SET007 Axioms: SET007+359.ax


%------------------------------------------------------------------------------
% File     : SET007+359 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Petri Net Concepts
% 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    : petri [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   84 (   2 unit)
%            Number of atoms       :  329 (  56 equality)
%            Maximal formula depth :   14 (   6 average)
%            Number of connectives :  278 (  33 ~  ;   0  |; 112  &)
%                                         (  21 <=>; 112 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   1 propositional; 0-4 arity)
%            Number of functors    :   41 (   1 constant; 0-4 arity)
%            Number of variables   :  204 (   0 singleton; 173 !;  31 ?)
%            Maximal term depth    :    4 (   2 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_petri,axiom,(
    ? [A] :
      ( l1_petri(A)
      & v1_petri(A) ) )).

fof(rc2_petri,axiom,(
    ? [A] :
      ( l1_petri(A)
      & v3_petri(A) ) )).

fof(rc3_petri,axiom,(
    ? [A] :
      ( l1_petri(A)
      & v5_petri(A) ) )).

fof(t2_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => ! [C] :
              ( r2_hidden(C,k5_petri(A,B))
            <=> ? [D] :
                  ( m1_petri(D,u2_petri(A),u1_petri(A),u4_petri(A))
                  & ? [E] :
                      ( m1_subset_1(E,u1_petri(A))
                      & r2_hidden(E,B)
                      & D = k4_tarski(C,E) ) ) ) ) ) )).

fof(t4_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => ! [C] :
              ( r2_hidden(C,k6_petri(A,B))
            <=> ? [D] :
                  ( m1_petri(D,u1_petri(A),u2_petri(A),u3_petri(A))
                  & ? [E] :
                      ( m1_subset_1(E,u1_petri(A))
                      & r2_hidden(E,B)
                      & D = k4_tarski(E,C) ) ) ) ) ) )).

fof(t6_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(A)))
         => ! [C] :
              ( r2_hidden(C,k7_petri(A,B))
            <=> ? [D] :
                  ( m1_petri(D,u1_petri(A),u2_petri(A),u3_petri(A))
                  & ? [E] :
                      ( m1_subset_1(E,u2_petri(A))
                      & r2_hidden(E,B)
                      & D = k4_tarski(C,E) ) ) ) ) ) )).

fof(t8_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(A)))
         => ! [C] :
              ( r2_hidden(C,k8_petri(A,B))
            <=> ? [D] :
                  ( m1_petri(D,u2_petri(A),u1_petri(A),u4_petri(A))
                  & ? [E] :
                      ( m1_subset_1(E,u2_petri(A))
                      & r2_hidden(E,B)
                      & D = k4_tarski(E,C) ) ) ) ) ) )).

fof(t9_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => k5_petri(A,k1_subset_1(u1_petri(A))) = k1_xboole_0 ) )).

fof(t10_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => k6_petri(A,k1_subset_1(u1_petri(A))) = k1_xboole_0 ) )).

fof(t11_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => k7_petri(A,k1_subset_1(u2_petri(A))) = k1_xboole_0 ) )).

fof(t12_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => k8_petri(A,k1_subset_1(u2_petri(A))) = k1_xboole_0 ) )).

fof(d5_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => ( v2_petri(B,A)
          <=> m1_subset_1(k5_petri(A,B),k1_zfmisc_1(k6_petri(A,B))) ) ) ) )).

fof(d6_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ( v3_petri(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
            & v2_petri(B,A) ) ) ) )).

fof(d7_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => ( v4_petri(B,A)
          <=> m1_subset_1(k6_petri(A,B),k1_zfmisc_1(k5_petri(A,B))) ) ) ) )).

fof(d8_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ( v5_petri(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
            & v4_petri(B,A) ) ) ) )).

fof(d9_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => k10_petri(A) = g1_petri(u1_petri(A),u2_petri(A),k9_petri(u2_petri(A),u1_petri(A),u4_petri(A)),k9_petri(u1_petri(A),u2_petri(A),u3_petri(A))) ) )).

fof(t13_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => k10_petri(k10_petri(A)) = g1_petri(u1_petri(A),u2_petri(A),u3_petri(A),u4_petri(A)) ) )).

fof(t14_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ( u1_petri(A) = u1_petri(k10_petri(A))
        & u2_petri(A) = u2_petri(k10_petri(A))
        & k9_petri(u1_petri(A),u2_petri(A),u3_petri(A)) = u4_petri(k10_petri(A))
        & k9_petri(u2_petri(A),u1_petri(A),u4_petri(A)) = u3_petri(k10_petri(A)) ) ) )).

fof(d10_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => k11_petri(A,B) = B ) ) )).

fof(d11_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,u1_petri(A))
         => k12_petri(A,B) = B ) ) )).

fof(d12_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(k10_petri(A))))
         => k13_petri(A,B) = B ) ) )).

fof(d13_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,u1_petri(k10_petri(A)))
         => k14_petri(A,B) = B ) ) )).

fof(d14_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(A)))
         => k15_petri(A,B) = B ) ) )).

fof(d15_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,u2_petri(A))
         => k16_petri(A,B) = B ) ) )).

fof(d16_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(k10_petri(A))))
         => k17_petri(A,B) = B ) ) )).

fof(d17_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,u2_petri(k10_petri(A)))
         => k18_petri(A,B) = B ) ) )).

fof(t15_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => k6_petri(k10_petri(A),k11_petri(A,B)) = k5_petri(A,B) ) ) )).

fof(t16_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => k5_petri(k10_petri(A),k11_petri(A,B)) = k6_petri(A,B) ) ) )).

fof(t17_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => ( v2_petri(B,A)
          <=> v4_petri(k11_petri(A,B),k10_petri(A)) ) ) ) )).

fof(t18_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => ( v4_petri(B,A)
          <=> v2_petri(k11_petri(A,B),k10_petri(A)) ) ) ) )).

fof(t19_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,u2_petri(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_petri(A)))
             => ( r2_hidden(B,k6_petri(A,C))
              <=> ~ r1_xboole_0(k7_petri(A,k6_domain_1(u2_petri(A),B)),C) ) ) ) ) )).

fof(t20_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,u2_petri(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_petri(A)))
             => ( r2_hidden(B,k5_petri(A,C))
              <=> ~ r1_xboole_0(k8_petri(A,k6_domain_1(u2_petri(A),B)),C) ) ) ) ) )).

fof(dt_m1_petri,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B) )
     => ! [D] :
          ( m1_petri(D,A,B,C)
         => m1_subset_1(D,k2_zfmisc_1(A,B)) ) ) )).

fof(existence_m1_petri,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B) )
     => ? [D] : m1_petri(D,A,B,C) ) )).

fof(redefinition_m1_petri,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B) )
     => ! [D] :
          ( m1_petri(D,A,B,C)
        <=> m1_subset_1(D,C) ) ) )).

fof(dt_l1_petri,axiom,(
    $true )).

fof(existence_l1_petri,axiom,(
    ? [A] : l1_petri(A) )).

fof(abstractness_v1_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ( v1_petri(A)
       => A = g1_petri(u1_petri(A),u2_petri(A),u3_petri(A),u4_petri(A)) ) ) )).

fof(dt_k1_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u3_petri(A)) )
     => m1_subset_1(k1_petri(A,B),u1_petri(A)) ) )).

fof(redefinition_k1_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u3_petri(A)) )
     => k1_petri(A,B) = k1_mcart_1(B) ) )).

fof(dt_k2_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u3_petri(A)) )
     => m1_subset_1(k2_petri(A,B),u2_petri(A)) ) )).

fof(redefinition_k2_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u3_petri(A)) )
     => k2_petri(A,B) = k2_mcart_1(B) ) )).

fof(dt_k3_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u4_petri(A)) )
     => m1_subset_1(k3_petri(A,B),u2_petri(A)) ) )).

fof(redefinition_k3_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u4_petri(A)) )
     => k3_petri(A,B) = k1_mcart_1(B) ) )).

fof(dt_k4_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u4_petri(A)) )
     => m1_subset_1(k4_petri(A,B),u1_petri(A)) ) )).

fof(redefinition_k4_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u4_petri(A)) )
     => k4_petri(A,B) = k2_mcart_1(B) ) )).

fof(dt_k5_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_petri(A))) )
     => m1_subset_1(k5_petri(A,B),k1_zfmisc_1(u2_petri(A))) ) )).

fof(dt_k6_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_petri(A))) )
     => m1_subset_1(k6_petri(A,B),k1_zfmisc_1(u2_petri(A))) ) )).

fof(dt_k7_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u2_petri(A))) )
     => m1_subset_1(k7_petri(A,B),k1_zfmisc_1(u1_petri(A))) ) )).

fof(dt_k8_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u2_petri(A))) )
     => m1_subset_1(k8_petri(A,B),k1_zfmisc_1(u1_petri(A))) ) )).

fof(dt_k9_petri,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B) )
     => ( ~ v1_xboole_0(k9_petri(A,B,C))
        & m2_relset_1(k9_petri(A,B,C),B,A) ) ) )).

fof(involutiveness_k9_petri,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B) )
     => k9_petri(A,B,k9_petri(A,B,C)) = C ) )).

fof(redefinition_k9_petri,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B) )
     => k9_petri(A,B,C) = k4_relat_1(C) ) )).

fof(dt_k10_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ( v1_petri(k10_petri(A))
        & l1_petri(k10_petri(A)) ) ) )).

fof(dt_k11_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_petri(A))) )
     => m1_subset_1(k11_petri(A,B),k1_zfmisc_1(u1_petri(k10_petri(A)))) ) )).

fof(dt_k12_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u1_petri(A)) )
     => m1_subset_1(k12_petri(A,B),u1_petri(k10_petri(A))) ) )).

fof(dt_k13_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u1_petri(k10_petri(A)))) )
     => m1_subset_1(k13_petri(A,B),k1_zfmisc_1(u1_petri(A))) ) )).

fof(dt_k14_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u1_petri(k10_petri(A))) )
     => m1_subset_1(k14_petri(A,B),u1_petri(A)) ) )).

fof(dt_k15_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u2_petri(A))) )
     => m1_subset_1(k15_petri(A,B),k1_zfmisc_1(u2_petri(k10_petri(A)))) ) )).

fof(dt_k16_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u2_petri(A)) )
     => m1_subset_1(k16_petri(A,B),u2_petri(k10_petri(A))) ) )).

fof(dt_k17_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,k1_zfmisc_1(u2_petri(k10_petri(A)))) )
     => m1_subset_1(k17_petri(A,B),k1_zfmisc_1(u2_petri(A))) ) )).

fof(dt_k18_petri,axiom,(
    ! [A,B] :
      ( ( l1_petri(A)
        & m1_subset_1(B,u2_petri(k10_petri(A))) )
     => m1_subset_1(k18_petri(A,B),u2_petri(A)) ) )).

fof(dt_u1_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ~ v1_xboole_0(u1_petri(A)) ) )).

fof(dt_u2_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ~ v1_xboole_0(u2_petri(A)) ) )).

fof(dt_u3_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ( ~ v1_xboole_0(u3_petri(A))
        & m2_relset_1(u3_petri(A),u1_petri(A),u2_petri(A)) ) ) )).

fof(dt_u4_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ( ~ v1_xboole_0(u4_petri(A))
        & m2_relset_1(u4_petri(A),u2_petri(A),u1_petri(A)) ) ) )).

fof(dt_g1_petri,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B)
        & ~ v1_xboole_0(D)
        & m1_relset_1(D,B,A) )
     => ( v1_petri(g1_petri(A,B,C,D))
        & l1_petri(g1_petri(A,B,C,D)) ) ) )).

fof(free_g1_petri,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_relset_1(C,A,B)
        & ~ v1_xboole_0(D)
        & m1_relset_1(D,B,A) )
     => ! [E,F,G,H] :
          ( g1_petri(A,B,C,D) = g1_petri(E,F,G,H)
         => ( A = E
            & B = F
            & C = G
            & D = H ) ) ) )).

fof(d1_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => k5_petri(A,B) = a_2_0_petri(A,B) ) ) )).

fof(d2_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => k6_petri(A,B) = a_2_1_petri(A,B) ) ) )).

fof(t1_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => k5_petri(A,B) = a_2_2_petri(A,B) ) ) )).

fof(t3_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_petri(A)))
         => k6_petri(A,B) = a_2_3_petri(A,B) ) ) )).

fof(d3_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(A)))
         => k7_petri(A,B) = a_2_4_petri(A,B) ) ) )).

fof(d4_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(A)))
         => k8_petri(A,B) = a_2_5_petri(A,B) ) ) )).

fof(t5_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(A)))
         => k7_petri(A,B) = a_2_6_petri(A,B) ) ) )).

fof(t7_petri,axiom,(
    ! [A] :
      ( l1_petri(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u2_petri(A)))
         => k8_petri(A,B) = a_2_7_petri(A,B) ) ) )).

fof(fraenkel_a_2_0_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_petri(B))) )
     => ( r2_hidden(A,a_2_0_petri(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u2_petri(B))
            & A = D
            & ? [E] :
                ( m1_petri(E,u2_petri(B),u1_petri(B),u4_petri(B))
                & ? [F] :
                    ( m1_subset_1(F,u1_petri(B))
                    & r2_hidden(F,C)
                    & E = k1_domain_1(u2_petri(B),u1_petri(B),D,F) ) ) ) ) ) )).

fof(fraenkel_a_2_1_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_petri(B))) )
     => ( r2_hidden(A,a_2_1_petri(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u2_petri(B))
            & A = D
            & ? [E] :
                ( m1_petri(E,u1_petri(B),u2_petri(B),u3_petri(B))
                & ? [F] :
                    ( m1_subset_1(F,u1_petri(B))
                    & r2_hidden(F,C)
                    & E = k1_domain_1(u1_petri(B),u2_petri(B),F,D) ) ) ) ) ) )).

fof(fraenkel_a_2_2_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_petri(B))) )
     => ( r2_hidden(A,a_2_2_petri(B,C))
      <=> ? [D] :
            ( m1_petri(D,u2_petri(B),u1_petri(B),u4_petri(B))
            & A = k3_petri(B,D)
            & r2_hidden(k4_petri(B,D),C) ) ) ) )).

fof(fraenkel_a_2_3_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u1_petri(B))) )
     => ( r2_hidden(A,a_2_3_petri(B,C))
      <=> ? [D] :
            ( m1_petri(D,u1_petri(B),u2_petri(B),u3_petri(B))
            & A = k2_petri(B,D)
            & r2_hidden(k1_petri(B,D),C) ) ) ) )).

fof(fraenkel_a_2_4_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u2_petri(B))) )
     => ( r2_hidden(A,a_2_4_petri(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_petri(B))
            & A = D
            & ? [E] :
                ( m1_petri(E,u1_petri(B),u2_petri(B),u3_petri(B))
                & ? [F] :
                    ( m1_subset_1(F,u2_petri(B))
                    & r2_hidden(F,C)
                    & E = k1_domain_1(u1_petri(B),u2_petri(B),D,F) ) ) ) ) ) )).

fof(fraenkel_a_2_5_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u2_petri(B))) )
     => ( r2_hidden(A,a_2_5_petri(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_petri(B))
            & A = D
            & ? [E] :
                ( m1_petri(E,u2_petri(B),u1_petri(B),u4_petri(B))
                & ? [F] :
                    ( m1_subset_1(F,u2_petri(B))
                    & r2_hidden(F,C)
                    & E = k1_domain_1(u2_petri(B),u1_petri(B),F,D) ) ) ) ) ) )).

fof(fraenkel_a_2_6_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u2_petri(B))) )
     => ( r2_hidden(A,a_2_6_petri(B,C))
      <=> ? [D] :
            ( m1_petri(D,u1_petri(B),u2_petri(B),u3_petri(B))
            & A = k1_petri(B,D)
            & r2_hidden(k2_petri(B,D),C) ) ) ) )).

fof(fraenkel_a_2_7_petri,axiom,(
    ! [A,B,C] :
      ( ( l1_petri(B)
        & m1_subset_1(C,k1_zfmisc_1(u2_petri(B))) )
     => ( r2_hidden(A,a_2_7_petri(B,C))
      <=> ? [D] :
            ( m1_petri(D,u2_petri(B),u1_petri(B),u4_petri(B))
            & A = k4_petri(B,D)
            & r2_hidden(k3_petri(B,D),C) ) ) ) )).
%------------------------------------------------------------------------------