                   => k1_funct_1(A,k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C))) = k1_funct_1(f4_s8_trees_2,k4_tarski(k3_trees_2(f1_s8_trees_2,A,B),C)) ) ) ) ) ) )).

fof(s9_trees_2,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v3_trees_2(A)
      & m3_trees_2(A,f1_s9_trees_2)
      & k1_funct_1(A,k1_xboole_0) = f2_s9_trees_2
      & ! [B] :
          ( m1_trees_1(B,k1_relat_1(A))
         => ( k1_trees_2(k1_relat_1(A),B) = a_2_4_trees_2(A,B)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( ~ r1_xreal_0(f3_s9_trees_2(k3_trees_2(f1_s9_trees_2,A,B)),C)
                 => k1_funct_1(A,k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,C))) = k1_funct_1(f4_s9_trees_2,k4_tarski(k3_trees_2(f1_s9_trees_2,A,B),C)) ) ) ) ) ) )).

fof(fraenkel_a_2_0_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_trees_1(B)
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_0_trees_2(B,C))
      <=> ? [D] :
            ( m1_trees_1(D,B)
            & A = D
            & k3_finseq_1(D) = C ) ) ) )).

fof(fraenkel_a_2_1_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_trees_1(B)
        & m1_trees_1(C,B) )
     => ( r2_hidden(A,a_2_1_trees_2(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D))
            & r2_hidden(k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D)),B) ) ) ) )).

fof(fraenkel_a_1_0_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_trees_1(B) )
     => ( r2_hidden(A,a_1_0_trees_2(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k2_trees_2(B,C) ) ) ) )).

fof(fraenkel_a_1_1_trees_2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_finset_1(B)
        & v1_trees_1(B) )
     => ( r2_hidden(A,a_1_1_trees_2(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k2_trees_2(B,C)
            & r1_xreal_0(C,k6_trees_1(B)) ) ) ) )).

fof(fraenkel_a_2_2_trees_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_trees_1(B)
        & m1_trees_2(C,B) )
     => ( r2_hidden(A,a_2_2_trees_2(B,C))
      <=> ? [D] :
            ( m1_trees_1(D,B)
            & A = D
            & ? [E] :
                ( m2_finseq_1(E,k5_numbers)
                & r2_hidden(E,C)
                & r1_tarski(D,E) ) ) ) ) )).

fof(fraenkel_a_0_0_trees_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_trees_2)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s2_trees_2)
          & A = f3_s2_trees_2(B)
          & r2_hidden(B,f2_s2_trees_2) ) ) )).

fof(fraenkel_a_0_1_trees_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_trees_2)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s3_trees_2)
          & A = f4_s3_trees_2(B)
          & r2_hidden(B,f2_s3_trees_2) ) ) )).

fof(fraenkel_a_2_3_trees_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,f1_s8_trees_2)
        & m1_trees_1(C,k1_relat_1(B)) )
     => ( r2_hidden(A,a_2_3_trees_2(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D))
            & r2_hidden(D,f3_s8_trees_2(k3_trees_2(f1_s8_trees_2,B,C))) ) ) ) )).

fof(fraenkel_a_2_4_trees_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,f1_s9_trees_2)
        & m1_trees_1(C,k1_relat_1(B)) )
     => ( r2_hidden(A,a_2_4_trees_2(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k8_finseq_1(k5_numbers,C,k12_finseq_1(k5_numbers,D))
            & ~ r1_xreal_0(f3_s9_trees_2(k3_trees_2(f1_s9_trees_2,B,C)),D) ) ) ) )).
%------------------------------------------------------------------------------