SET007 Axioms: SET007+680.ax


%------------------------------------------------------------------------------
% File     : SET007+680 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Dynkin's Lemma in Measure Theory
% 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    : dynkin [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   54 (   5 unit)
%            Number of atoms       :  302 (  21 equality)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :  306 (  58 ~  ;   0  |; 100  &)
%                                         (  10 <=>; 138 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-3 arity)
%            Number of functors    :   34 (   5 constant; 0-4 arity)
%            Number of variables   :  154 (   0 singleton; 152 !;   2 ?)
%            Maximal term depth    :    6 (   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(fc1_dynkin,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => v1_finset_1(k1_algseq_1(A)) ) )).

fof(fc2_dynkin,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(k1_dynkin(A,B,C))
      & v1_funct_1(k1_dynkin(A,B,C)) ) )).

fof(cc1_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_dynkin(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(t1_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( r2_hidden(C,k2_relat_1(B))
            <=> ? [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                  & k1_prob_1(A,B,D) = C ) ) ) ) )).

fof(t2_dynkin,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => v1_finset_1(k2_algseq_1(A)) ) )).

fof(d1_dynkin,axiom,(
    ! [A,B,C] : k1_dynkin(A,B,C) = k1_funct_4(k2_funcop_1(k5_numbers,C),k4_funct_4(np__0,np__1,A,B)) )).

fof(t3_dynkin,axiom,(
    $true )).

fof(t4_dynkin,axiom,(
    $true )).

fof(t5_dynkin,axiom,(
    ! [A,B,C] :
      ( k1_funct_1(k1_dynkin(A,B,C),np__0) = A
      & k1_funct_1(k1_dynkin(A,B,C),np__1) = B
      & ! [D] :
          ( m2_subset_1(D,k1_numbers,k5_numbers)
         => ~ ( D != np__0
              & D != np__1
              & k1_funct_1(k1_dynkin(A,B,C),D) != C ) ) ) )).

fof(t6_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k3_tarski(k2_relat_1(k1_dynkin(B,C,k1_xboole_0))) = k4_subset_1(A,B,C) ) ) ) )).

fof(d2_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k5_numbers,k1_zfmisc_1(A))
                    & m2_relset_1(D,k5_numbers,k1_zfmisc_1(A)) )
                 => ( D = k4_dynkin(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => k1_prob_1(A,D,E) = k5_subset_1(A,C,k1_prob_1(A,B,E)) ) ) ) ) ) ) )).

fof(d3_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ( v1_prob_2(B)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ~ r1_xreal_0(D,C)
                     => r1_xboole_0(k1_prob_1(A,B,C),k1_prob_1(A,B,D)) ) ) ) ) ) ) )).

fof(t7_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( r1_tarski(B,k1_setfam_1(A))
        <=> ! [C] :
              ( m1_subset_1(C,A)
             => r1_tarski(B,C) ) ) ) )).

fof(d4_dynkin,axiom,(
    $true )).

fof(d5_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k5_dynkin(A,B,C) = k4_xboole_0(k1_prob_1(A,B,C),k3_tarski(k2_relat_1(k7_relat_1(B,k2_algseq_1(C))))) ) ) ) )).

fof(d6_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
                & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
             => ( C = k6_dynkin(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k1_prob_1(A,C,D) = k5_dynkin(A,B,D) ) ) ) ) ) )).

fof(t8_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k1_prob_1(A,k6_dynkin(A,B),C) = k4_xboole_0(k1_prob_1(A,B,C),k3_tarski(k2_relat_1(k7_relat_1(B,k2_algseq_1(C))))) ) ) ) )).

fof(t9_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => v1_prob_2(k6_dynkin(A,B)) ) ) )).

fof(t10_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => k3_tarski(k2_relat_1(k6_dynkin(A,B))) = k3_tarski(k2_relat_1(B)) ) ) )).

fof(t11_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( r1_xboole_0(B,C)
               => v1_prob_2(k3_dynkin(A,B,C,k1_subset_1(A))) ) ) ) ) )).

fof(t12_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ( v1_prob_2(B)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => v1_prob_2(k4_dynkin(A,B,C)) ) ) ) ) )).

fof(t13_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
            & m2_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k5_subset_1(A,C,k2_prob_1(A,B)) = k2_prob_1(A,k4_dynkin(A,B,C)) ) ) ) )).

fof(d7_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( m1_dynkin(B,A)
          <=> ( ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
                    & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
                 => ( ( r1_tarski(k2_relat_1(C),B)
                      & v1_prob_2(C) )
                   => r2_hidden(k2_prob_1(A,C),B) ) )
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(A))
                 => ( r2_hidden(C,B)
                   => r2_hidden(k3_subset_1(A,C),B) ) )
              & r2_hidden(k1_xboole_0,B) ) ) ) ) )).

fof(t14_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_dynkin(k1_zfmisc_1(A),A) ) )).

fof(t15_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( ! [C] :
                ( r2_hidden(C,A)
               => m1_dynkin(C,B) )
           => m1_dynkin(k1_setfam_1(A),B) ) ) ) )).

fof(t16_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
                 => ( ( m1_dynkin(D,A)
                      & v2_finsub_1(D)
                      & r2_hidden(B,D)
                      & r2_hidden(C,D) )
                   => r2_hidden(k6_subset_1(A,B,C),D) ) ) ) ) ) )).

fof(t17_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
                 => ( ( m1_dynkin(D,A)
                      & v2_finsub_1(D)
                      & r2_hidden(B,D)
                      & r2_hidden(C,D) )
                   => r2_hidden(k4_subset_1(A,B,C),D) ) ) ) ) ) )).

fof(t18_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( ( m1_dynkin(B,A)
              & v2_finsub_1(B) )
           => ! [C] :
                ( v1_finset_1(C)
               => ( r1_tarski(C,B)
                 => r2_hidden(k3_tarski(C),B) ) ) ) ) ) )).

fof(t19_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( ( m1_dynkin(B,A)
              & v2_finsub_1(B) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
                  & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
               => ( r1_tarski(k2_relat_1(C),B)
                 => r1_tarski(k2_relat_1(k6_dynkin(A,C)),B) ) ) ) ) ) )).

fof(t20_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( ( m1_dynkin(B,A)
              & v2_finsub_1(B) )
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A))
                  & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) )
               => ( r1_tarski(k2_relat_1(C),B)
                 => r2_hidden(k3_tarski(k2_relat_1(C)),B) ) ) ) ) ) )).

fof(t21_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_dynkin(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_zfmisc_1(A),B)
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(A),B)
                 => ( r1_xboole_0(C,D)
                   => r2_hidden(k4_subset_1(A,C,D),B) ) ) ) ) ) )).

fof(t22_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_dynkin(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_zfmisc_1(A),B)
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(A),B)
                 => ( r1_tarski(C,D)
                   => r2_hidden(k6_subset_1(A,D,C),B) ) ) ) ) ) )).

fof(t23_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( ( m1_dynkin(B,A)
              & v2_finsub_1(B) )
           => m1_prob_1(B,A) ) ) ) )).

fof(d8_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ! [C] :
              ( m1_dynkin(C,A)
             => ( C = k7_dynkin(A,B)
              <=> ( r1_tarski(B,C)
                  & ! [D] :
                      ( m1_dynkin(D,A)
                     => ( r1_tarski(B,D)
                       => r1_tarski(C,D) ) ) ) ) ) ) ) )).

fof(d9_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ( D = k8_dynkin(A,B,C)
              <=> ! [E] :
                    ( m1_subset_1(E,k1_zfmisc_1(A))
                   => ( r2_hidden(E,D)
                    <=> r2_hidden(k5_subset_1(A,E,C),B) ) ) ) ) ) ) )).

fof(t24_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( ( r2_hidden(C,B)
                      & r2_hidden(D,k7_dynkin(A,B))
                      & v2_finsub_1(B) )
                   => r2_hidden(k5_subset_1(A,C,D),k7_dynkin(A,B)) ) ) ) ) ) )).

fof(t25_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( ( r2_hidden(C,k7_dynkin(A,B))
                      & r2_hidden(D,k7_dynkin(A,B))
                      & v2_finsub_1(B) )
                   => r2_hidden(k5_subset_1(A,C,D),k7_dynkin(A,B)) ) ) ) ) ) )).

fof(t26_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( v2_finsub_1(B)
           => v2_finsub_1(k7_dynkin(A,B)) ) ) ) )).

fof(t27_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( v2_finsub_1(B)
           => ! [C] :
                ( m1_dynkin(C,A)
               => ( r1_tarski(B,C)
                 => r1_tarski(k11_prob_1(A,B),C) ) ) ) ) ) )).

fof(dt_m1_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_dynkin(B,A)
         => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(existence_m1_dynkin,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] : m1_dynkin(B,A) ) )).

fof(redefinition_v1_dynkin,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ( v1_dynkin(B,A)
      <=> v1_prob_2(B) ) ) )).

fof(dt_k1_dynkin,axiom,(
    $true )).

fof(dt_k2_dynkin,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A) )
     => ( v1_funct_1(k2_dynkin(A,B,C,D))
        & v1_funct_2(k2_dynkin(A,B,C,D),k5_numbers,A)
        & m2_relset_1(k2_dynkin(A,B,C,D),k5_numbers,A) ) ) )).

fof(redefinition_k2_dynkin,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A) )
     => k2_dynkin(A,B,C,D) = k1_dynkin(B,C,D) ) )).

fof(dt_k3_dynkin,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,k1_zfmisc_1(A)) )
     => ( v1_funct_1(k3_dynkin(A,B,C,D))
        & v1_funct_2(k3_dynkin(A,B,C,D),k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(k3_dynkin(A,B,C,D),k5_numbers,k1_zfmisc_1(A)) ) ) )).

fof(redefinition_k3_dynkin,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,k1_zfmisc_1(A)) )
     => k3_dynkin(A,B,C,D) = k1_dynkin(B,C,D) ) )).

fof(dt_k4_dynkin,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A))
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => ( v1_funct_1(k4_dynkin(A,B,C))
        & v1_funct_2(k4_dynkin(A,B,C),k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(k4_dynkin(A,B,C),k5_numbers,k1_zfmisc_1(A)) ) ) )).

fof(dt_k5_dynkin,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A))
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k5_dynkin(A,B,C),k1_zfmisc_1(A)) ) )).

fof(dt_k6_dynkin,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => ( v1_funct_1(k6_dynkin(A,B))
        & v1_funct_2(k6_dynkin(A,B),k5_numbers,k1_zfmisc_1(A))
        & m2_relset_1(k6_dynkin(A,B),k5_numbers,k1_zfmisc_1(A)) ) ) )).

fof(dt_k7_dynkin,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => m1_dynkin(k7_dynkin(A,B),A) ) )).

fof(dt_k8_dynkin,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => m1_subset_1(k8_dynkin(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(dt_k9_dynkin,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_dynkin(B,A)
        & m1_subset_1(C,B) )
     => m1_dynkin(k9_dynkin(A,B,C),A) ) )).

fof(redefinition_k9_dynkin,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_dynkin(B,A)
        & m1_subset_1(C,B) )
     => k9_dynkin(A,B,C) = k8_dynkin(A,B,C) ) )).
%------------------------------------------------------------------------------