SET007 Axioms: SET007+758.ax


%------------------------------------------------------------------------------
% File     : SET007+758 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Chains on a Grating in Euclidean Space
% 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    : chain_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  137 (   2 unit)
%            Number of atoms       : 1152 ( 124 equality)
%            Maximal formula depth :   28 (  11 average)
%            Number of connectives : 1272 ( 257 ~  ;  19  |; 480  &)
%                                         (  53 <=>; 463 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   44 (   1 propositional; 0-5 arity)
%            Number of functors    :   63 (  17 constant; 0-5 arity)
%            Number of variables   :  557 (   0 singleton; 513 !;  44 ?)
%            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(rc1_chain_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xcmplx_0(A)
      & v1_finset_1(A)
      & v1_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_int_1(A)
      & v1_rat_1(A) ) )).

fof(fc1_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => ( ~ v1_xboole_0(k1_finseq_1(A))
        & v1_finset_1(k1_finseq_1(A)) ) ) )).

fof(rc2_chain_1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A)
      & ~ v1_realset1(A) ) )).

fof(fc2_chain_1,axiom,(
    ! [A,B] :
      ( ~ v1_realset1(A)
     => ( ~ v1_xboole_0(k2_xboole_0(A,B))
        & ~ v1_realset1(k2_xboole_0(A,B)) ) ) )).

fof(fc3_chain_1,axiom,(
    ! [A,B] :
      ( ~ v1_realset1(A)
     => ( ~ v1_xboole_0(k2_xboole_0(B,A))
        & ~ v1_realset1(k2_xboole_0(B,A)) ) ) )).

fof(fc4_chain_1,axiom,
    ( ~ v1_xboole_0(k1_numbers)
    & ~ v1_realset1(k1_numbers)
    & v1_membered(k1_numbers)
    & v2_membered(k1_numbers) )).

fof(rc3_chain_1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B)
          & ~ v1_realset1(B) ) ) )).

fof(t1_chain_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ~ r1_xreal_0(B,C) ) ) ) ) ) )).

fof(t2_chain_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ? [C] :
              ( m1_subset_1(C,k1_numbers)
              & ~ r1_xreal_0(C,A)
              & ~ r1_xreal_0(C,B) ) ) ) )).

fof(d1_chain_1,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ( v1_xboole_0(A)
      <=> r1_xreal_0(A,np__0) ) ) )).

fof(d2_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_xboole_0(A)
      <=> ~ r1_xreal_0(np__1,A) ) ) )).

fof(d3_chain_1,axiom,(
    ! [A] :
      ( v1_realset1(A)
    <=> ! [B,C] :
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => B = C ) ) )).

fof(t3_chain_1,axiom,(
    $true )).

fof(t4_chain_1,axiom,(
    ! [A,B] :
      ( v1_realset1(k2_tarski(A,B))
    <=> A = B ) )).

fof(t5_chain_1,axiom,(
    ! [A,B] :
      ~ ( v1_realset1(A)
        & ~ v1_realset1(k2_xboole_0(A,k1_tarski(B)))
        & ! [C] : A != k1_tarski(C) ) )).

fof(t6_chain_1,axiom,(
    ! [A] :
      ( k1_card_1(A) = np__2
    <=> ? [B,C] :
          ( r2_hidden(B,A)
          & r2_hidden(C,A)
          & B != C
          & ! [D] :
              ~ ( r2_hidden(D,A)
                & D != B
                & D != C ) ) ) )).

fof(t7_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( v1_abian(A)
            <=> v1_abian(B) )
          <=> v1_abian(k1_nat_1(A,B)) ) ) ) )).

fof(t8_chain_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r1_xboole_0(A,B)
           => ( ( v1_abian(k4_card_1(A))
              <=> v1_abian(k4_card_1(B)) )
            <=> v1_abian(k4_card_1(k2_xboole_0(A,B))) ) ) ) ) )).

fof(t9_chain_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( ( v1_abian(k4_card_1(A))
            <=> v1_abian(k4_card_1(B)) )
          <=> v1_abian(k4_card_1(k5_xboole_0(A,B))) ) ) ) )).

fof(d4_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_finseq_2(B,k1_numbers) )
         => ( B = k1_euclid(A)
          <=> ! [C] :
                ( r2_hidden(C,B)
              <=> ( v1_funct_1(C)
                  & v1_funct_2(C,k2_finseq_1(A),k1_numbers)
                  & m2_relset_1(C,k2_finseq_1(A),k1_numbers) ) ) ) ) ) )).

fof(d5_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers))
            & m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers)) )
         => ( m1_chain_1(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k2_finseq_1(A))
               => ( ~ v1_realset1(k8_funct_2(k2_finseq_1(A),k1_zfmisc_1(k1_numbers),B,C))
                  & v1_finset_1(k8_funct_2(k2_finseq_1(A),k1_zfmisc_1(k1_numbers),B,C)) ) ) ) ) ) )).

fof(t10_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m1_chain_1(C,A)
             => ( r2_hidden(B,k4_card_3(C))
              <=> ! [D] :
                    ( m1_subset_1(D,k2_finseq_1(A))
                   => r2_hidden(k2_chain_1(A,B,D),k3_chain_1(A,C,D)) ) ) ) ) ) )).

fof(t11_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => v1_finset_1(k4_card_3(B)) ) ) )).

fof(t12_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & r2_hidden(B,A)
          & ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r2_hidden(C,A)
               => r1_xreal_0(C,B) ) ) ) ) )).

fof(t13_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & r2_hidden(B,A)
          & ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r2_hidden(C,A)
               => r1_xreal_0(B,C) ) ) ) ) )).

fof(t14_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & ? [C] :
              ( m1_subset_1(C,k1_numbers)
              & r2_hidden(B,A)
              & r2_hidden(C,A)
              & ~ r1_xreal_0(C,B)
              & ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( r2_hidden(D,A)
                      & ~ r1_xreal_0(D,B)
                      & ~ r1_xreal_0(C,D) ) ) ) ) ) )).

fof(t15_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & r2_hidden(B,A)
          & ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( r2_hidden(C,A)
               => r1_xreal_0(C,B) ) ) ) ) )).

fof(t16_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ? [B] :
          ( m1_subset_1(B,k1_numbers)
          & ? [C] :
              ( m1_subset_1(C,k1_numbers)
              & r2_hidden(B,A)
              & r2_hidden(C,A)
              & ~ r1_xreal_0(B,C)
              & ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( r2_hidden(D,A)
                   => ( r1_xreal_0(C,D)
                      & r1_xreal_0(D,B) ) ) ) ) ) ) )).

fof(d6_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k2_zfmisc_1(k1_numbers,k1_numbers))
         => ( m2_chain_1(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & B = k1_domain_1(k1_numbers,k1_numbers,C,D)
                    & r2_hidden(C,A)
                    & r2_hidden(D,A)
                    & ( ( ~ r1_xreal_0(D,C)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( r2_hidden(E,A)
                                & ~ r1_xreal_0(E,C)
                                & ~ r1_xreal_0(D,E) ) ) )
                      | ( ~ r1_xreal_0(C,D)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ( r2_hidden(E,A)
                             => ( r1_xreal_0(E,C)
                                & r1_xreal_0(D,E) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A)
              <=> ( r2_hidden(B,A)
                  & r2_hidden(C,A)
                  & ( ( ~ r1_xreal_0(C,B)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( r2_hidden(D,A)
                              & ~ r1_xreal_0(D,B)
                              & ~ r1_xreal_0(C,D) ) ) )
                    | ( ~ r1_xreal_0(B,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ( r2_hidden(D,A)
                           => ( r1_xreal_0(D,B)
                              & r1_xreal_0(C,D) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( A = k7_domain_1(k1_numbers,B,C)
                       => ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,D,E),A)
                        <=> ( ( D = B
                              & E = C )
                            | ( D = C
                              & E = B ) ) ) ) ) ) ) ) ) )).

fof(t19_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( r2_hidden(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) )).

fof(t20_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( r2_hidden(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) )).

fof(t21_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A)
                      & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,D),A) )
                   => C = D ) ) ) ) ) )).

fof(t22_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,B,C),A)
                      & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,D,C),A) )
                   => B = D ) ) ) ) ) )).

fof(t23_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ( ( m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,C,B),A)
                          & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,E,D),A) )
                       => ( r1_xreal_0(C,B)
                          | r1_xreal_0(E,D)
                          | ( C = E
                            & B = D ) ) ) ) ) ) ) ) )).

fof(t24_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ( r2_hidden(B,k4_chain_1(A,C,D))
                  <=> ~ ( ~ ! [E] :
                              ( m1_subset_1(E,k2_finseq_1(A))
                             => ( r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E))
                                & r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,D,E)) ) )
                        & ! [E] :
                            ( m1_subset_1(E,k2_finseq_1(A))
                           => ~ ( ~ r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,D,E))
                                & ( r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,D,E))
                                  | r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E)) ) ) ) ) ) ) ) ) ) )).

fof(t25_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ( ! [E] :
                        ( m1_subset_1(E,k2_finseq_1(A))
                       => r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) )
                   => ( r2_hidden(D,k4_chain_1(A,B,C))
                    <=> ! [E] :
                          ( m1_subset_1(E,k2_finseq_1(A))
                         => ( r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,D,E))
                            & r1_xreal_0(k2_chain_1(A,D,E),k2_chain_1(A,C,E)) ) ) ) ) ) ) ) ) )).

fof(t26_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ( ~ ! [E] :
                          ( m1_subset_1(E,k2_finseq_1(A))
                         => r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E)) )
                   => ( r2_hidden(D,k4_chain_1(A,C,B))
                    <=> ? [E] :
                          ( m1_subset_1(E,k2_finseq_1(A))
                          & ~ r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E))
                          & ( r1_xreal_0(k2_chain_1(A,D,E),k2_chain_1(A,B,E))
                            | r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,D,E)) ) ) ) ) ) ) ) ) )).

fof(t27_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ( r2_hidden(B,k4_chain_1(A,B,C))
                & r2_hidden(C,k4_chain_1(A,B,C)) ) ) ) ) )).

fof(t28_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => k4_chain_1(A,B,B) = k6_domain_1(k1_euclid(A),B) ) ) )).

fof(t29_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ! [E] :
                      ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                     => ( ! [F] :
                            ( m1_subset_1(F,k2_finseq_1(A))
                           => r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) )
                       => ( r1_tarski(k4_chain_1(A,D,E),k4_chain_1(A,B,C))
                        <=> ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => ( r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,D,F))
                                & r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F))
                                & r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,C,F)) ) ) ) ) ) ) ) ) ) )).

fof(t30_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ! [E] :
                      ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                     => ( ! [F] :
                            ( m1_subset_1(F,k2_finseq_1(A))
                           => ~ r1_xreal_0(k2_chain_1(A,C,F),k2_chain_1(A,B,F)) )
                       => ( r1_tarski(k4_chain_1(A,C,B),k4_chain_1(A,D,E))
                        <=> ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => ( r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,E,F))
                                & ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F))
                                & r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,C,F)) ) ) ) ) ) ) ) ) ) )).

fof(t31_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ! [E] :
                      ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                     => ( ( ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) )
                          & ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => ~ r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,D,F)) ) )
                       => ( r1_tarski(k4_chain_1(A,B,C),k4_chain_1(A,E,D))
                        <=> ~ ! [F] :
                                ( m1_subset_1(F,k2_finseq_1(A))
                               => ( ~ r1_xreal_0(k2_chain_1(A,C,F),k2_chain_1(A,D,F))
                                  & ~ r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,B,F)) ) ) ) ) ) ) ) ) ) )).

fof(t32_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ! [E] :
                      ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                     => ( ( ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) )
                          | ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => ~ r1_xreal_0(k2_chain_1(A,B,F),k2_chain_1(A,C,F)) ) )
                       => ( k4_chain_1(A,B,C) = k4_chain_1(A,D,E)
                        <=> ( B = D
                            & C = E ) ) ) ) ) ) ) ) )).

fof(t33_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ( r1_xreal_0(A,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k1_euclid(B)))
                   => ( r2_hidden(D,k5_chain_1(B,C,A))
                    <=> ? [E] :
                          ( m2_finseq_2(E,k1_numbers,k1_euclid(B))
                          & ? [F] :
                              ( m2_finseq_2(F,k1_numbers,k1_euclid(B))
                              & D = k4_chain_1(B,E,F)
                              & ~ ( ! [G] :
                                      ( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(B)))
                                     => ~ ( k4_card_1(G) = A
                                          & ! [H] :
                                              ( m1_subset_1(H,k2_finseq_1(B))
                                             => ( ( r2_hidden(H,G)
                                                  & ~ r1_xreal_0(k2_chain_1(B,F,H),k2_chain_1(B,E,H))
                                                  & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,H),k2_chain_1(B,F,H)),k3_chain_1(B,C,H)) )
                                                | ( ~ r2_hidden(H,G)
                                                  & k2_chain_1(B,E,H) = k2_chain_1(B,F,H)
                                                  & r2_hidden(k2_chain_1(B,E,H),k3_chain_1(B,C,H)) ) ) ) ) )
                                  & ~ ( A = B
                                      & ! [G] :
                                          ( m1_subset_1(G,k2_finseq_1(B))
                                         => ( ~ r1_xreal_0(k2_chain_1(B,E,G),k2_chain_1(B,F,G))
                                            & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,G),k2_chain_1(B,F,G)),k3_chain_1(B,C,G)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t34_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(B))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(B))
                 => ! [E] :
                      ( m1_chain_1(E,B)
                     => ( r1_xreal_0(A,B)
                       => ( r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
                        <=> ~ ( ! [F] :
                                  ( m1_subset_1(F,k1_zfmisc_1(k2_finseq_1(B)))
                                 => ~ ( k4_card_1(F) = A
                                      & ! [G] :
                                          ( m1_subset_1(G,k2_finseq_1(B))
                                         => ( ( r2_hidden(G,F)
                                              & ~ r1_xreal_0(k2_chain_1(B,D,G),k2_chain_1(B,C,G))
                                              & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,G),k2_chain_1(B,D,G)),k3_chain_1(B,E,G)) )
                                            | ( ~ r2_hidden(G,F)
                                              & k2_chain_1(B,C,G) = k2_chain_1(B,D,G)
                                              & r2_hidden(k2_chain_1(B,C,G),k3_chain_1(B,E,G)) ) ) ) ) )
                              & ~ ( A = B
                                  & ! [F] :
                                      ( m1_subset_1(F,k2_finseq_1(B))
                                     => ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F))
                                        & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,F),k2_chain_1(B,D,F)),k3_chain_1(B,E,F)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t35_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(B))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(B))
                 => ! [E] :
                      ( m1_chain_1(E,B)
                     => ~ ( r1_xreal_0(A,B)
                          & r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
                          & ? [F] :
                              ( m1_subset_1(F,k2_finseq_1(B))
                              & ~ ( ~ r1_xreal_0(k2_chain_1(B,D,F),k2_chain_1(B,C,F))
                                  & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,F),k2_chain_1(B,D,F)),k3_chain_1(B,E,F)) )
                              & ~ ( k2_chain_1(B,C,F) = k2_chain_1(B,D,F)
                                  & r2_hidden(k2_chain_1(B,C,F),k3_chain_1(B,E,F)) ) )
                          & ~ ! [F] :
                                ( m1_subset_1(F,k2_finseq_1(B))
                               => ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F))
                                  & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,F),k2_chain_1(B,D,F)),k3_chain_1(B,E,F)) ) ) ) ) ) ) ) ) )).

fof(t36_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(B))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(B))
                 => ! [E] :
                      ( m1_chain_1(E,B)
                     => ( ( r1_xreal_0(A,B)
                          & r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A)) )
                       => ! [F] :
                            ( m1_subset_1(F,k2_finseq_1(B))
                           => ( r2_hidden(k2_chain_1(B,C,F),k3_chain_1(B,E,F))
                              & r2_hidden(k2_chain_1(B,D,F),k3_chain_1(B,E,F)) ) ) ) ) ) ) ) ) )).

fof(t37_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(B))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(B))
                 => ! [E] :
                      ( m1_chain_1(E,B)
                     => ~ ( r1_xreal_0(A,B)
                          & r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
                          & ~ ! [F] :
                                ( m1_subset_1(F,k2_finseq_1(B))
                               => r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F)) )
                          & ? [F] :
                              ( m1_subset_1(F,k2_finseq_1(B))
                              & r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F)) ) ) ) ) ) ) ) )).

fof(t38_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
             => ( r2_hidden(C,k5_chain_1(A,B,np__0))
              <=> ? [D] :
                    ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                    & C = k4_chain_1(A,D,D)
                    & ! [E] :
                        ( m1_subset_1(E,k2_finseq_1(A))
                       => r2_hidden(k2_chain_1(A,D,E),k3_chain_1(A,B,E)) ) ) ) ) ) ) )).

fof(t39_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m1_chain_1(D,A)
                 => ( r2_hidden(k4_chain_1(A,B,C),k5_chain_1(A,D,np__0))
                  <=> ( B = C
                      & ! [E] :
                          ( m1_subset_1(E,k2_finseq_1(A))
                         => r2_hidden(k2_chain_1(A,B,E),k3_chain_1(A,D,E)) ) ) ) ) ) ) ) )).

fof(t40_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
             => ( r2_hidden(C,k5_chain_1(A,B,A))
              <=> ? [D] :
                    ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                    & ? [E] :
                        ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                        & C = k4_chain_1(A,D,E)
                        & ! [F] :
                            ( m1_subset_1(F,k2_finseq_1(A))
                           => m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,F),k2_chain_1(A,E,F)),k3_chain_1(A,B,F)) )
                        & ( ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => ~ r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,D,F)) )
                          | ! [F] :
                              ( m1_subset_1(F,k2_finseq_1(A))
                             => ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F)) ) ) ) ) ) ) ) ) )).

fof(t41_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m1_chain_1(D,A)
                 => ( r2_hidden(k4_chain_1(A,B,C),k5_chain_1(A,D,A))
                  <=> ( ! [E] :
                          ( m1_subset_1(E,k2_finseq_1(A))
                         => m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,E),k2_chain_1(A,C,E)),k3_chain_1(A,D,E)) )
                      & ( ! [E] :
                            ( m1_subset_1(E,k2_finseq_1(A))
                           => ~ r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E)) )
                        | ! [E] :
                            ( m1_subset_1(E,k2_finseq_1(A))
                           => ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) ) ) ) ) ) ) ) ) )).

fof(t42_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ( B = k1_nat_1(A,np__1)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k1_euclid(B)))
                   => ( r2_hidden(D,k5_chain_1(B,C,A))
                    <=> ? [E] :
                          ( m2_finseq_2(E,k1_numbers,k1_euclid(B))
                          & ? [F] :
                              ( m2_finseq_2(F,k1_numbers,k1_euclid(B))
                              & ? [G] :
                                  ( m1_subset_1(G,k2_finseq_1(B))
                                  & D = k4_chain_1(B,E,F)
                                  & k2_chain_1(B,E,G) = k2_chain_1(B,F,G)
                                  & r2_hidden(k2_chain_1(B,E,G),k3_chain_1(B,C,G))
                                  & ! [H] :
                                      ( m1_subset_1(H,k2_finseq_1(B))
                                     => ( H != G
                                       => ( ~ r1_xreal_0(k2_chain_1(B,F,H),k2_chain_1(B,E,H))
                                          & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,H),k2_chain_1(B,F,H)),k3_chain_1(B,C,H)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t43_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(B))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(B))
                 => ! [E] :
                      ( m1_chain_1(E,B)
                     => ( B = k1_nat_1(A,np__1)
                       => ( r2_hidden(k4_chain_1(B,C,D),k5_chain_1(B,E,A))
                        <=> ? [F] :
                              ( m1_subset_1(F,k2_finseq_1(B))
                              & k2_chain_1(B,C,F) = k2_chain_1(B,D,F)
                              & r2_hidden(k2_chain_1(B,C,F),k3_chain_1(B,E,F))
                              & ! [G] :
                                  ( m1_subset_1(G,k2_finseq_1(B))
                                 => ( G != F
                                   => ( ~ r1_xreal_0(k2_chain_1(B,D,G),k2_chain_1(B,C,G))
                                      & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,C,G),k2_chain_1(B,D,G)),k3_chain_1(B,E,G)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t44_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_euclid(A)))
             => ( r2_hidden(C,k5_chain_1(A,B,np__1))
              <=> ? [D] :
                    ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                    & ? [E] :
                        ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                        & ? [F] :
                            ( m1_subset_1(F,k2_finseq_1(A))
                            & C = k4_chain_1(A,D,E)
                            & ~ ( r1_xreal_0(k2_chain_1(A,E,F),k2_chain_1(A,D,F))
                                & ~ ( A = np__1
                                    & ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F)) ) )
                            & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,F),k2_chain_1(A,E,F)),k3_chain_1(A,B,F))
                            & ! [G] :
                                ( m1_subset_1(G,k2_finseq_1(A))
                               => ( G != F
                                 => ( k2_chain_1(A,D,G) = k2_chain_1(A,E,G)
                                    & r2_hidden(k2_chain_1(A,D,G),k3_chain_1(A,B,G)) ) ) ) ) ) ) ) ) ) ) )).

fof(t45_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m1_chain_1(D,A)
                 => ( r2_hidden(k4_chain_1(A,B,C),k5_chain_1(A,D,np__1))
                  <=> ? [E] :
                        ( m1_subset_1(E,k2_finseq_1(A))
                        & ~ ( r1_xreal_0(k2_chain_1(A,C,E),k2_chain_1(A,B,E))
                            & ~ ( A = np__1
                                & ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) ) )
                        & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,E),k2_chain_1(A,C,E)),k3_chain_1(A,D,E))
                        & ! [F] :
                            ( m1_subset_1(F,k2_finseq_1(A))
                           => ( F != E
                             => ( k2_chain_1(A,B,F) = k2_chain_1(A,C,F)
                                & r2_hidden(k2_chain_1(A,B,F),k3_chain_1(A,D,F)) ) ) ) ) ) ) ) ) ) )).

fof(t46_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(C))
                 => ! [E] :
                      ( m2_finseq_2(E,k1_numbers,k1_euclid(C))
                     => ! [F] :
                          ( m2_finseq_2(F,k1_numbers,k1_euclid(C))
                         => ! [G] :
                              ( m2_finseq_2(G,k1_numbers,k1_euclid(C))
                             => ! [H] :
                                  ( m1_chain_1(H,C)
                                 => ( ( r1_xreal_0(A,C)
                                      & r1_xreal_0(B,C)
                                      & r2_hidden(k4_chain_1(C,D,E),k5_chain_1(C,H,A))
                                      & r2_hidden(k4_chain_1(C,F,G),k5_chain_1(C,H,B))
                                      & r1_tarski(k4_chain_1(C,D,E),k4_chain_1(C,F,G)) )
                                   => ! [I] :
                                        ( m1_subset_1(I,k2_finseq_1(C))
                                       => ~ ( ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,F,I)
                                                & k2_chain_1(C,E,I) = k2_chain_1(C,G,I) )
                                            & ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,F,I)
                                                & k2_chain_1(C,E,I) = k2_chain_1(C,F,I) )
                                            & ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,G,I)
                                                & k2_chain_1(C,E,I) = k2_chain_1(C,G,I) )
                                            & ~ ( r1_xreal_0(k2_chain_1(C,D,I),k2_chain_1(C,E,I))
                                                & ~ r1_xreal_0(k2_chain_1(C,F,I),k2_chain_1(C,G,I))
                                                & r1_xreal_0(k2_chain_1(C,G,I),k2_chain_1(C,D,I))
                                                & r1_xreal_0(k2_chain_1(C,E,I),k2_chain_1(C,F,I)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t47_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(C))
                 => ! [E] :
                      ( m2_finseq_2(E,k1_numbers,k1_euclid(C))
                     => ! [F] :
                          ( m2_finseq_2(F,k1_numbers,k1_euclid(C))
                         => ! [G] :
                              ( m2_finseq_2(G,k1_numbers,k1_euclid(C))
                             => ! [H] :
                                  ( m1_chain_1(H,C)
                                 => ~ ( ~ r1_xreal_0(B,A)
                                      & r1_xreal_0(B,C)
                                      & r2_hidden(k4_chain_1(C,D,E),k5_chain_1(C,H,A))
                                      & r2_hidden(k4_chain_1(C,F,G),k5_chain_1(C,H,B))
                                      & r1_tarski(k4_chain_1(C,D,E),k4_chain_1(C,F,G))
                                      & ! [I] :
                                          ( m1_subset_1(I,k2_finseq_1(C))
                                         => ( ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,F,I)
                                                & k2_chain_1(C,E,I) = k2_chain_1(C,F,I) )
                                            & ~ ( k2_chain_1(C,D,I) = k2_chain_1(C,G,I)
                                                & k2_chain_1(C,E,I) = k2_chain_1(C,G,I) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t48_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                 => ! [E] :
                      ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                     => ! [F] :
                          ( m1_chain_1(F,A)
                         => ! [G] :
                              ( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(A)))
                             => ! [H] :
                                  ( m1_subset_1(H,k1_zfmisc_1(k2_finseq_1(A)))
                                 => ( ( r1_tarski(k4_chain_1(A,B,C),k4_chain_1(A,D,E))
                                      & ! [I] :
                                          ( m1_subset_1(I,k2_finseq_1(A))
                                         => ( ( r2_hidden(I,G)
                                              & ~ r1_xreal_0(k2_chain_1(A,C,I),k2_chain_1(A,B,I))
                                              & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,I),k2_chain_1(A,C,I)),k3_chain_1(A,F,I)) )
                                            | ( ~ r2_hidden(I,G)
                                              & k2_chain_1(A,B,I) = k2_chain_1(A,C,I)
                                              & r2_hidden(k2_chain_1(A,B,I),k3_chain_1(A,F,I)) ) ) )
                                      & ! [I] :
                                          ( m1_subset_1(I,k2_finseq_1(A))
                                         => ( ( r2_hidden(I,H)
                                              & ~ r1_xreal_0(k2_chain_1(A,E,I),k2_chain_1(A,D,I))
                                              & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,I),k2_chain_1(A,E,I)),k3_chain_1(A,F,I)) )
                                            | ( ~ r2_hidden(I,H)
                                              & k2_chain_1(A,D,I) = k2_chain_1(A,E,I)
                                              & r2_hidden(k2_chain_1(A,D,I),k3_chain_1(A,F,I)) ) ) ) )
                                   => ( r1_tarski(G,H)
                                      & ! [I] :
                                          ( m1_subset_1(I,k2_finseq_1(A))
                                         => ( ~ ( ~ r2_hidden(I,G)
                                                & r2_hidden(I,H) )
                                           => ( k2_chain_1(A,B,I) = k2_chain_1(A,D,I)
                                              & k2_chain_1(A,C,I) = k2_chain_1(A,E,I) ) ) )
                                      & ! [I] :
                                          ( m1_subset_1(I,k2_finseq_1(A))
                                         => ~ ( ~ r2_hidden(I,G)
                                              & r2_hidden(I,H)
                                              & ~ ( k2_chain_1(A,B,I) = k2_chain_1(A,D,I)
                                                  & k2_chain_1(A,C,I) = k2_chain_1(A,D,I) )
                                              & ~ ( k2_chain_1(A,B,I) = k2_chain_1(A,E,I)
                                                  & k2_chain_1(A,C,I) = k2_chain_1(A,E,I) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d9_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k6_chain_1(A,B,C) = k1_xboole_0 ) ) ) )).

fof(d10_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => k7_chain_1(A,B) = k5_chain_1(A,B,A) ) ) )).

fof(d11_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,A))
             => ( C = k9_chain_1(A,B)
              <=> ? [D] :
                    ( m2_finseq_2(D,k1_numbers,k1_euclid(A))
                    & ? [E] :
                        ( m2_finseq_2(E,k1_numbers,k1_euclid(A))
                        & C = k4_chain_1(A,D,E)
                        & ! [F] :
                            ( m1_subset_1(F,k2_finseq_1(A))
                           => ( ~ r1_xreal_0(k2_chain_1(A,D,F),k2_chain_1(A,E,F))
                              & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,D,F),k2_chain_1(A,E,F)),k3_chain_1(A,B,F)) ) ) ) ) ) ) ) ) )).

fof(t49_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m1_chain_1(D,A)
                 => ( m2_subset_1(k4_chain_1(A,B,C),k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,D,A))
                   => ( k4_chain_1(A,B,C) = k9_chain_1(A,D)
                    <=> ! [E] :
                          ( m1_subset_1(E,k2_finseq_1(A))
                         => ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E)) ) ) ) ) ) ) ) )).

fof(t50_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => ! [D] :
                  ( m1_chain_1(D,A)
                 => ( k4_chain_1(A,B,C) = k9_chain_1(A,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,k2_finseq_1(A))
                       => ( ~ r1_xreal_0(k2_chain_1(A,B,E),k2_chain_1(A,C,E))
                          & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(A,B,E),k2_chain_1(A,C,E)),k3_chain_1(A,D,E)) ) ) ) ) ) ) ) )).

fof(t51_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
                 => ! [E] :
                      ( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(A,np__1)))
                     => ( r2_hidden(E,k10_chain_1(B,C,A,D))
                      <=> r1_tarski(D,E) ) ) ) ) ) ) )).

fof(d14_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C)))
                     => ( r1_chain_1(A,B,C,D,E)
                      <=> E = k11_chain_1(A,B,C,D) ) ) ) ) ) ) )).

fof(t52_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
                     => ( r2_hidden(D,k11_chain_1(B,C,A,E))
                      <=> ( r1_xreal_0(k1_nat_1(A,np__1),B)
                          & ~ v1_abian(k4_card_1(k5_subset_1(k5_chain_1(B,C,k1_nat_1(A,np__1)),k10_chain_1(B,C,A,D),E))) ) ) ) ) ) ) ) )).

fof(t53_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ( ~ r1_xreal_0(k1_nat_1(A,np__1),B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
                   => k11_chain_1(B,C,A,D) = k6_chain_1(B,C,A) ) ) ) ) ) )).

fof(t54_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ( r1_xreal_0(k1_nat_1(A,np__1),B)
               => ! [D] :
                    ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
                   => ! [E] :
                        ( m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(A,np__1)))
                       => ( r2_hidden(D,k11_chain_1(B,C,A,k6_domain_1(k5_chain_1(B,C,k1_nat_1(A,np__1)),E)))
                        <=> r1_tarski(D,E) ) ) ) ) ) ) ) )).

fof(t55_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ( B = k1_nat_1(A,np__1)
               => ! [D] :
                    ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,A))
                   => k4_card_1(k10_chain_1(B,C,A,D)) = np__2 ) ) ) ) ) )).

fof(t56_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,k1_nat_1(np__0,np__1)))
             => k4_card_1(k11_chain_1(A,B,np__0,k6_domain_1(k5_chain_1(A,B,k1_nat_1(np__0,np__1)),C))) = np__2 ) ) ) )).

fof(t57_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ( k7_chain_1(A,B) = k3_subset_1(k5_chain_1(A,B,A),k6_chain_1(A,B,A))
            & k6_chain_1(A,B,A) = k3_subset_1(k5_chain_1(A,B,A),k7_chain_1(A,B)) ) ) ) )).

fof(t58_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
                 => k8_chain_1(B,C,A,D,k6_chain_1(B,C,A)) = D ) ) ) ) )).

fof(t59_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
                 => k8_chain_1(B,C,A,D,D) = k6_chain_1(B,C,A) ) ) ) ) )).

fof(t60_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k5_chain_1(A,B,A)))
             => k3_subset_1(k5_chain_1(A,B,A),C) = k8_chain_1(A,B,A,C,k7_chain_1(A,B)) ) ) ) )).

fof(t61_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => k11_chain_1(B,C,A,k6_chain_1(B,C,k1_nat_1(A,np__1))) = k6_chain_1(B,C,A) ) ) ) )).

fof(t62_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_chain_1(B,k1_nat_1(A,np__1))
         => k11_chain_1(k1_nat_1(A,np__1),B,A,k7_chain_1(k1_nat_1(A,np__1),B)) = k6_chain_1(k1_nat_1(A,np__1),B,A) ) ) )).

fof(t63_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
                     => k11_chain_1(B,C,A,k8_chain_1(B,C,k1_nat_1(A,np__1),D,E)) = k8_chain_1(B,C,A,k11_chain_1(B,C,A,D),k11_chain_1(B,C,A,E)) ) ) ) ) ) )).

fof(t64_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_chain_1(B,k1_nat_1(A,np__1))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k5_chain_1(k1_nat_1(A,np__1),B,k1_nat_1(A,np__1))))
             => k11_chain_1(k1_nat_1(A,np__1),B,A,k3_subset_1(k5_chain_1(k1_nat_1(A,np__1),B,k1_nat_1(A,np__1)),C)) = k11_chain_1(k1_nat_1(A,np__1),B,A,C) ) ) ) )).

fof(t65_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(k1_nat_1(A,np__1),np__1))))
                 => k11_chain_1(B,C,A,k11_chain_1(B,C,k1_nat_1(A,np__1),D)) = k6_chain_1(B,C,A) ) ) ) ) )).

fof(d15_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
                 => ( m3_chain_1(D,A,B,C)
                  <=> ~ ( ~ ( C = np__0
                            & v1_abian(k4_card_1(D)) )
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ~ ( C = k1_nat_1(E,np__1)
                                & ? [F] :
                                    ( m1_subset_1(F,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(E,np__1))))
                                    & F = D
                                    & k11_chain_1(A,B,E,F) = k6_chain_1(A,B,E) ) ) ) ) ) ) ) ) ) )).

fof(t66_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(A,np__1))))
                 => ( m3_chain_1(D,B,C,k1_nat_1(A,np__1))
                  <=> k11_chain_1(B,C,A,D) = k6_chain_1(B,C,A) ) ) ) ) ) )).

fof(t67_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ( ~ r1_xreal_0(A,B)
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
                   => m3_chain_1(D,B,C,A) ) ) ) ) ) )).

fof(t68_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k5_chain_1(A,B,np__0)))
             => ( m3_chain_1(C,A,B,np__0)
              <=> v1_abian(k4_card_1(C)) ) ) ) ) )).

fof(d16_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m3_chain_1(D,A,B,k1_nat_1(C,np__1))
                 => k11_chain_1(A,B,C,D) = k6_chain_1(A,B,C) ) ) ) ) )).

fof(t69_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m3_chain_1(C,A,B,A)
             => m3_chain_1(k3_subset_1(k5_chain_1(A,B,A),C),A,B,A) ) ) ) )).

fof(d17_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( ~ v3_struct_0(D)
                    & v1_rlvect_1(D)
                    & v3_rlvect_1(D)
                    & v4_rlvect_1(D)
                    & v5_rlvect_1(D)
                    & v6_rlvect_1(D)
                    & l1_rlvect_1(D) )
                 => ( D = k16_chain_1(A,B,C)
                  <=> ( u1_struct_0(D) = k1_chain_1(k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,C))
                      & k1_rlvect_1(D) = k12_chain_1(A,B,C)
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(D))
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(D))
                             => ! [G] :
                                  ( m1_subset_1(G,k1_zfmisc_1(k5_chain_1(A,B,C)))
                                 => ! [H] :
                                      ( m1_subset_1(H,k1_zfmisc_1(k5_chain_1(A,B,C)))
                                     => ( ( E = G
                                          & F = H )
                                       => k4_rlvect_1(D,E,F) = k8_chain_1(A,B,C,G,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t70_chain_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m1_chain_1(C,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(B,C,A)))
                <=> m1_subset_1(D,u1_struct_0(k16_chain_1(B,C,A))) ) ) ) ) )).

fof(d18_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_mod_4(D,k16_chain_1(A,B,k1_nat_1(C,np__1)),k16_chain_1(A,B,C))
                 => ( D = k17_chain_1(A,B,C)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k16_chain_1(A,B,k1_nat_1(C,np__1))))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1))))
                           => ( E = F
                             => k8_funct_2(u1_struct_0(k16_chain_1(A,B,k1_nat_1(C,np__1))),u1_struct_0(k16_chain_1(A,B,C)),D,E) = k15_chain_1(A,B,C,F) ) ) ) ) ) ) ) ) )).

fof(s2_chain_1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s2_chain_1)
         => ( r2_hidden(A,f2_s2_chain_1)
           => p1_s2_chain_1(k6_domain_1(f1_s2_chain_1,A)) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s2_chain_1)
         => ! [B] :
              ( ( ~ v1_xboole_0(B)
                & v1_finset_1(B)
                & m1_subset_1(B,k1_zfmisc_1(f1_s2_chain_1)) )
             => ( ( r2_hidden(A,f2_s2_chain_1)
                  & r1_tarski(B,f2_s2_chain_1)
                  & p1_s2_chain_1(B) )
               => ( r2_hidden(A,B)
                  | p1_s2_chain_1(k4_subset_1(f1_s2_chain_1,B,k6_domain_1(f1_s2_chain_1,A))) ) ) ) ) )
   => p1_s2_chain_1(f2_s2_chain_1) )).

fof(s3_chain_1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s3_chain_1)
         => ! [B] :
              ( m1_subset_1(B,f1_s3_chain_1)
             => ( ( r2_hidden(A,f2_s3_chain_1)
                  & r2_hidden(B,f2_s3_chain_1) )
               => ( A = B
                  | p1_s3_chain_1(k7_domain_1(f1_s3_chain_1,A,B)) ) ) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s3_chain_1)
         => ! [B] :
              ( ( v1_finset_1(B)
                & ~ v1_realset1(B)
                & m1_subset_1(B,k1_zfmisc_1(f1_s3_chain_1)) )
             => ( ( r2_hidden(A,f2_s3_chain_1)
                  & r1_tarski(B,f2_s3_chain_1)
                  & p1_s3_chain_1(B) )
               => ( r2_hidden(A,B)
                  | p1_s3_chain_1(k4_subset_1(f1_s3_chain_1,B,k6_domain_1(f1_s3_chain_1,A))) ) ) ) ) )
   => p1_s3_chain_1(f2_s3_chain_1) )).

fof(s4_chain_1,axiom,
    ( ( p1_s4_chain_1(k6_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1))
      & ! [A] :
          ( m2_subset_1(A,k1_zfmisc_1(k1_euclid(f1_s4_chain_1)),k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1))
         => ( r2_hidden(A,f4_s4_chain_1)
           => p1_s4_chain_1(k6_domain_1(k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1),A)) ) )
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1)))
         => ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(k5_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1)))
             => ( ( r1_tarski(A,f4_s4_chain_1)
                  & r1_tarski(B,f4_s4_chain_1)
                  & p1_s4_chain_1(A)
                  & p1_s4_chain_1(B) )
               => p1_s4_chain_1(k8_chain_1(f1_s4_chain_1,f2_s4_chain_1,f3_s4_chain_1,A,B)) ) ) ) )
   => p1_s4_chain_1(f4_s4_chain_1) )).

fof(dt_m1_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ( v1_funct_1(B)
            & v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers))
            & m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k1_numbers)) ) ) ) )).

fof(existence_m1_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers) )
     => ? [B] : m1_chain_1(B,A) ) )).

fof(dt_m2_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m2_chain_1(B,A)
         => m1_subset_1(B,k2_zfmisc_1(k1_numbers,k1_numbers)) ) ) )).

fof(existence_m2_chain_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & ~ v1_realset1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ? [B] : m2_chain_1(B,A) ) )).

fof(dt_m3_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => ! [D] :
          ( m3_chain_1(D,A,B,C)
         => m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C))) ) ) )).

fof(existence_m3_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => ? [D] : m3_chain_1(D,A,B,C) ) )).

fof(dt_k1_chain_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => m1_subset_1(k1_chain_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(redefinition_k1_chain_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k1_chain_1(A,B) = k1_zfmisc_1(B) ) )).

fof(dt_k2_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k2_finseq_1(A)) )
     => m1_subset_1(k2_chain_1(A,B,C),k1_numbers) ) )).

fof(redefinition_k2_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k2_finseq_1(A)) )
     => k2_chain_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k3_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k2_finseq_1(A)) )
     => ( v1_finset_1(k3_chain_1(A,B,C))
        & ~ v1_realset1(k3_chain_1(A,B,C))
        & m1_subset_1(k3_chain_1(A,B,C),k1_zfmisc_1(k1_numbers)) ) ) )).

fof(redefinition_k3_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k2_finseq_1(A)) )
     => k3_chain_1(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k4_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k1_euclid(A))
        & m1_subset_1(C,k1_euclid(A)) )
     => ( ~ v1_xboole_0(k4_chain_1(A,B,C))
        & m1_subset_1(k4_chain_1(A,B,C),k1_zfmisc_1(k1_euclid(A))) ) ) )).

fof(dt_k5_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => ( ~ v1_xboole_0(k5_chain_1(A,B,C))
        & v1_finset_1(k5_chain_1(A,B,C))
        & m1_subset_1(k5_chain_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(k1_euclid(A)))) ) ) )).

fof(dt_k6_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => m1_subset_1(k6_chain_1(A,B,C),k1_zfmisc_1(k5_chain_1(A,B,C))) ) )).

fof(dt_k7_chain_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A) )
     => m1_subset_1(k7_chain_1(A,B),k1_zfmisc_1(k5_chain_1(A,B,A))) ) )).

fof(dt_k8_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
        & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) )
     => m1_subset_1(k8_chain_1(A,B,C,D,E),k1_zfmisc_1(k5_chain_1(A,B,C))) ) )).

fof(commutativity_k8_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
        & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) )
     => k8_chain_1(A,B,C,D,E) = k8_chain_1(A,B,C,E,D) ) )).

fof(redefinition_k8_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,C)))
        & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(A,B,C))) )
     => k8_chain_1(A,B,C,D,E) = k5_xboole_0(D,E) ) )).

fof(dt_k9_chain_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A) )
     => m2_subset_1(k9_chain_1(A,B),k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,A)) ) )).

fof(dt_k10_chain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k5_chain_1(A,B,C)) )
     => m1_subset_1(k10_chain_1(A,B,C,D),k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) ) )).

fof(dt_k11_chain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) )
     => m1_subset_1(k11_chain_1(A,B,C,D),k1_zfmisc_1(k5_chain_1(A,B,C))) ) )).

fof(dt_k12_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => m3_chain_1(k12_chain_1(A,B,C),A,B,C) ) )).

fof(redefinition_k12_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => k12_chain_1(A,B,C) = k6_chain_1(A,B,C) ) )).

fof(dt_k13_chain_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A) )
     => m3_chain_1(k13_chain_1(A,B),A,B,A) ) )).

fof(redefinition_k13_chain_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A) )
     => k13_chain_1(A,B) = k7_chain_1(A,B) ) )).

fof(dt_k14_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m3_chain_1(D,A,B,C)
        & m3_chain_1(E,A,B,C) )
     => m3_chain_1(k14_chain_1(A,B,C,D,E),A,B,C) ) )).

fof(commutativity_k14_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m3_chain_1(D,A,B,C)
        & m3_chain_1(E,A,B,C) )
     => k14_chain_1(A,B,C,D,E) = k14_chain_1(A,B,C,E,D) ) )).

fof(redefinition_k14_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m3_chain_1(D,A,B,C)
        & m3_chain_1(E,A,B,C) )
     => k14_chain_1(A,B,C,D,E) = k5_xboole_0(D,E) ) )).

fof(dt_k15_chain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) )
     => m3_chain_1(k15_chain_1(A,B,C,D),A,B,C) ) )).

fof(redefinition_k15_chain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1)))) )
     => k15_chain_1(A,B,C,D) = k11_chain_1(A,B,C,D) ) )).

fof(dt_k16_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => ( ~ v3_struct_0(k16_chain_1(A,B,C))
        & v1_rlvect_1(k16_chain_1(A,B,C))
        & v3_rlvect_1(k16_chain_1(A,B,C))
        & v4_rlvect_1(k16_chain_1(A,B,C))
        & v5_rlvect_1(k16_chain_1(A,B,C))
        & v6_rlvect_1(k16_chain_1(A,B,C))
        & l1_rlvect_1(k16_chain_1(A,B,C)) ) ) )).

fof(dt_k17_chain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_chain_1(B,A)
        & m1_subset_1(C,k5_numbers) )
     => m1_mod_4(k17_chain_1(A,B,C),k16_chain_1(A,B,k1_nat_1(C,np__1)),k16_chain_1(A,B,C)) ) )).

fof(d7_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k1_numbers,k1_euclid(A))
         => ! [C] :
              ( m2_finseq_2(C,k1_numbers,k1_euclid(A))
             => k4_chain_1(A,B,C) = a_3_0_chain_1(A,B,C) ) ) ) )).

fof(d8_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(C,A)
               => k5_chain_1(A,B,C) = a_3_1_chain_1(A,B,C) ) ) ) ) )).

fof(d12_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_zfmisc_1(k1_euclid(A)),k5_chain_1(A,B,C))
                 => k10_chain_1(A,B,C,D) = a_4_0_chain_1(A,B,C,D) ) ) ) ) )).

fof(d13_chain_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m1_chain_1(B,A)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k5_chain_1(A,B,k1_nat_1(C,np__1))))
                 => k11_chain_1(A,B,C,D) = a_4_1_chain_1(A,B,C,D) ) ) ) ) )).

fof(s1_chain_1,axiom,(
    r1_tarski(a_0_0_chain_1,f1_s1_chain_1) )).

fof(fraenkel_a_3_0_chain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers)
        & m2_finseq_2(C,k1_numbers,k1_euclid(B))
        & m2_finseq_2(D,k1_numbers,k1_euclid(B)) )
     => ( r2_hidden(A,a_3_0_chain_1(B,C,D))
      <=> ? [E] :
            ( m2_finseq_2(E,k1_numbers,k1_euclid(B))
            & A = E
            & ~ ( ~ ! [F] :
                      ( m1_subset_1(F,k2_finseq_1(B))
                     => ( r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,E,F))
                        & r1_xreal_0(k2_chain_1(B,E,F),k2_chain_1(B,D,F)) ) )
                & ! [F] :
                    ( m1_subset_1(F,k2_finseq_1(B))
                   => ~ ( ~ r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,D,F))
                        & ( r1_xreal_0(k2_chain_1(B,E,F),k2_chain_1(B,D,F))
                          | r1_xreal_0(k2_chain_1(B,C,F),k2_chain_1(B,E,F)) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_1_chain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_chain_1(C,B)
        & m2_subset_1(D,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_3_1_chain_1(B,C,D))
      <=> ? [E,F] :
            ( m2_finseq_2(E,k1_numbers,k1_euclid(B))
            & m2_finseq_2(F,k1_numbers,k1_euclid(B))
            & A = k4_chain_1(B,E,F)
            & ~ ( ! [G] :
                    ( m1_subset_1(G,k1_zfmisc_1(k2_finseq_1(B)))
                   => ~ ( k4_card_1(G) = D
                        & ! [H] :
                            ( m1_subset_1(H,k2_finseq_1(B))
                           => ( ( r2_hidden(H,G)
                                & ~ r1_xreal_0(k2_chain_1(B,F,H),k2_chain_1(B,E,H))
                                & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,H),k2_chain_1(B,F,H)),k3_chain_1(B,C,H)) )
                              | ( ~ r2_hidden(H,G)
                                & k2_chain_1(B,E,H) = k2_chain_1(B,F,H)
                                & r2_hidden(k2_chain_1(B,E,H),k3_chain_1(B,C,H)) ) ) ) ) )
                & ~ ( D = B
                    & ! [G] :
                        ( m1_subset_1(G,k2_finseq_1(B))
                       => ( ~ r1_xreal_0(k2_chain_1(B,E,G),k2_chain_1(B,F,G))
                          & m2_chain_1(k1_domain_1(k1_numbers,k1_numbers,k2_chain_1(B,E,G),k2_chain_1(B,F,G)),k3_chain_1(B,C,G)) ) ) ) ) ) ) ) )).

fof(fraenkel_a_4_0_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_chain_1(C,B)
        & m2_subset_1(D,k1_numbers,k5_numbers)
        & m2_subset_1(E,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,D)) )
     => ( r2_hidden(A,a_4_0_chain_1(B,C,D,E))
      <=> ? [F] :
            ( m2_subset_1(F,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,k1_nat_1(D,np__1)))
            & A = F
            & r1_tarski(E,F) ) ) ) )).

fof(fraenkel_a_4_1_chain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & m2_subset_1(B,k1_numbers,k5_numbers)
        & m1_chain_1(C,B)
        & m2_subset_1(D,k1_numbers,k5_numbers)
        & m1_subset_1(E,k1_zfmisc_1(k5_chain_1(B,C,k1_nat_1(D,np__1)))) )
     => ( r2_hidden(A,a_4_1_chain_1(B,C,D,E))
      <=> ? [F] :
            ( m2_subset_1(F,k1_zfmisc_1(k1_euclid(B)),k5_chain_1(B,C,D))
            & A = F
            & r1_xreal_0(k1_nat_1(D,np__1),B)
            & ~ v1_abian(k4_card_1(k5_subset_1(k5_chain_1(B,C,k1_nat_1(D,np__1)),k10_chain_1(B,C,D,F),E))) ) ) ) )).

fof(fraenkel_a_0_0_chain_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_chain_1)
    <=> ? [B,C] :
          ( m1_subset_1(B,f2_s1_chain_1)
          & m1_subset_1(C,f2_s1_chain_1)
          & A = f3_s1_chain_1(B,C)
          & p1_s1_chain_1(B,C) ) ) )).
%------------------------------------------------------------------------------