SET007 Axioms: SET007+183.ax


%------------------------------------------------------------------------------
% File     : SET007+183 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Subtrees
% 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_9 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   95 (   3 unit)
%            Number of atoms       :  589 (  59 equality)
%            Maximal formula depth :   14 (   7 average)
%            Number of connectives :  575 (  81 ~  ;   1  |; 319  &)
%                                         (  28 <=>; 146 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-4 arity)
%            Number of functors    :   55 (   6 constant; 0-3 arity)
%            Number of variables   :  207 (   0 singleton; 181 !;  26 ?)
%            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(cc1_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_trees_2(A) ) ) )).

fof(cc2_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v1_trees_9(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) ) ) )).

fof(rc1_trees_9,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v3_trees_2(A)
      & v1_trees_9(A) ) )).

fof(rc2_trees_9,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v3_trees_2(A)
      & ~ v1_trees_9(A) ) )).

fof(fc1_trees_9,axiom,(
    ! [A] :
      ( v1_relat_1(k1_trees_4(A))
      & v1_funct_1(k1_trees_4(A))
      & v1_finset_1(k1_trees_4(A))
      & v3_trees_2(k1_trees_4(A))
      & v1_trees_9(k1_trees_4(A)) ) )).

fof(cc3_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_trees_2(A) )
     => ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v2_trees_9(A) ) ) )).

fof(rc3_trees_9,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A)
      & v1_trees_1(A)
      & v1_trees_2(A)
      & v2_trees_9(A) ) )).

fof(cc4_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v3_trees_9(A) ) ) )).

fof(cc5_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v3_trees_9(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v4_trees_9(A) ) ) )).

fof(rc4_trees_9,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v3_trees_2(A)
      & v3_trees_9(A)
      & v4_trees_9(A) ) )).

fof(fc2_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v3_trees_9(A) )
     => ( ~ v1_xboole_0(k1_relat_1(A))
        & v1_trees_1(k1_relat_1(A))
        & v1_trees_2(k1_relat_1(A))
        & v2_trees_9(k1_relat_1(A)) ) ) )).

fof(fc3_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v4_trees_9(A) )
     => ( ~ v1_xboole_0(k1_relat_1(A))
        & v1_trees_1(k1_relat_1(A))
        & v2_trees_9(k1_relat_1(A)) ) ) )).

fof(fc4_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v2_trees_9(A)
        & m1_subset_1(B,A) )
     => v1_finset_1(k1_trees_2(A,B)) ) )).

fof(rc5_trees_9,axiom,(
    ! [A] :
    ? [B] :
      ( m1_finseq_1(B,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v2_funct_1(B)
      & v1_xboole_0(B)
      & v1_finset_1(B)
      & v1_finseq_1(B) ) )).

fof(fc5_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A)
        & m1_subset_1(B,k1_relat_1(A)) )
     => ( v1_relat_1(k5_trees_2(A,B))
        & v1_funct_1(k5_trees_2(A,B))
        & v1_finset_1(k5_trees_2(A,B))
        & v3_trees_2(k5_trees_2(A,B))
        & v3_trees_9(k5_trees_2(A,B))
        & v4_trees_9(k5_trees_2(A,B)) ) ) )).

fof(cc6_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_trees_3(A))) )
     => ! [C] :
          ( m1_subset_1(C,B)
         => v1_finset_1(C) ) ) )).

fof(fc6_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( ~ v1_xboole_0(k3_trees_9(A))
        & v3_trees_3(k3_trees_9(A)) ) ) )).

fof(cc7_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m1_subset_1(B,k3_trees_9(A))
         => v1_finset_1(B) ) ) )).

fof(fc7_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( v1_relat_1(k6_trees_9(A))
        & ~ v1_xboole_0(k6_trees_9(A)) ) ) )).

fof(fc8_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => ( ~ v1_xboole_0(k9_trees_9(A))
        & v3_trees_3(k9_trees_9(A)) ) ) )).

fof(fc9_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( ~ v1_xboole_0(k1_tarski(A))
        & v1_finset_1(k1_tarski(A))
        & v3_trees_3(k1_tarski(A)) ) ) )).

fof(rc6_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ? [B] :
          ( m1_subset_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_xboole_0(B)
          & v1_finset_1(B)
          & v1_finseq_1(B) ) ) )).

fof(t1_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => k5_trees_2(A,k6_finseq_1(k5_numbers)) = A ) )).

fof(t2_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k5_numbers)
             => ( r2_hidden(k8_finseq_1(k5_numbers,B,C),A)
               => k4_trees_1(A,k8_finseq_1(k5_numbers,B,C)) = k4_trees_1(k4_trees_1(A,B),C) ) ) ) ) )).

fof(t3_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k5_numbers)
             => ( r2_hidden(k8_finseq_1(k5_numbers,B,C),k1_relat_1(A))
               => k5_trees_2(A,k8_finseq_1(k5_numbers,B,C)) = k5_trees_2(k5_trees_2(A,B),C) ) ) ) ) )).

fof(d1_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( v1_trees_9(A)
      <=> k1_relat_1(A) = k2_trees_1(np__0) ) ) )).

fof(t4_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( v1_trees_9(A)
      <=> r2_hidden(k1_xboole_0,k3_trees_1(k1_relat_1(A))) ) ) )).

fof(t5_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => ( k4_trees_1(A,B) = k2_trees_1(np__0)
          <=> r2_hidden(B,k3_trees_1(A)) ) ) ) )).

fof(t6_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m1_trees_1(B,k1_relat_1(A))
         => ( v1_trees_9(k5_trees_2(A,B))
          <=> r2_hidden(B,k3_trees_1(k1_relat_1(A))) ) ) ) )).

fof(d2_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( v2_trees_9(A)
      <=> ! [B] :
            ( m1_trees_1(B,A)
           => v1_finset_1(k1_trees_2(A,B)) ) ) ) )).

fof(d3_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( v3_trees_9(A)
      <=> v1_trees_2(k1_relat_1(A)) ) ) )).

fof(d4_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ( v4_trees_9(A)
      <=> v2_trees_9(k1_relat_1(A)) ) ) )).

fof(t7_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v2_trees_9(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))
              <=> ~ r1_xreal_0(k4_card_1(k1_trees_2(A,B)),C) ) ) ) ) )).

fof(d5_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v2_trees_9(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => ! [C] :
              ( ( v2_funct_1(C)
                & m2_finseq_1(C,A) )
             => ( C = k1_trees_9(A,B)
              <=> ( k3_finseq_1(C) = k4_card_1(k1_trees_2(A,B))
                  & k2_relat_1(C) = k1_trees_2(A,B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ~ r1_xreal_0(k3_finseq_1(C),D)
                       => k1_funct_1(C,k1_nat_1(D,np__1)) = k8_finseq_1(k5_numbers,B,k12_finseq_1(k5_numbers,D)) ) ) ) ) ) ) ) )).

fof(d6_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v4_trees_9(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( r2_hidden(B,k1_relat_1(A))
           => ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v1_finseq_1(C) )
               => ( C = k2_trees_9(A,B)
                <=> ? [D] :
                      ( m1_trees_1(D,k1_relat_1(A))
                      & D = B
                      & C = k5_relat_1(k1_trees_9(k1_relat_1(A),D),A) ) ) ) ) ) ) )).

fof(t8_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v4_trees_9(A) )
     => ? [B,C] :
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & v1_finseq_1(C)
          & v6_trees_3(C)
          & A = k4_trees_4(B,C) ) ) )).

fof(t9_trees_9,axiom,(
    $true )).

fof(t10_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & v1_trees_1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => ( A = k4_trees_1(A,B)
           => B = k1_xboole_0 ) ) ) )).

fof(t11_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ( r2_hidden(A,k3_trees_9(B))
      <=> ? [C] :
            ( m1_trees_1(C,k1_relat_1(B))
            & A = k5_trees_2(B,C) ) ) ) )).

fof(t12_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => r2_hidden(A,k3_trees_9(A)) ) )).

fof(t13_trees_9,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) )
         => ( ( v1_finset_1(A)
              & k3_trees_9(A) = k3_trees_9(B) )
           => A = B ) ) ) )).

fof(t14_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m1_trees_1(B,k1_relat_1(A))
         => r1_tarski(k3_trees_9(k5_trees_2(A,B)),k3_trees_9(A)) ) ) )).

fof(t15_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ( r2_hidden(A,k6_trees_9(B))
      <=> ? [C] :
            ( m1_trees_1(C,k1_relat_1(B))
            & A = k4_tarski(C,k5_trees_2(B,C)) ) ) ) )).

fof(t16_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => r2_hidden(k4_tarski(k1_xboole_0,A),k6_trees_9(A)) ) )).

fof(t17_trees_9,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) )
         => ( k6_trees_9(A) = k6_trees_9(B)
           => A = B ) ) ) )).

fof(t18_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ! [C] :
          ( r2_hidden(A,k7_trees_9(B,C))
        <=> ? [D] :
              ( m1_trees_1(D,k1_relat_1(B))
              & A = k5_trees_2(B,D)
              & ~ ( r2_hidden(D,k3_trees_1(k1_relat_1(B)))
                  & ~ r2_hidden(k1_funct_1(B,D),C) ) ) ) ) )).

fof(t19_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( v1_xboole_0(k7_trees_9(A,B))
        <=> ( v1_trees_9(A)
            & ~ r2_hidden(k1_funct_1(A,k1_xboole_0),B) ) ) ) )).

fof(d10_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ! [B,C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A)))
            & m2_relset_1(C,k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A))) )
         => ( C = k8_trees_9(A,B)
          <=> ! [D] :
                ( ( v1_relat_1(D)
                  & v1_funct_1(D)
                  & v3_trees_2(D) )
               => ( r2_hidden(D,k7_trees_9(A,B))
                 => ! [E] :
                      ( m2_finseq_1(E,k3_trees_9(A))
                     => ( E = k1_funct_1(C,D)
                       => D = k4_trees_4(k1_funct_1(D,k1_xboole_0),E) ) ) ) ) ) ) ) )).

fof(t20_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v3_trees_3(B) )
     => ( r2_hidden(A,k9_trees_9(B))
      <=> ? [C] :
            ( m3_trees_3(C,B)
            & ? [D] :
                ( m1_trees_1(D,k1_relat_1(C))
                & A = k5_trees_2(C,D) ) ) ) ) )).

fof(t21_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_trees_3(B) )
         => ( r2_hidden(A,B)
           => r2_hidden(A,k9_trees_9(B)) ) ) ) )).

fof(t22_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_trees_3(B) )
         => ( r1_tarski(A,B)
           => r1_tarski(k9_trees_9(A),k9_trees_9(B)) ) ) ) )).

fof(t23_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => k9_trees_9(k1_tarski(A)) = k3_trees_9(A) ) )).

fof(t25_trees_9,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v3_trees_3(C) )
     => ( r2_hidden(A,k12_trees_9(C,B))
      <=> ? [D] :
            ( m3_trees_3(D,C)
            & ? [E] :
                ( m1_trees_1(E,k1_relat_1(D))
                & A = k5_trees_2(D,E)
                & ~ ( r2_hidden(E,k3_trees_1(k1_relat_1(D)))
                    & ~ r2_hidden(k1_funct_1(D,E),B) ) ) ) ) ) )).

fof(t26_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v3_trees_3(B) )
     => ( v1_xboole_0(k12_trees_9(B,A))
      <=> ! [C] :
            ( m3_trees_3(C,B)
           => ( v1_trees_9(C)
              & ~ r2_hidden(k1_funct_1(C,k1_xboole_0),A) ) ) ) ) )).

fof(t27_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] : k12_trees_9(k1_tarski(A),B) = k7_trees_9(A,B) ) )).

fof(d13_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => ( ! [B] :
            ( m3_trees_3(B,A)
           => v1_finset_1(B) )
       => ! [B,C] :
            ( ( v1_funct_1(C)
              & v1_funct_2(C,k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A)))
              & m2_relset_1(C,k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A))) )
           => ( C = k13_trees_9(A,B)
            <=> ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v3_trees_2(D) )
                 => ( r2_hidden(D,k12_trees_9(A,B))
                   => ! [E] :
                        ( m2_finseq_1(E,k9_trees_9(A))
                       => ( E = k1_funct_1(C,D)
                         => D = k4_trees_4(k1_funct_1(D,k1_xboole_0),E) ) ) ) ) ) ) ) ) )).

fof(t29_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m1_trees_1(B,k1_relat_1(A))
         => ( k3_finseq_1(k2_trees_9(A,B)) = k3_finseq_1(k1_trees_9(k1_relat_1(A),B))
            & k4_finseq_1(k2_trees_9(A,B)) = k4_finseq_1(k1_trees_9(k1_relat_1(A),B)) ) ) ) )).

fof(t30_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v5_trees_3(A) )
     => ! [B] :
          ( ( v1_xboole_0(B)
            & m1_trees_1(B,k13_trees_3(A)) )
         => k4_card_1(k1_trees_2(k13_trees_3(A),B)) = k3_finseq_1(A) ) ) )).

fof(t31_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ! [B,C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v1_finseq_1(C)
            & v6_trees_3(C) )
         => ( A = k4_trees_4(B,C)
           => ! [D] :
                ( ( v1_xboole_0(D)
                  & m1_trees_1(D,k1_relat_1(A)) )
               => k2_trees_9(A,D) = k16_trees_3(C) ) ) ) ) )).

fof(t32_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ! [B] :
          ( m1_trees_1(B,k1_relat_1(A))
         => ! [C] :
              ( m1_trees_1(C,k1_relat_1(k5_trees_2(A,B)))
             => k2_trees_9(A,k8_finseq_1(k5_numbers,B,C)) = k2_trees_9(k5_trees_2(A,B),C) ) ) ) )).

fof(s1_trees_9,axiom,
    ( ( ! [A] :
          ~ ( r2_hidden(A,f2_s1_trees_9)
            & ! [B] :
                ( m2_subset_1(B,k1_numbers,k5_numbers)
               => A != f1_s1_trees_9(B) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( r2_hidden(f1_s1_trees_9(B),f2_s1_trees_9)
               => ( r1_xreal_0(B,A)
                  | r2_hidden(f1_s1_trees_9(A),f2_s1_trees_9) ) ) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( f1_s1_trees_9(A) = f1_s1_trees_9(B)
               => A = B ) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r2_hidden(f1_s1_trees_9(A),f2_s1_trees_9)
        <=> ~ r1_xreal_0(k4_card_1(f2_s1_trees_9),A) ) ) )).

fof(dt_m1_trees_9,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m4_trees_3(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ! [D] :
          ( m1_trees_9(D,A,B,C)
         => m5_trees_3(D,A,B) ) ) )).

fof(existence_m1_trees_9,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m4_trees_3(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ? [D] : m1_trees_9(D,A,B,C) ) )).

fof(redefinition_m1_trees_9,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m4_trees_3(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ! [D] :
          ( m1_trees_9(D,A,B,C)
        <=> m1_subset_1(D,C) ) ) )).

fof(dt_k1_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v2_trees_9(A)
        & m1_subset_1(B,A) )
     => ( v2_funct_1(k1_trees_9(A,B))
        & m2_finseq_1(k1_trees_9(A,B),A) ) ) )).

fof(dt_k2_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A)
        & v4_trees_9(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( v1_relat_1(k2_trees_9(A,B))
        & v1_funct_1(k2_trees_9(A,B))
        & v1_finseq_1(k2_trees_9(A,B)) ) ) )).

fof(dt_k3_trees_9,axiom,(
    $true )).

fof(dt_k4_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A) )
     => ( ~ v1_xboole_0(k4_trees_9(A,B))
        & m1_subset_1(k4_trees_9(A,B),k1_zfmisc_1(k4_trees_3(A))) ) ) )).

fof(redefinition_k4_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A) )
     => k4_trees_9(A,B) = k3_trees_9(B) ) )).

fof(dt_k5_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_finset_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A) )
     => ( ~ v1_xboole_0(k5_trees_9(A,B))
        & m1_subset_1(k5_trees_9(A,B),k1_zfmisc_1(k5_trees_3(A))) ) ) )).

fof(redefinition_k5_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_finset_1(B)
        & v3_trees_2(B)
        & m3_trees_2(B,A) )
     => k5_trees_9(A,B) = k3_trees_9(B) ) )).

fof(dt_k6_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => m1_subset_1(k6_trees_9(A),k1_zfmisc_1(k2_zfmisc_1(k1_relat_1(A),k3_trees_9(A)))) ) )).

fof(dt_k7_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => m1_subset_1(k7_trees_9(A,B),k1_zfmisc_1(k3_trees_9(A))) ) )).

fof(dt_k8_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A)
        & v3_trees_2(A) )
     => ( v1_funct_1(k8_trees_9(A,B))
        & v1_funct_2(k8_trees_9(A,B),k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A)))
        & m2_relset_1(k8_trees_9(A,B),k7_trees_9(A,B),k3_finseq_2(k3_trees_9(A))) ) ) )).

fof(dt_k9_trees_9,axiom,(
    $true )).

fof(dt_k10_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k4_trees_3(A))) )
     => ( ~ v1_xboole_0(k10_trees_9(A,B))
        & m1_subset_1(k10_trees_9(A,B),k1_zfmisc_1(k4_trees_3(A))) ) ) )).

fof(redefinition_k10_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k4_trees_3(A))) )
     => k10_trees_9(A,B) = k9_trees_9(B) ) )).

fof(dt_k11_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_trees_3(A))) )
     => ( ~ v1_xboole_0(k11_trees_9(A,B))
        & m1_subset_1(k11_trees_9(A,B),k1_zfmisc_1(k5_trees_3(A))) ) ) )).

fof(redefinition_k11_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k5_trees_3(A))) )
     => k11_trees_9(A,B) = k9_trees_9(B) ) )).

fof(dt_k12_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => m1_subset_1(k12_trees_9(A,B),k1_zfmisc_1(k9_trees_9(A))) ) )).

fof(dt_k13_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => ( v1_funct_1(k13_trees_9(A,B))
        & v1_funct_2(k13_trees_9(A,B),k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A)))
        & m2_relset_1(k13_trees_9(A,B),k12_trees_9(A,B),k3_finseq_2(k9_trees_9(A))) ) ) )).

fof(d7_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => k3_trees_9(A) = a_1_0_trees_9(A) ) )).

fof(d8_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => k6_trees_9(A) = a_1_1_trees_9(A) ) )).

fof(d9_trees_9,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v3_trees_2(A) )
     => ! [B] : k7_trees_9(A,B) = a_2_0_trees_9(A,B) ) )).

fof(d11_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => k9_trees_9(A) = a_1_2_trees_9(A) ) )).

fof(t24_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => k9_trees_9(A) = k3_tarski(a_1_3_trees_9(A)) ) )).

fof(d12_trees_9,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_trees_3(A) )
     => ! [B] : k12_trees_9(A,B) = a_2_1_trees_9(A,B) ) )).

fof(t28_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v3_trees_3(B) )
     => k12_trees_9(B,A) = k3_tarski(a_2_2_trees_9(A,B)) ) )).

fof(fraenkel_a_1_0_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ( r2_hidden(A,a_1_0_trees_9(B))
      <=> ? [C] :
            ( m1_trees_1(C,k1_relat_1(B))
            & A = k5_trees_2(B,C) ) ) ) )).

fof(fraenkel_a_1_1_trees_9,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ( r2_hidden(A,a_1_1_trees_9(B))
      <=> ? [C] :
            ( m1_trees_1(C,k1_relat_1(B))
            & A = k4_tarski(C,k5_trees_2(B,C)) ) ) ) )).

fof(fraenkel_a_2_0_trees_9,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v3_trees_2(B) )
     => ( r2_hidden(A,a_2_0_trees_9(B,C))
      <=> ? [D] :
            ( m1_trees_1(D,k1_relat_1(B))
            & A = k5_trees_2(B,D)
            & ~ ( r2_hidden(D,k3_trees_1(k1_relat_1(B)))
                & ~ r2_hidden(k1_funct_1(B,D),C) ) ) ) ) )).

fof(fraenkel_a_1_2_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v3_trees_3(B) )
     => ( r2_hidden(A,a_1_2_trees_9(B))
      <=> ? [C,D] :
            ( m3_trees_3(C,B)
            & m1_trees_1(D,k1_relat_1(C))
            & A = k5_trees_2(C,D) ) ) ) )).

fof(fraenkel_a_1_3_trees_9,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v3_trees_3(B) )
     => ( r2_hidden(A,a_1_3_trees_9(B))
      <=> ? [C] :
            ( m3_trees_3(C,B)
            & A = k3_trees_9(C) ) ) ) )).

fof(fraenkel_a_2_1_trees_9,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v3_trees_3(B) )
     => ( r2_hidden(A,a_2_1_trees_9(B,C))
      <=> ? [D,E] :
            ( m3_trees_3(D,B)
            & m1_trees_1(E,k1_relat_1(D))
            & A = k5_trees_2(D,E)
            & ~ ( r2_hidden(E,k3_trees_1(k1_relat_1(D)))
                & ~ r2_hidden(k1_funct_1(D,E),C) ) ) ) ) )).

fof(fraenkel_a_2_2_trees_9,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v3_trees_3(C) )
     => ( r2_hidden(A,a_2_2_trees_9(B,C))
      <=> ? [D] :
            ( m3_trees_3(D,C)
            & A = k7_trees_9(D,B) ) ) ) )).
%------------------------------------------------------------------------------