SET007 Axioms: SET007+859.ax


%------------------------------------------------------------------------------
% File     : SET007+859 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Substitution in First-Order Formulas: Elementary Properties
% 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    : substut1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  146 (  10 unit)
%            Number of atoms       :  733 ( 163 equality)
%            Maximal formula depth :   25 (   6 average)
%            Number of connectives :  628 (  41 ~  ;   0  |; 230  &)
%                                         (  27 <=>; 330 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   35 (   1 propositional; 0-3 arity)
%            Number of functors    :  111 (  24 constant; 0-4 arity)
%            Number of variables   :  307 (   0 singleton; 269 !;  38 ?)
%            Maximal term depth    :    8 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_substut1,axiom,(
    ~ v1_xboole_0(k1_substut1) )).

fof(fc2_substut1,axiom,
    ( ~ v1_xboole_0(k1_substut1)
    & v1_fraenkel(k1_substut1) )).

fof(rc1_substut1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_substut1)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A) ) )).

fof(fc3_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => v1_finset_1(k9_substut1(A)) ) )).

fof(rc2_substut1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_substut1(A) ) )).

fof(fc4_substut1,axiom,
    ( ~ v1_xboole_0(k16_substut1)
    & v1_substut1(k16_substut1) )).

fof(fc5_substut1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k7_qc_lang1(A))
        & m1_qc_lang1(C,A)
        & m1_subset_1(D,k1_substut1) )
     => v4_substut1(k17_substut1(B,C,D)) ) )).

fof(fc6_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => v5_substut1(k20_substut1(A)) ) )).

fof(fc7_substut1,axiom,(
    ~ v1_xboole_0(k38_substut1) )).

fof(d1_substut1,axiom,(
    k1_substut1 = k4_partfun1(k2_qc_lang1,k2_qc_lang1) )).

fof(d2_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_substut1)
     => k2_substut1(A) = A ) )).

fof(t1_substut1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_substut1)
     => ( r2_hidden(A,k1_relat_1(B))
       => r2_hidden(k1_funct_1(B,A),k2_qc_lang1) ) ) )).

fof(d3_substut1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k1_substut1)
         => ! [C] :
              ( m2_finseq_1(C,k1_qc_lang1)
             => ( C = k3_substut1(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k3_finseq_1(A)) )
                       => ( ( r2_hidden(k1_funct_1(A,D),k1_relat_1(B))
                           => k1_funct_1(C,D) = k1_funct_1(B,k1_funct_1(A,D)) )
                          & ( ~ r2_hidden(k1_funct_1(A,D),k1_relat_1(B))
                           => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) ) )).

fof(d4_substut1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_qc_lang1)
     => k4_substut1(A) = A ) )).

fof(d5_substut1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k2_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k1_substut1)
         => k5_substut1(A,B) = k3_substut1(k4_substut1(A),B) ) ) )).

fof(d8_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k2_qc_lang1))
         => ( B = k9_substut1(A)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
                & m2_relset_1(C,k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1))
                & B = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,A)
                & ! [D] :
                    ( m1_subset_1(D,k8_qc_lang1)
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(k2_qc_lang1))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(k2_qc_lang1))
                           => ( ( D = k9_cqc_lang
                               => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k1_subset_1(k2_qc_lang1) )
                              & ( v2_qc_lang1(D)
                               => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k8_substut1(k16_qc_lang1(D)) )
                              & ( ( v3_qc_lang1(D)
                                  & E = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k17_qc_lang1(D)) )
                               => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = E )
                              & ( ( v4_qc_lang1(D)
                                  & E = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k18_qc_lang1(D))
                                  & F = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k19_qc_lang1(D)) )
                               => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k4_subset_1(k2_qc_lang1,E,F) )
                              & ( ( v5_qc_lang1(D)
                                  & E = k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,k21_qc_lang1(D)) )
                               => k8_funct_2(k8_qc_lang1,k1_zfmisc_1(k2_qc_lang1),C,D) = k4_subset_1(k2_qc_lang1,E,k6_domain_1(k2_qc_lang1,k20_qc_lang1(D))) ) ) ) ) ) ) ) ) ) )).

fof(t2_substut1,axiom,(
    k9_substut1(k9_cqc_lang) = k1_xboole_0 )).

fof(t3_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v2_qc_lang1(A)
       => k9_substut1(A) = k8_substut1(k16_qc_lang1(A)) ) ) )).

fof(t4_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v3_qc_lang1(A)
       => k9_substut1(A) = k9_substut1(k17_qc_lang1(A)) ) ) )).

fof(t5_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v4_qc_lang1(A)
       => k9_substut1(A) = k4_subset_1(k2_qc_lang1,k9_substut1(k18_qc_lang1(A)),k9_substut1(k19_qc_lang1(A))) ) ) )).

fof(t6_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v5_qc_lang1(A)
       => k9_substut1(A) = k4_subset_1(k2_qc_lang1,k9_substut1(k21_qc_lang1(A)),k6_domain_1(k2_qc_lang1,k20_qc_lang1(A))) ) ) )).

fof(d11_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] :
          ( ( v1_finset_1(B)
            & m1_subset_1(B,k1_substut1) )
         => k12_substut1(A,B) = k4_xboole_0(k5_numbers,k4_subset_1(k5_numbers,k10_substut1(A),k11_substut1(B))) ) ) )).

fof(d12_substut1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1) )
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => k13_substut1(A,B) = k10_cqc_sim1(k12_substut1(B,A)) ) ) )).

fof(d13_substut1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => ! [C] :
              ( ( v1_finset_1(C)
                & m1_subset_1(C,k1_substut1) )
             => ( ? [D] :
                    ( m1_subset_1(D,k1_substut1)
                    & C = k7_substut1(A,k14_qc_lang1(A,B),D) )
               => ( ( r2_hidden(A,k2_relat_1(C))
                   => k14_substut1(A,B,C) = k2_xboole_0(C,k6_domain_1(k2_zfmisc_1(k2_qc_lang1,k2_qc_lang1),k1_domain_1(k2_qc_lang1,k2_qc_lang1,A,k2_qc_lang3(k13_substut1(C,B))))) )
                  & ( ~ r2_hidden(A,k2_relat_1(C))
                   => k14_substut1(A,B,C) = k2_xboole_0(C,k6_domain_1(k2_zfmisc_1(k2_qc_lang1,k2_qc_lang1),k1_domain_1(k2_qc_lang1,k2_qc_lang1,A,A))) ) ) ) ) ) ) )).

fof(d14_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k1_substut1)
         => ! [C] :
              ( r1_substut1(A,B,C)
            <=> ( ( v5_qc_lang1(A)
                 => C = k14_substut1(k20_qc_lang1(A),k21_qc_lang1(A),k7_substut1(k20_qc_lang1(A),A,B)) )
                & ( ~ v5_qc_lang1(A)
                 => C = k1_xboole_0 ) ) ) ) ) )).

fof(d15_substut1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( A = k15_substut1
      <=> ! [B] :
            ( r2_hidden(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k8_qc_lang1)
                & ? [D] :
                    ( m1_subset_1(D,k1_substut1)
                    & ? [E] :
                        ( B = k4_tarski(k1_domain_1(k8_qc_lang1,k1_substut1,C,D),E)
                        & r1_substut1(C,D,E) ) ) ) ) ) ) )).

fof(t7_substut1,axiom,
    ( m1_subset_1(k2_zfmisc_1(k8_qc_lang1,k1_substut1),k1_zfmisc_1(k2_zfmisc_1(k13_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)),k1_substut1)))
    & ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ! [B] :
            ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
           => ! [C] :
                ( m1_qc_lang1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,k1_substut1)
                   => r2_hidden(k4_tarski(k7_finseq_1(k12_finseq_1(k7_qc_lang1(A),B),C),D),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) ) )
    & ! [A] :
        ( m1_subset_1(A,k1_substut1)
       => r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k2_zfmisc_1(k5_numbers,k5_numbers))),k1_substut1,k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__0,np__0)),A),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) )
    & ! [A] :
        ( m2_finseq_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
       => ! [B] :
            ( m1_subset_1(B,k1_substut1)
           => ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,A,B),k2_zfmisc_1(k8_qc_lang1,k1_substut1))
             => r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__1,np__0)),A),B),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) )
    & ! [A] :
        ( m2_finseq_1(A,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
       => ! [B] :
            ( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
           => ! [C] :
                ( m1_subset_1(C,k1_substut1)
               => ( ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,A,C),k2_zfmisc_1(k8_qc_lang1,k1_substut1))
                    & r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,B,C),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) )
                 => r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__2,np__0)),A),B),C),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) ) )
    & ! [A] :
        ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
       => ! [B] :
            ( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
           => ! [C] :
                ( m1_subset_1(C,k1_substut1)
               => ( r2_hidden(k4_tarski(B,k1_funct_1(k15_substut1,k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,A)),B),C))),k2_zfmisc_1(k8_qc_lang1,k1_substut1))
                 => r2_hidden(k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,A)),B),C),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) ) ) ) )).

fof(d16_substut1,axiom,(
    ! [A] :
      ( v1_substut1(A)
    <=> ( m1_subset_1(A,k1_zfmisc_1(k2_zfmisc_1(k13_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)),k1_substut1)))
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
               => ! [D] :
                    ( m1_qc_lang1(D,B)
                   => ! [E] :
                        ( m1_subset_1(E,k1_substut1)
                       => r2_hidden(k4_tarski(k7_finseq_1(k12_finseq_1(k7_qc_lang1(B),C),D),E),A) ) ) ) )
        & ! [B] :
            ( m1_subset_1(B,k1_substut1)
           => r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k2_zfmisc_1(k5_numbers,k5_numbers))),k1_substut1,k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__0,np__0)),B),A) )
        & ! [B] :
            ( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
           => ! [C] :
                ( m1_subset_1(C,k1_substut1)
               => ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,B,C),A)
                 => r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__1,np__0)),B),C),A) ) ) )
        & ! [B] :
            ( m2_finseq_1(B,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
           => ! [C] :
                ( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
               => ! [D] :
                    ( m1_subset_1(D,k1_substut1)
                   => ( ( r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,B,D),A)
                        & r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,C,D),A) )
                     => r2_hidden(k1_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))),k1_substut1,k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k8_finseq_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__2,np__0)),B),C),D),A) ) ) ) )
        & ! [B] :
            ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
           => ! [C] :
                ( m2_finseq_1(C,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
               => ! [D] :
                    ( m1_subset_1(D,k1_substut1)
                   => ( r2_hidden(k4_tarski(C,k1_funct_1(k15_substut1,k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,B)),C),D))),A)
                     => r2_hidden(k4_tarski(k7_finseq_1(k7_finseq_1(k12_finseq_1(k2_zfmisc_1(k5_numbers,k5_numbers),k1_domain_1(k5_numbers,k5_numbers,np__3,np__0)),k12_finseq_1(k2_qc_lang1,B)),C),D),A) ) ) ) ) ) ) )).

fof(d17_substut1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( A = k16_substut1
      <=> ( v1_substut1(A)
          & ! [B] :
              ( ~ v1_xboole_0(B)
             => ( v1_substut1(B)
               => r1_tarski(A,B) ) ) ) ) ) )).

fof(t8_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ? [B] :
          ( m1_subset_1(B,k8_qc_lang1)
          & ? [C] :
              ( m1_subset_1(C,k1_substut1)
              & A = k1_domain_1(k8_qc_lang1,k1_substut1,B,C) ) ) ) )).

fof(d18_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_qc_lang1)
     => ! [B] :
          ( m2_finseq_1(B,k1_qc_lang1)
         => ! [C] :
              ( m1_subset_1(C,k1_substut1)
             => ( k6_qc_lang1(A) = k3_finseq_1(B)
               => k17_substut1(A,B,C) = k1_domain_1(k8_qc_lang1,k1_substut1,k9_qc_lang1(A,B),C) ) ) ) ) )).

fof(t9_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_substut1)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
             => ! [D] :
                  ( m1_qc_lang1(D,B)
                 => k17_substut1(C,D,A) = k1_domain_1(k8_qc_lang1,k1_substut1,k9_qc_lang1(C,D),A) ) ) ) ) )).

fof(d19_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v2_substut1(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_substut1)
            & A = k1_domain_1(k7_cqc_lang,k1_substut1,k9_cqc_lang,B) ) ) ) )).

fof(t10_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => A = k1_domain_1(k8_qc_lang1,k1_substut1,k18_substut1(A),k19_substut1(A)) ) )).

fof(d20_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => k20_substut1(A) = k1_domain_1(k8_qc_lang1,k1_substut1,k12_qc_lang1(k18_substut1(A)),k19_substut1(A)) ) )).

fof(d21_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( k19_substut1(A) = k19_substut1(B)
           => k21_substut1(A,B) = k1_domain_1(k8_qc_lang1,k1_substut1,k13_qc_lang1(k18_substut1(A),k18_substut1(B)),k19_substut1(A)) ) ) ) )).

fof(d22_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ( v3_substut1(A)
      <=> ? [B] :
            ( m1_subset_1(B,k1_substut1)
            & k19_substut1(k22_substut1(A)) = k1_funct_1(k15_substut1,k1_domain_1(k8_qc_lang1,k1_substut1,k14_qc_lang1(k23_substut1(A),k18_substut1(k22_substut1(A))),B)) ) ) ) )).

fof(d23_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ( v3_substut1(A)
       => ! [B] :
            ( m1_subset_1(B,k1_substut1)
           => ( m1_substut1(B,A)
            <=> k19_substut1(k22_substut1(A)) = k1_funct_1(k15_substut1,k1_domain_1(k8_qc_lang1,k1_substut1,k14_qc_lang1(k23_substut1(A),k18_substut1(k22_substut1(A))),B)) ) ) ) ) )).

fof(d24_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ! [B] :
          ( m1_substut1(B,A)
         => ( v3_substut1(A)
           => k24_substut1(A,B) = k1_domain_1(k8_qc_lang1,k1_substut1,k14_qc_lang1(k23_substut1(A),k18_substut1(k22_substut1(A))),B) ) ) ) )).

fof(d25_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v4_substut1(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ? [C] :
                ( m2_subset_1(C,k5_qc_lang1,k7_qc_lang1(B))
                & ? [D] :
                    ( m1_qc_lang1(D,B)
                    & ? [E] :
                        ( m1_subset_1(E,k1_substut1)
                        & A = k17_substut1(C,D,E) ) ) ) ) ) ) )).

fof(t11_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v4_substut1(A)
       => v2_qc_lang1(k18_substut1(A)) ) ) )).

fof(d26_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v5_substut1(A)
      <=> ? [B] :
            ( m1_subset_1(B,k16_substut1)
            & A = k20_substut1(B) ) ) ) )).

fof(d27_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v6_substut1(A)
      <=> ? [B] :
            ( m1_subset_1(B,k16_substut1)
            & ? [C] :
                ( m1_subset_1(C,k16_substut1)
                & A = k21_substut1(B,C)
                & k19_substut1(B) = k19_substut1(C) ) ) ) ) )).

fof(d28_substut1,axiom,(
    ! [A] :
      ( v7_substut1(A)
    <=> ? [B] :
          ( m1_subset_1(B,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
          & ? [C] :
              ( m1_substut1(C,B)
              & A = k24_substut1(B,C)
              & v3_substut1(B) ) ) ) )).

fof(t12_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ~ ( ~ v2_substut1(A)
          & ~ v4_substut1(A)
          & ~ v5_substut1(A)
          & ~ v6_substut1(A)
          & ~ v7_substut1(A) ) ) )).

fof(d29_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v4_substut1(A)
       => ! [B] :
            ( m2_finseq_1(B,k1_qc_lang1)
           => ( B = k26_substut1(A)
            <=> ? [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                  & ? [D] :
                      ( m2_subset_1(D,k5_qc_lang1,k7_qc_lang1(C))
                      & ? [E] :
                          ( m1_qc_lang1(E,C)
                          & ? [F] :
                              ( m1_subset_1(F,k1_substut1)
                              & B = E
                              & A = k17_substut1(D,E,F) ) ) ) ) ) ) ) ) )).

fof(d30_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v5_substut1(A)
       => ! [B] :
            ( m1_subset_1(B,k16_substut1)
           => ( B = k27_substut1(A)
            <=> A = k20_substut1(B) ) ) ) ) )).

fof(d31_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v6_substut1(A)
       => ! [B] :
            ( m1_subset_1(B,k16_substut1)
           => ( B = k28_substut1(A)
            <=> ? [C] :
                  ( m1_subset_1(C,k16_substut1)
                  & A = k21_substut1(B,C)
                  & k19_substut1(B) = k19_substut1(C) ) ) ) ) ) )).

fof(d32_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v6_substut1(A)
       => ! [B] :
            ( m1_subset_1(B,k16_substut1)
           => ( B = k29_substut1(A)
            <=> ? [C] :
                  ( m1_subset_1(C,k16_substut1)
                  & A = k21_substut1(C,B)
                  & k19_substut1(C) = k19_substut1(B) ) ) ) ) ) )).

fof(d33_substut1,axiom,(
    ! [A] :
      ( v7_substut1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
         => ( B = k30_substut1(A)
          <=> ? [C] :
                ( m1_subset_1(C,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
                & ? [D] :
                    ( m1_substut1(D,C)
                    & A = k24_substut1(C,D)
                    & k23_substut1(C) = B
                    & v3_substut1(C) ) ) ) ) ) )).

fof(d34_substut1,axiom,(
    ! [A] :
      ( v7_substut1(A)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( B = k31_substut1(A)
          <=> ? [C] :
                ( m1_subset_1(C,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
                & ? [D] :
                    ( m1_substut1(D,C)
                    & A = k24_substut1(C,D)
                    & k22_substut1(C) = B
                    & v3_substut1(C) ) ) ) ) ) )).

fof(t13_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( k19_substut1(A) = k19_substut1(B)
           => v6_substut1(k21_substut1(A,B)) ) ) ) )).

fof(t14_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ! [B] :
          ( m1_substut1(B,A)
         => ( v3_substut1(A)
           => v7_substut1(k24_substut1(A,B)) ) ) ) )).

fof(t15_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( k20_substut1(A) = k20_substut1(B)
           => A = B ) ) ) )).

fof(t16_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => k27_substut1(k20_substut1(A)) = A ) )).

fof(t17_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ! [C] :
              ( m1_subset_1(C,k16_substut1)
             => ! [D] :
                  ( m1_subset_1(D,k16_substut1)
                 => ( ( k19_substut1(A) = k19_substut1(B)
                      & k19_substut1(C) = k19_substut1(D)
                      & k21_substut1(A,B) = k21_substut1(C,D) )
                   => ( A = C
                      & B = D ) ) ) ) ) ) )).

fof(t18_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( k19_substut1(A) = k19_substut1(B)
           => k28_substut1(k21_substut1(A,B)) = A ) ) ) )).

fof(t19_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( k19_substut1(A) = k19_substut1(B)
           => k29_substut1(k21_substut1(A,B)) = B ) ) ) )).

fof(t20_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ! [B] :
          ( m1_subset_1(B,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
         => ! [C] :
              ( m1_substut1(C,A)
             => ! [D] :
                  ( m1_substut1(D,B)
                 => ( ( v3_substut1(A)
                      & v3_substut1(B)
                      & k24_substut1(A,C) = k24_substut1(B,D) )
                   => A = B ) ) ) ) ) )).

fof(t21_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ! [B] :
          ( m1_substut1(B,A)
         => ( v3_substut1(A)
           => k31_substut1(k24_substut1(A,B)) = k22_substut1(A) ) ) ) )).

fof(t22_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ~ ( v5_substut1(A)
          & r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k27_substut1(A))))) ) ) )).

fof(t23_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v6_substut1(A)
       => ( ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k28_substut1(A)))))
          & ~ r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k29_substut1(A))))) ) ) ) )).

fof(t24_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ~ ( v7_substut1(A)
          & r1_xreal_0(k3_finseq_1(k10_qc_lang1(k18_substut1(A))),k3_finseq_1(k10_qc_lang1(k18_substut1(k31_substut1(A))))) ) ) )).

fof(t25_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( ( v2_substut1(A)
         => k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__0 )
        & ~ ( v4_substut1(A)
            & ! [B] :
                ( m2_subset_1(B,k1_numbers,k5_numbers)
               => ~ m2_subset_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1),k5_qc_lang1,k7_qc_lang1(B)) ) )
        & ( v5_substut1(A)
         => k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__1 )
        & ( v6_substut1(A)
         => k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__2 )
        & ( v7_substut1(A)
         => k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) = np__3 ) ) ) )).

fof(t26_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v4_substut1(A)
       => ( k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__0
          & k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__1
          & k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__2
          & k1_mcart_1(k1_funct_1(k10_qc_lang1(k18_substut1(A)),np__1)) != np__3 ) ) ) )).

fof(t27_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( ~ ( v4_substut1(A)
            & v5_substut1(A) )
        & ~ ( v4_substut1(A)
            & v6_substut1(A) )
        & ~ ( v4_substut1(A)
            & v7_substut1(A) )
        & ~ ( v5_substut1(A)
            & v6_substut1(A) )
        & ~ ( v5_substut1(A)
            & v7_substut1(A) )
        & ~ ( v6_substut1(A)
            & v7_substut1(A) )
        & ~ ( v2_substut1(A)
            & v4_substut1(A) )
        & ~ ( v2_substut1(A)
            & v5_substut1(A) )
        & ~ ( v2_substut1(A)
            & v6_substut1(A) )
        & ~ ( v2_substut1(A)
            & v7_substut1(A) ) ) ) )).

fof(d35_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => k32_substut1(A) = A ) )).

fof(d36_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => ( ( r2_hidden(k20_qc_lang1(k33_substut1(A)),k2_relat_1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A))))
         => k35_substut1(A) = k2_qc_lang3(k13_substut1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A)),k21_qc_lang1(k33_substut1(A)))) )
        & ( ~ r2_hidden(k20_qc_lang1(k33_substut1(A)),k2_relat_1(k7_substut1(k20_qc_lang1(k33_substut1(A)),k33_substut1(A),k34_substut1(A))))
         => k35_substut1(A) = k20_qc_lang1(k33_substut1(A)) ) ) ) )).

fof(d37_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => k36_substut1(A,B) = k14_qc_lang1(k35_substut1(k32_substut1(A)),B) ) ) )).

fof(d38_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => ( B = k37_substut1(A)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k16_substut1,k8_qc_lang1)
                & m2_relset_1(C,k16_substut1,k8_qc_lang1)
                & B = k8_funct_2(k16_substut1,k8_qc_lang1,C,A)
                & ! [D] :
                    ( m1_subset_1(D,k16_substut1)
                   => ( ( v2_substut1(D)
                       => k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k9_cqc_lang )
                      & ( v4_substut1(D)
                       => k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k9_qc_lang1(k15_qc_lang1(k18_substut1(D)),k3_substut1(k26_substut1(D),k19_substut1(D))) )
                      & ( v5_substut1(D)
                       => k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k12_qc_lang1(k8_funct_2(k16_substut1,k8_qc_lang1,C,k27_substut1(D))) )
                      & ( v6_substut1(D)
                       => k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k13_qc_lang1(k8_funct_2(k16_substut1,k8_qc_lang1,C,k28_substut1(D)),k8_funct_2(k16_substut1,k8_qc_lang1,C,k29_substut1(D))) )
                      & ( v7_substut1(D)
                       => k8_funct_2(k16_substut1,k8_qc_lang1,C,D) = k36_substut1(D,k8_funct_2(k16_substut1,k8_qc_lang1,C,k31_substut1(D))) ) ) ) ) ) ) ) )).

fof(t28_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v5_substut1(A)
       => k37_substut1(A) = k12_qc_lang1(k37_substut1(k27_substut1(A))) ) ) )).

fof(t29_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => k37_substut1(k20_substut1(A)) = k12_qc_lang1(k37_substut1(A)) ) )).

fof(t30_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v6_substut1(A)
       => k37_substut1(A) = k13_qc_lang1(k37_substut1(k28_substut1(A)),k37_substut1(k29_substut1(A))) ) ) )).

fof(t31_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( k19_substut1(A) = k19_substut1(B)
           => k37_substut1(k21_substut1(A,B)) = k13_qc_lang1(k37_substut1(A),k37_substut1(B)) ) ) ) )).

fof(t32_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v7_substut1(A)
       => k37_substut1(A) = k36_substut1(A,k37_substut1(k31_substut1(A))) ) ) )).

fof(t33_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( v2_substut1(A)
       => m2_subset_1(k37_substut1(A),k8_qc_lang1,k7_cqc_lang) ) ) )).

fof(t34_substut1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( v1_cqc_lang(B,A)
              & m1_qc_lang1(B,A) )
          <=> ( m2_finseq_1(B,k2_qc_lang1)
              & k3_finseq_1(B) = A ) ) ) ) )).

fof(t35_substut1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
         => ! [C] :
              ( ( v1_cqc_lang(C,A)
                & m1_qc_lang1(C,A) )
             => ! [D] :
                  ( m1_subset_1(D,k1_substut1)
                 => m2_subset_1(k37_substut1(k17_substut1(B,C,D)),k8_qc_lang1,k7_cqc_lang) ) ) ) ) )).

fof(t36_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ( m2_subset_1(k37_substut1(A),k8_qc_lang1,k7_cqc_lang)
       => m2_subset_1(k37_substut1(k20_substut1(A)),k8_qc_lang1,k7_cqc_lang) ) ) )).

fof(t37_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ( ( k19_substut1(A) = k19_substut1(B)
              & m2_subset_1(k37_substut1(A),k8_qc_lang1,k7_cqc_lang)
              & m2_subset_1(k37_substut1(B),k8_qc_lang1,k7_cqc_lang) )
           => m2_subset_1(k37_substut1(k21_substut1(A,B)),k8_qc_lang1,k7_cqc_lang) ) ) ) )).

fof(t38_substut1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ! [C] :
              ( m1_substut1(C,k25_substut1(B,A))
             => ( ( m2_subset_1(k37_substut1(B),k8_qc_lang1,k7_cqc_lang)
                  & v3_substut1(k25_substut1(B,A)) )
               => m2_subset_1(k37_substut1(k24_substut1(k25_substut1(B,A),C)),k8_qc_lang1,k7_cqc_lang) ) ) ) ) )).

fof(s1_substut1,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
             => ! [C] :
                  ( m1_qc_lang1(C,A)
                 => ! [D] :
                      ( m1_subset_1(D,k1_substut1)
                     => p1_s1_substut1(k17_substut1(B,C,D)) ) ) ) )
      & ! [A] :
          ( m1_subset_1(A,k16_substut1)
         => ( v2_substut1(A)
           => p1_s1_substut1(A) ) )
      & ! [A] :
          ( m1_subset_1(A,k16_substut1)
         => ( p1_s1_substut1(A)
           => p1_s1_substut1(k20_substut1(A)) ) )
      & ! [A] :
          ( m1_subset_1(A,k16_substut1)
         => ! [B] :
              ( m1_subset_1(B,k16_substut1)
             => ( ( k19_substut1(A) = k19_substut1(B)
                  & p1_s1_substut1(A)
                  & p1_s1_substut1(B) )
               => p1_s1_substut1(k21_substut1(A,B)) ) ) )
      & ! [A] :
          ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
         => ! [B] :
              ( m1_subset_1(B,k16_substut1)
             => ! [C] :
                  ( m1_substut1(C,k25_substut1(B,A))
                 => ( ( v3_substut1(k25_substut1(B,A))
                      & p1_s1_substut1(B) )
                   => p1_s1_substut1(k24_substut1(k25_substut1(B,A),C)) ) ) ) ) )
   => ! [A] :
        ( m1_subset_1(A,k16_substut1)
       => p1_s1_substut1(A) ) )).

fof(s2_substut1,axiom,
    ( ! [A] :
        ( m1_subset_1(A,k16_substut1)
       => ( ( v4_substut1(A)
           => p1_s2_substut1(A) )
          & ( v2_substut1(A)
           => p1_s2_substut1(A) )
          & ( ( v5_substut1(A)
              & p1_s2_substut1(k27_substut1(A)) )
           => p1_s2_substut1(A) )
          & ( ( v6_substut1(A)
              & p1_s2_substut1(k28_substut1(A))
              & p1_s2_substut1(k29_substut1(A)) )
           => p1_s2_substut1(A) )
          & ( ( v7_substut1(A)
              & p1_s2_substut1(k31_substut1(A)) )
           => p1_s2_substut1(A) ) ) )
   => ! [A] :
        ( m1_subset_1(A,k16_substut1)
       => p1_s2_substut1(A) ) )).

fof(s3_substut1,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k16_substut1,f1_s3_substut1)
      & m2_relset_1(A,k16_substut1,f1_s3_substut1)
      & ! [B] :
          ( m1_subset_1(B,k16_substut1)
         => ! [C] :
              ( m1_subset_1(C,f1_s3_substut1)
             => ! [D] :
                  ( m1_subset_1(D,f1_s3_substut1)
                 => ( ( v2_substut1(B)
                     => k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f2_s3_substut1 )
                    & ( v4_substut1(B)
                     => k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f3_s3_substut1(B) )
                    & ( ( v5_substut1(B)
                        & C = k8_funct_2(k16_substut1,f1_s3_substut1,A,k27_substut1(B)) )
                     => k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f4_s3_substut1(C) )
                    & ( ( v6_substut1(B)
                        & C = k8_funct_2(k16_substut1,f1_s3_substut1,A,k28_substut1(B))
                        & D = k8_funct_2(k16_substut1,f1_s3_substut1,A,k29_substut1(B)) )
                     => k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f5_s3_substut1(C,D) )
                    & ( ( v7_substut1(B)
                        & C = k8_funct_2(k16_substut1,f1_s3_substut1,A,k31_substut1(B)) )
                     => k8_funct_2(k16_substut1,f1_s3_substut1,A,B) = f6_s3_substut1(B,C) ) ) ) ) ) ) )).

fof(s4_substut1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,k16_substut1)
         => ! [B] :
              ( m1_subset_1(B,f1_s4_substut1)
             => ! [C] :
                  ( m1_subset_1(C,f1_s4_substut1)
                 => ( ( v2_substut1(A)
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f4_s4_substut1 )
                    & ( v4_substut1(A)
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f5_s4_substut1(A) )
                    & ( ( v5_substut1(A)
                        & B = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k27_substut1(A)) )
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f6_s4_substut1(B) )
                    & ( ( v6_substut1(A)
                        & B = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k28_substut1(A))
                        & C = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k29_substut1(A)) )
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f7_s4_substut1(B,C) )
                    & ( ( v7_substut1(A)
                        & B = k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,k31_substut1(A)) )
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f2_s4_substut1,A) = f8_s4_substut1(A,B) ) ) ) ) )
      & ! [A] :
          ( m1_subset_1(A,k16_substut1)
         => ! [B] :
              ( m1_subset_1(B,f1_s4_substut1)
             => ! [C] :
                  ( m1_subset_1(C,f1_s4_substut1)
                 => ( ( v2_substut1(A)
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f4_s4_substut1 )
                    & ( v4_substut1(A)
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f5_s4_substut1(A) )
                    & ( ( v5_substut1(A)
                        & B = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k27_substut1(A)) )
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f6_s4_substut1(B) )
                    & ( ( v6_substut1(A)
                        & B = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k28_substut1(A))
                        & C = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k29_substut1(A)) )
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f7_s4_substut1(B,C) )
                    & ( ( v7_substut1(A)
                        & B = k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,k31_substut1(A)) )
                     => k8_funct_2(k16_substut1,f1_s4_substut1,f3_s4_substut1,A) = f8_s4_substut1(A,B) ) ) ) ) ) )
   => f2_s4_substut1 = f3_s4_substut1 )).

fof(s5_substut1,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k16_substut1,k38_substut1)
       => ! [B] :
            ( m2_subset_1(B,k16_substut1,k38_substut1)
           => ! [C] :
                ( m2_subset_1(C,k1_qc_lang1,k2_qc_lang1)
               => ! [D] :
                    ( m1_substut1(D,k25_substut1(A,C))
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ! [F] :
                            ( ( v1_cqc_lang(F,E)
                              & m1_qc_lang1(F,E) )
                           => ! [G] :
                                ( m2_subset_1(G,k5_qc_lang1,k7_qc_lang1(E))
                               => ! [H] :
                                    ( m1_subset_1(H,k1_substut1)
                                   => ( p1_s5_substut1(k17_substut1(G,F,H))
                                      & ( v2_substut1(A)
                                       => p1_s5_substut1(A) )
                                      & ( p1_s5_substut1(A)
                                       => p1_s5_substut1(k20_substut1(A)) )
                                      & ( ( k19_substut1(A) = k19_substut1(B)
                                          & p1_s5_substut1(A)
                                          & p1_s5_substut1(B) )
                                       => p1_s5_substut1(k21_substut1(A,B)) )
                                      & ( ( v3_substut1(k25_substut1(A,C))
                                          & p1_s5_substut1(A) )
                                       => p1_s5_substut1(k24_substut1(k25_substut1(A,C),D)) ) ) ) ) ) ) ) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k16_substut1,k38_substut1)
       => p1_s5_substut1(A) ) )).

fof(dt_m1_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ! [B] :
          ( m1_substut1(B,A)
         => m1_subset_1(B,k1_substut1) ) ) )).

fof(existence_m1_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => ? [B] : m1_substut1(B,A) ) )).

fof(dt_k1_substut1,axiom,(
    $true )).

fof(dt_k2_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_substut1)
     => ( v1_funct_1(k2_substut1(A))
        & m2_relset_1(k2_substut1(A),k2_qc_lang1,k2_qc_lang1) ) ) )).

fof(dt_k3_substut1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k1_qc_lang1)
        & m1_subset_1(B,k1_substut1) )
     => m2_finseq_1(k3_substut1(A,B),k1_qc_lang1) ) )).

fof(dt_k4_substut1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k2_qc_lang1)
     => m2_finseq_1(k4_substut1(A),k1_qc_lang1) ) )).

fof(dt_k5_substut1,axiom,(
    ! [A,B] :
      ( ( m1_finseq_1(A,k2_qc_lang1)
        & m1_subset_1(B,k1_substut1) )
     => m2_finseq_1(k5_substut1(A,B),k2_qc_lang1) ) )).

fof(dt_k6_substut1,axiom,(
    ! [A,B] :
      ( m1_subset_1(A,k1_substut1)
     => m1_subset_1(k6_substut1(A,B),k1_substut1) ) )).

fof(redefinition_k6_substut1,axiom,(
    ! [A,B] :
      ( m1_subset_1(A,k1_substut1)
     => k6_substut1(A,B) = k7_relat_1(A,B) ) )).

fof(dt_k7_substut1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k2_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1)
        & m1_subset_1(C,k1_substut1) )
     => ( v1_finset_1(k7_substut1(A,B,C))
        & m1_subset_1(k7_substut1(A,B,C),k1_substut1) ) ) )).

fof(dt_k8_substut1,axiom,(
    ! [A] :
      ( m1_finseq_1(A,k1_qc_lang1)
     => m1_subset_1(k8_substut1(A),k1_zfmisc_1(k2_qc_lang1)) ) )).

fof(dt_k9_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => m1_subset_1(k9_substut1(A),k1_zfmisc_1(k2_qc_lang1)) ) )).

fof(dt_k10_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => ( v1_finset_1(k10_substut1(A))
        & m1_subset_1(k10_substut1(A),k1_zfmisc_1(k5_numbers)) ) ) )).

fof(dt_k11_substut1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1) )
     => ( v1_finset_1(k11_substut1(A))
        & m1_subset_1(k11_substut1(A),k1_zfmisc_1(k5_numbers)) ) ) )).

fof(dt_k12_substut1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k8_qc_lang1)
        & v1_finset_1(B)
        & m1_subset_1(B,k1_substut1) )
     => ( ~ v1_xboole_0(k12_substut1(A,B))
        & m1_subset_1(k12_substut1(A,B),k1_zfmisc_1(k5_numbers)) ) ) )).

fof(dt_k13_substut1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m2_subset_1(k13_substut1(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k14_substut1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k2_qc_lang1)
        & m1_subset_1(B,k8_qc_lang1)
        & v1_finset_1(C)
        & m1_subset_1(C,k1_substut1) )
     => m1_subset_1(k14_substut1(A,B,C),k1_substut1) ) )).

fof(dt_k15_substut1,axiom,
    ( v1_relat_1(k15_substut1)
    & v1_funct_1(k15_substut1) )).

fof(dt_k16_substut1,axiom,(
    ~ v1_xboole_0(k16_substut1) )).

fof(dt_k17_substut1,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_qc_lang1)
        & m1_finseq_1(B,k1_qc_lang1)
        & m1_subset_1(C,k1_substut1) )
     => m1_subset_1(k17_substut1(A,B,C),k16_substut1) ) )).

fof(dt_k18_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k18_substut1(A),k8_qc_lang1) ) )).

fof(redefinition_k18_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => k18_substut1(A) = k1_mcart_1(A) ) )).

fof(dt_k19_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k19_substut1(A),k1_substut1) ) )).

fof(redefinition_k19_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => k19_substut1(A) = k2_mcart_1(A) ) )).

fof(dt_k20_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k20_substut1(A),k16_substut1) ) )).

fof(dt_k21_substut1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k16_substut1)
        & m1_subset_1(B,k16_substut1) )
     => m1_subset_1(k21_substut1(A,B),k16_substut1) ) )).

fof(dt_k22_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => m1_subset_1(k22_substut1(A),k16_substut1) ) )).

fof(redefinition_k22_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => k22_substut1(A) = k1_mcart_1(A) ) )).

fof(dt_k23_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => m2_subset_1(k23_substut1(A),k1_qc_lang1,k2_qc_lang1) ) )).

fof(redefinition_k23_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
     => k23_substut1(A) = k2_mcart_1(A) ) )).

fof(dt_k24_substut1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k2_zfmisc_1(k16_substut1,k2_qc_lang1))
        & m1_substut1(B,A) )
     => m1_subset_1(k24_substut1(A,B),k16_substut1) ) )).

fof(dt_k25_substut1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k16_substut1)
        & m1_subset_1(B,k2_qc_lang1) )
     => m1_subset_1(k25_substut1(A,B),k2_zfmisc_1(k16_substut1,k2_qc_lang1)) ) )).

fof(redefinition_k25_substut1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k16_substut1)
        & m1_subset_1(B,k2_qc_lang1) )
     => k25_substut1(A,B) = k4_tarski(A,B) ) )).

fof(dt_k26_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m2_finseq_1(k26_substut1(A),k1_qc_lang1) ) )).

fof(dt_k27_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k27_substut1(A),k16_substut1) ) )).

fof(dt_k28_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k28_substut1(A),k16_substut1) ) )).

fof(dt_k29_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k29_substut1(A),k16_substut1) ) )).

fof(dt_k30_substut1,axiom,(
    ! [A] : m2_subset_1(k30_substut1(A),k1_qc_lang1,k2_qc_lang1) )).

fof(dt_k31_substut1,axiom,(
    ! [A] : m1_subset_1(k31_substut1(A),k16_substut1) )).

fof(dt_k32_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k32_substut1(A),k2_zfmisc_1(k8_qc_lang1,k1_substut1)) ) )).

fof(dt_k33_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => m1_subset_1(k33_substut1(A),k8_qc_lang1) ) )).

fof(redefinition_k33_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => k33_substut1(A) = k1_mcart_1(A) ) )).

fof(dt_k34_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => m1_subset_1(k34_substut1(A),k1_substut1) ) )).

fof(redefinition_k34_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => k34_substut1(A) = k2_mcart_1(A) ) )).

fof(dt_k35_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_zfmisc_1(k8_qc_lang1,k1_substut1))
     => m2_subset_1(k35_substut1(A),k1_qc_lang1,k2_qc_lang1) ) )).

fof(dt_k36_substut1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k16_substut1)
        & m1_subset_1(B,k8_qc_lang1) )
     => m1_subset_1(k36_substut1(A,B),k8_qc_lang1) ) )).

fof(dt_k37_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k16_substut1)
     => m1_subset_1(k37_substut1(A),k8_qc_lang1) ) )).

fof(dt_k38_substut1,axiom,(
    m1_subset_1(k38_substut1,k1_zfmisc_1(k16_substut1)) )).

fof(dt_k39_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k38_substut1)
     => m2_subset_1(k39_substut1(A),k8_qc_lang1,k7_cqc_lang) ) )).

fof(redefinition_k39_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k38_substut1)
     => k39_substut1(A) = k37_substut1(A) ) )).

fof(d6_substut1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_qc_lang1,k2_qc_lang1)
     => ! [B] :
          ( m1_subset_1(B,k8_qc_lang1)
         => ! [C] :
              ( m1_subset_1(C,k1_substut1)
             => k7_substut1(A,B,C) = k6_substut1(C,a_3_0_substut1(A,B,C)) ) ) ) )).

fof(d7_substut1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_qc_lang1)
     => k8_substut1(A) = a_1_0_substut1(A) ) )).

fof(d9_substut1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k8_qc_lang1)
     => k10_substut1(A) = a_1_1_substut1(A) ) )).

fof(d10_substut1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & m1_subset_1(A,k1_substut1) )
     => k11_substut1(A) = a_1_2_substut1(A) ) )).

fof(d39_substut1,axiom,(
    k38_substut1 = a_0_0_substut1 )).

fof(fraenkel_a_3_0_substut1,axiom,(
    ! [A,B,C,D] :
      ( ( m2_subset_1(B,k1_qc_lang1,k2_qc_lang1)
        & m1_subset_1(C,k8_qc_lang1)
        & m1_subset_1(D,k1_substut1) )
     => ( r2_hidden(A,a_3_0_substut1(B,C,D))
      <=> ? [E] :
            ( m2_subset_1(E,k1_qc_lang1,k2_qc_lang1)
            & A = E
            & r2_hidden(E,k24_qc_lang1(C))
            & m1_subset_1(E,k1_relat_1(D))
            & E != B
            & E != k1_funct_1(D,E) ) ) ) )).

fof(fraenkel_a_1_0_substut1,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,k1_qc_lang1)
     => ( r2_hidden(A,a_1_0_substut1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k1_funct_1(B,C)
            & r1_xreal_0(np__1,C)
            & r1_xreal_0(C,k3_finseq_1(B))
            & r2_hidden(k1_funct_1(B,C),k2_qc_lang1) ) ) ) )).

fof(fraenkel_a_1_1_substut1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k8_qc_lang1)
     => ( r2_hidden(A,a_1_1_substut1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & r2_hidden(k2_qc_lang3(C),k9_substut1(B)) ) ) ) )).

fof(fraenkel_a_1_2_substut1,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(B)
        & m1_subset_1(B,k1_substut1) )
     => ( r2_hidden(A,a_1_2_substut1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & r2_hidden(k2_qc_lang3(C),k2_relat_1(B)) ) ) ) )).

fof(fraenkel_a_0_0_substut1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_substut1)
    <=> ? [B] :
          ( m1_subset_1(B,k16_substut1)
          & A = B
          & m2_subset_1(k18_substut1(B),k8_qc_lang1,k7_cqc_lang) ) ) )).
%------------------------------------------------------------------------------