SET007 Axioms: SET007+102.ax


%------------------------------------------------------------------------------
% File     : SET007+102 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Partial Functions from a Domain to a Domain
% 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    : partfun2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   95 (  22 unit)
%            Number of atoms       :  666 (  91 equality)
%            Maximal formula depth :   19 (  10 average)
%            Number of connectives :  709 ( 138 ~  ;   1  |; 191  &)
%                                         (  31 <=>; 348 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   1 propositional; 0-4 arity)
%            Number of functors    :   37 (   8 constant; 0-6 arity)
%            Number of variables   :  351 (   0 singleton; 340 !;  11 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_partfun2,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0) )).

fof(rc1_partfun2,axiom,(
    ! [A,B] :
    ? [C] :
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C)
      & v2_funct_1(C) ) )).

fof(t1_partfun2,axiom,(
    $true )).

fof(t2_partfun2,axiom,(
    $true )).

fof(t3_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,B,A) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,B,A) )
                 => ( ( k4_relset_1(B,A,C) = k4_relset_1(B,A,D)
                      & ! [E] :
                          ( m1_subset_1(E,B)
                         => ( r2_hidden(E,k4_relset_1(B,A,C))
                           => k4_finseq_4(B,A,C,E) = k4_finseq_4(B,A,D,E) ) ) )
                   => C = D ) ) ) ) ) )).

fof(t4_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,C,B) )
             => ( r2_hidden(A,k2_relat_1(D))
              <=> ? [E] :
                    ( m1_subset_1(E,C)
                    & r2_hidden(E,k4_relset_1(C,B,D))
                    & A = k4_finseq_4(C,B,D,E) ) ) ) ) ) )).

fof(t5_partfun2,axiom,(
    $true )).

fof(t6_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,C,A) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,C,B) )
                         => ( F = k1_partfun1(C,A,A,B,D,E)
                          <=> ( ! [G] :
                                  ( m1_subset_1(G,C)
                                 => ( r2_hidden(G,k4_relset_1(C,B,F))
                                  <=> ( r2_hidden(G,k4_relset_1(C,A,D))
                                      & r2_hidden(k4_finseq_4(C,A,D,G),k4_relset_1(A,B,E)) ) ) )
                              & ! [G] :
                                  ( m1_subset_1(G,C)
                                 => ( r2_hidden(G,k4_relset_1(C,B,F))
                                   => k4_finseq_4(C,B,F,G) = k4_finseq_4(A,B,E,k4_finseq_4(C,A,D,G)) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_partfun2,axiom,(
    $true )).

fof(t8_partfun2,axiom,(
    $true )).

fof(t9_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,B,C) )
                         => ( ( r2_hidden(D,k4_relset_1(A,B,E))
                              & r2_hidden(k4_finseq_4(A,B,E,D),k4_relset_1(B,C,F)) )
                           => k4_finseq_4(A,C,k1_partfun1(A,B,B,C,E,F),D) = k4_finseq_4(B,C,F,k4_finseq_4(A,B,E,D)) ) ) ) ) ) ) ) )).

fof(t10_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,B,C) )
                         => ( ( r1_tarski(k2_relat_1(E),k4_relset_1(B,C,F))
                              & r2_hidden(D,k4_relset_1(A,B,E)) )
                           => k4_finseq_4(A,C,k1_partfun1(A,B,B,C,E,F),D) = k4_finseq_4(B,C,F,k4_finseq_4(A,B,E,D)) ) ) ) ) ) ) ) )).

fof(t11_partfun2,axiom,(
    $true )).

fof(t12_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,A) )
             => ( C = k1_partfun2(A,B)
              <=> ( k4_relset_1(A,A,C) = B
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => ( r2_hidden(D,B)
                       => k4_finseq_4(A,A,C,D) = D ) ) ) ) ) ) ) )).

fof(t13_partfun2,axiom,(
    $true )).

fof(t14_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,A) )
                 => ( r2_hidden(C,k5_subset_1(A,k4_relset_1(A,A,D),B))
                   => k4_finseq_4(A,A,D,C) = k4_finseq_4(A,A,k1_partfun1(A,A,A,A,k1_partfun2(A,B),D),C) ) ) ) ) ) )).

fof(t15_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,A) )
                 => ( r2_hidden(C,k4_relset_1(A,A,k1_partfun1(A,A,A,A,D,k1_partfun2(A,B))))
                  <=> ( r2_hidden(C,k4_relset_1(A,A,D))
                      & r2_hidden(k4_finseq_4(A,A,D,C),B) ) ) ) ) ) ) )).

fof(t16_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,B,A) )
             => ( ! [D] :
                    ( m1_subset_1(D,B)
                   => ! [E] :
                        ( m1_subset_1(E,B)
                       => ( ( r2_hidden(D,k4_relset_1(B,A,C))
                            & r2_hidden(E,k4_relset_1(B,A,C))
                            & k4_finseq_4(B,A,C,D) = k4_finseq_4(B,A,C,E) )
                         => D = E ) ) )
               => v2_funct_1(C) ) ) ) ) )).

fof(t17_partfun2,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ~ v1_xboole_0(D)
         => ! [E] :
              ( ( v1_funct_1(E)
                & m2_relset_1(E,C,D) )
             => ( ( v2_funct_1(E)
                  & r2_hidden(A,k4_relset_1(C,D,E))
                  & r2_hidden(B,k4_relset_1(C,D,E))
                  & k4_finseq_4(C,D,E,A) = k4_finseq_4(C,D,E,B) )
               => A = B ) ) ) ) )).

fof(t18_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v2_funct_1(C)
                & m2_relset_1(C,A,B) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,B,A) )
                 => ( D = k2_partfun2(A,B,C)
                  <=> ( k4_relset_1(B,A,D) = k2_relat_1(C)
                      & ! [E] :
                          ( m1_subset_1(E,B)
                         => ! [F] :
                              ( m1_subset_1(F,A)
                             => ( ( ( r2_hidden(E,k2_relat_1(C))
                                    & F = k4_finseq_4(B,A,D,E) )
                                 => ( r2_hidden(F,k4_relset_1(A,B,C))
                                    & E = k4_finseq_4(A,B,C,F) ) )
                                & ( ( r2_hidden(F,k4_relset_1(A,B,C))
                                    & E = k4_finseq_4(A,B,C,F) )
                                 => ( r2_hidden(E,k2_relat_1(C))
                                    & F = k4_finseq_4(B,A,D,E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t19_partfun2,axiom,(
    $true )).

fof(t20_partfun2,axiom,(
    $true )).

fof(t21_partfun2,axiom,(
    $true )).

fof(t22_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v2_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ( r2_hidden(C,k4_relset_1(A,B,D))
                   => ( C = k4_finseq_4(B,A,k2_partfun2(A,B,D),k4_finseq_4(A,B,D,C))
                      & C = k4_finseq_4(A,A,k1_partfun1(A,B,B,A,D,k2_partfun2(A,B,D)),C) ) ) ) ) ) ) )).

fof(t23_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v2_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ( r2_hidden(C,k2_relat_1(D))
                   => ( C = k4_finseq_4(A,B,D,k4_finseq_4(B,A,k2_partfun2(A,B,D),C))
                      & C = k4_finseq_4(B,B,k1_partfun1(B,A,A,B,k2_partfun2(A,B,D),D),C) ) ) ) ) ) ) )).

fof(t24_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,B) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,B,A) )
                 => ( ( v2_funct_1(C)
                      & k4_relset_1(A,B,C) = k2_relat_1(D)
                      & k2_relat_1(C) = k4_relset_1(B,A,D)
                      & ! [E] :
                          ( m1_subset_1(E,A)
                         => ! [F] :
                              ( m1_subset_1(F,B)
                             => ( ( r2_hidden(E,k4_relset_1(A,B,C))
                                  & r2_hidden(F,k4_relset_1(B,A,D)) )
                               => ( k4_finseq_4(A,B,C,E) = F
                                <=> k4_finseq_4(B,A,D,F) = E ) ) ) ) )
                   => D = k2_funct_1(C) ) ) ) ) ) )).

fof(t25_partfun2,axiom,(
    $true )).

fof(t26_partfun2,axiom,(
    $true )).

fof(t27_partfun2,axiom,(
    $true )).

fof(t28_partfun2,axiom,(
    $true )).

fof(t29_partfun2,axiom,(
    $true )).

fof(t30_partfun2,axiom,(
    $true )).

fof(t31_partfun2,axiom,(
    $true )).

fof(t32_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,C,B) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,C,B) )
                 => ( D = k2_partfun1(C,B,E,A)
                  <=> ( k4_relset_1(C,B,D) = k3_xboole_0(k4_relset_1(C,B,E),A)
                      & ! [F] :
                          ( m1_subset_1(F,C)
                         => ( r2_hidden(F,k4_relset_1(C,B,D))
                           => k4_finseq_4(C,B,D,F) = k4_finseq_4(C,B,E,F) ) ) ) ) ) ) ) ) )).

fof(t33_partfun2,axiom,(
    $true )).

fof(t34_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,B)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,B,C) )
                 => ( r2_hidden(D,k3_xboole_0(k4_relset_1(B,C,E),A))
                   => k4_finseq_4(B,C,k2_partfun1(B,C,E,A),D) = k4_finseq_4(B,C,E,D) ) ) ) ) ) )).

fof(t35_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,B)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,B,C) )
                 => ( ( r2_hidden(D,k4_relset_1(B,C,E))
                      & r2_hidden(D,A) )
                   => k4_finseq_4(B,C,k2_partfun1(B,C,E,A),D) = k4_finseq_4(B,C,E,D) ) ) ) ) ) )).

fof(t36_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,B)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,B,C) )
                 => ( ( r2_hidden(D,k4_relset_1(B,C,E))
                      & r2_hidden(D,A) )
                   => r2_hidden(k4_finseq_4(B,C,E,D),k2_relat_1(k2_partfun1(B,C,E,A))) ) ) ) ) ) )).

fof(t37_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,C,B) )
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,C,B) )
                 => ( D = k3_partfun2(C,B,A,E)
                  <=> ( ! [F] :
                          ( m1_subset_1(F,C)
                         => ( r2_hidden(F,k4_relset_1(C,B,D))
                          <=> ( r2_hidden(F,k4_relset_1(C,B,E))
                              & r2_hidden(k4_finseq_4(C,B,E,F),A) ) ) )
                      & ! [F] :
                          ( m1_subset_1(F,C)
                         => ( r2_hidden(F,k4_relset_1(C,B,D))
                           => k4_finseq_4(C,B,D,F) = k4_finseq_4(C,B,E,F) ) ) ) ) ) ) ) ) )).

fof(t38_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,B)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,B,C) )
                 => ( r2_hidden(D,k4_relset_1(B,C,k3_partfun2(B,C,A,E)))
                  <=> ( r2_hidden(D,k4_relset_1(B,C,E))
                      & r2_hidden(k4_finseq_4(B,C,E,D),A) ) ) ) ) ) ) )).

fof(t39_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,B)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,B,C) )
                 => ( r2_hidden(D,k4_relset_1(B,C,k3_partfun2(B,C,A,E)))
                   => k4_finseq_4(B,C,k3_partfun2(B,C,A,E),D) = k4_finseq_4(B,C,E,D) ) ) ) ) ) )).

fof(t40_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(B))
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,C,B) )
                 => ( D = k9_relat_1(E,A)
                  <=> ! [F] :
                        ( m1_subset_1(F,B)
                       => ( r2_hidden(F,D)
                        <=> ? [G] :
                              ( m1_subset_1(G,C)
                              & r2_hidden(G,k4_relset_1(C,B,E))
                              & r2_hidden(G,A)
                              & F = k4_finseq_4(C,B,E,G) ) ) ) ) ) ) ) ) )).

fof(t41_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,C)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,B,C) )
                 => ( r2_hidden(D,k10_relset_1(B,C,E,A))
                  <=> ? [F] :
                        ( m1_subset_1(F,B)
                        & r2_hidden(F,k4_relset_1(B,C,E))
                        & r2_hidden(F,A)
                        & D = k4_finseq_4(B,C,E,F) ) ) ) ) ) ) )).

fof(t42_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ( r2_hidden(C,k4_relset_1(A,B,D))
                   => k9_relat_1(D,k1_tarski(C)) = k1_tarski(k4_finseq_4(A,B,D,C)) ) ) ) ) ) )).

fof(t43_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( ( r2_hidden(C,k4_relset_1(A,B,E))
                          & r2_hidden(D,k4_relset_1(A,B,E)) )
                       => k9_relat_1(E,k2_tarski(C,D)) = k2_tarski(k4_finseq_4(A,B,E,C),k4_finseq_4(A,B,E,D)) ) ) ) ) ) ) )).

fof(t44_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(C))
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,C,B) )
                 => ( D = k3_funct_2(C,B,E,A)
                  <=> ! [F] :
                        ( m1_subset_1(F,C)
                       => ( r2_hidden(F,D)
                        <=> ( r2_hidden(F,k4_relset_1(C,B,E))
                            & r2_hidden(k4_finseq_4(C,B,E,F),A) ) ) ) ) ) ) ) ) )).

fof(t45_partfun2,axiom,(
    $true )).

fof(t46_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,B,A) )
             => ? [D] :
                  ( v1_funct_1(D)
                  & v1_funct_2(D,B,A)
                  & m2_relset_1(D,B,A)
                  & ! [E] :
                      ( m1_subset_1(E,B)
                     => ( r2_hidden(E,k4_relset_1(B,A,C))
                       => k8_funct_2(B,A,D,E) = k4_finseq_4(B,A,C,E) ) ) ) ) ) ) )).

fof(t47_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,B,A) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,B,A) )
                 => ( r1_partfun1(C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,B)
                       => ( r2_hidden(E,k5_subset_1(B,k4_relset_1(B,A,C),k4_relset_1(B,A,D)))
                         => k4_finseq_4(B,A,C,E) = k4_finseq_4(B,A,D,E) ) ) ) ) ) ) ) )).

fof(t48_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( m1_subset_1(E,B)
                     => ( r2_hidden(D,C)
                       => k4_finseq_4(A,B,k4_partfun2(A,B,C,E),D) = E ) ) ) ) ) ) )).

fof(t49_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,B,A) )
                 => ( ! [E] :
                        ( m1_subset_1(E,B)
                       => ( r2_hidden(E,k4_relset_1(B,A,D))
                         => k4_finseq_4(B,A,D,E) = C ) )
                   => D = k4_partfun2(B,A,k4_relset_1(B,A,D),C) ) ) ) ) ) )).

fof(t50_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,B)
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,B,C) )
                         => ( r2_hidden(E,k4_relset_1(B,C,F))
                           => k1_partfun1(A,B,B,C,k4_partfun2(A,B,D,E),F) = k4_partfun2(A,C,D,k4_finseq_4(B,C,F,E)) ) ) ) ) ) ) ) )).

fof(t51_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_partfun1(k1_partfun2(A,B),A,A)
          <=> B = A ) ) ) )).

fof(t52_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => ~ ( v1_partfun1(k4_partfun2(A,B,C,D),A,B)
                      & C = k1_xboole_0 ) ) ) ) ) )).

fof(t53_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(B))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( v1_partfun1(k4_partfun2(B,A,C,D),B,A)
                  <=> C = B ) ) ) ) ) )).

fof(d1_partfun2,axiom,(
    $true )).

fof(d2_partfun2,axiom,(
    $true )).

fof(d3_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,A,B) )
             => ! [D] :
                  ( r1_partfun2(A,B,C,D)
                <=> ? [E] :
                      ( m1_subset_1(E,B)
                      & ! [F] :
                          ( m1_subset_1(F,A)
                         => ( r2_hidden(F,k3_xboole_0(D,k4_relset_1(A,B,C)))
                           => k4_finseq_4(A,B,C,F) = E ) ) ) ) ) ) ) )).

fof(t54_partfun2,axiom,(
    $true )).

fof(t55_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,C,B) )
             => ( r1_partfun2(C,B,D,A)
              <=> ! [E] :
                    ( m1_subset_1(E,C)
                   => ! [F] :
                        ( m1_subset_1(F,C)
                       => ( ( r2_hidden(E,k3_xboole_0(A,k4_relset_1(C,B,D)))
                            & r2_hidden(F,k3_xboole_0(A,k4_relset_1(C,B,D))) )
                         => k4_finseq_4(C,B,D,E) = k4_finseq_4(C,B,D,F) ) ) ) ) ) ) ) )).

fof(t56_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,B,C) )
             => ( ~ r1_xboole_0(A,k4_relset_1(B,C,D))
               => ( r1_partfun2(B,C,D,A)
                <=> ? [E] :
                      ( m1_subset_1(E,C)
                      & k2_relat_1(k2_partfun1(B,C,D,A)) = k1_tarski(E) ) ) ) ) ) ) )).

fof(t57_partfun2,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ~ v1_xboole_0(D)
         => ! [E] :
              ( ( v1_funct_1(E)
                & m2_relset_1(E,C,D) )
             => ( ( r1_partfun2(C,D,E,A)
                  & r1_tarski(B,A) )
               => r1_partfun2(C,D,E,B) ) ) ) ) )).

fof(t58_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,B,C) )
             => ( r1_xboole_0(A,k4_relset_1(B,C,D))
               => r1_partfun2(B,C,D,A) ) ) ) ) )).

fof(t59_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( k2_partfun1(A,B,E,C) = k4_partfun2(A,B,k4_relset_1(A,B,k2_partfun1(A,B,E,C)),D)
                       => r1_partfun2(A,B,E,C) ) ) ) ) ) ) )).

fof(t60_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,B,C) )
             => r1_partfun2(B,C,D,k1_tarski(A)) ) ) ) )).

fof(t61_partfun2,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ~ v1_xboole_0(D)
         => ! [E] :
              ( ( v1_funct_1(E)
                & m2_relset_1(E,C,D) )
             => ( ( r1_partfun2(C,D,E,A)
                  & r1_partfun2(C,D,E,B) )
               => ( r1_xboole_0(k3_xboole_0(A,B),k4_relset_1(C,D,E))
                  | r1_partfun2(C,D,E,k2_xboole_0(A,B)) ) ) ) ) ) )).

fof(t62_partfun2,axiom,(
    ! [A,B,C] :
      ( ~ v1_xboole_0(C)
     => ! [D] :
          ( ~ v1_xboole_0(D)
         => ! [E] :
              ( ( v1_funct_1(E)
                & m2_relset_1(E,C,D) )
             => ( r1_partfun2(C,D,E,A)
               => r1_partfun2(C,D,k2_partfun1(C,D,E,B),A) ) ) ) ) )).

fof(t63_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => r1_partfun2(A,B,k4_partfun2(A,B,C,D),C) ) ) ) ) )).

fof(t64_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,B,A) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,B,A) )
                 => ( r1_tarski(C,D)
                  <=> ( r1_tarski(k4_relset_1(B,A,C),k4_relset_1(B,A,D))
                      & ! [E] :
                          ( m1_subset_1(E,B)
                         => ( r2_hidden(E,k4_relset_1(B,A,C))
                           => k4_finseq_4(B,A,C,E) = k4_finseq_4(B,A,D,E) ) ) ) ) ) ) ) ) )).

fof(t65_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( ( r2_hidden(C,k4_relset_1(A,B,E))
                          & D = k4_finseq_4(A,B,E,C) )
                      <=> r2_hidden(k4_tarski(C,D),E) ) ) ) ) ) ) )).

fof(t66_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( m1_subset_1(E,C)
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,A,B) )
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & m2_relset_1(G,B,C) )
                             => ( r2_hidden(k4_tarski(D,E),k1_partfun1(A,B,B,C,F,G))
                               => ( r2_hidden(k4_tarski(D,k4_finseq_4(A,B,F,D)),F)
                                  & r2_hidden(k4_tarski(k4_finseq_4(A,B,F,D),E),G) ) ) ) ) ) ) ) ) ) )).

fof(t67_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( E = k1_tarski(k4_tarski(C,D))
                       => k4_finseq_4(A,B,E,C) = D ) ) ) ) ) ) )).

fof(t68_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ( k4_relset_1(A,B,D) = k1_tarski(C)
                   => D = k1_tarski(k4_tarski(C,k4_finseq_4(A,B,D,C))) ) ) ) ) ) )).

fof(t69_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,A,B) )
                         => ( ( D = k2_relset_1(A,B,E,F)
                              & r2_hidden(C,k4_relset_1(A,B,D)) )
                           => ( k4_finseq_4(A,B,D,C) = k4_finseq_4(A,B,E,C)
                              & k4_finseq_4(A,B,D,C) = k4_finseq_4(A,B,F,C) ) ) ) ) ) ) ) ) )).

fof(t70_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,A,B) )
                         => ( ( r2_hidden(C,k4_relset_1(A,B,D))
                              & E = k1_relset_1(A,B,D,F) )
                           => k4_finseq_4(A,B,E,C) = k4_finseq_4(A,B,D,C) ) ) ) ) ) ) ) )).

fof(t71_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,A,B) )
                         => ( ( r2_hidden(C,k4_relset_1(A,B,D))
                              & E = k1_relset_1(A,B,F,D) )
                           => k4_finseq_4(A,B,E,C) = k4_finseq_4(A,B,D,C) ) ) ) ) ) ) ) )).

fof(t72_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,B) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & m2_relset_1(F,A,B) )
                         => ~ ( r2_hidden(C,k4_relset_1(A,B,D))
                              & D = k1_relset_1(A,B,E,F)
                              & k4_finseq_4(A,B,D,C) != k4_finseq_4(A,B,E,C)
                              & k4_finseq_4(A,B,D,C) != k4_finseq_4(A,B,F,C) ) ) ) ) ) ) ) )).

fof(t73_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( ( r2_hidden(D,k4_relset_1(A,B,E))
                          & r2_hidden(D,C) )
                      <=> r2_hidden(k4_tarski(D,k4_finseq_4(A,B,E,D)),k2_partfun1(A,B,E,C)) ) ) ) ) ) ) )).

fof(t74_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(B))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( ( r2_hidden(D,k4_relset_1(A,B,E))
                          & r2_hidden(k4_finseq_4(A,B,E,D),C) )
                      <=> r2_hidden(k4_tarski(D,k4_finseq_4(A,B,E,D)),k3_partfun2(A,B,C,E)) ) ) ) ) ) ) )).

fof(t75_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(B))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( r2_hidden(D,k3_funct_2(A,B,E,C))
                      <=> ( r2_hidden(k4_tarski(D,k4_finseq_4(A,B,E,D)),E)
                          & r2_hidden(k4_finseq_4(A,B,E,D),C) ) ) ) ) ) ) ) )).

fof(t76_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,C,B) )
             => ( r1_partfun2(C,B,D,A)
              <=> ? [E] :
                    ( m1_subset_1(E,B)
                    & ! [F] :
                        ( m1_subset_1(F,C)
                       => ( r2_hidden(F,k3_xboole_0(A,k4_relset_1(C,B,D)))
                         => k1_funct_1(D,F) = E ) ) ) ) ) ) ) )).

fof(t77_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ( v1_funct_1(D)
                & m2_relset_1(D,C,B) )
             => ( r1_partfun2(C,B,D,A)
              <=> ! [E] :
                    ( m1_subset_1(E,C)
                   => ! [F] :
                        ( m1_subset_1(F,C)
                       => ( ( r2_hidden(E,k3_xboole_0(A,k4_relset_1(C,B,D)))
                            & r2_hidden(F,k3_xboole_0(A,k4_relset_1(C,B,D))) )
                         => k1_funct_1(D,E) = k1_funct_1(D,F) ) ) ) ) ) ) ) )).

fof(t78_partfun2,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( m1_subset_1(D,B)
             => ! [E] :
                  ( ( v1_funct_1(E)
                    & m2_relset_1(E,C,B) )
                 => ~ ( r2_hidden(D,k9_relat_1(E,A))
                      & ! [F] :
                          ( m1_subset_1(F,C)
                         => ~ ( r2_hidden(F,k4_relset_1(C,B,E))
                              & r2_hidden(F,A)
                              & D = k1_funct_1(E,F) ) ) ) ) ) ) ) )).

fof(t79_partfun2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,B)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & m2_relset_1(E,A,B) )
                     => ( v2_funct_1(E)
                       => ( ( ( r2_hidden(D,k2_relat_1(E))
                              & C = k1_funct_1(k2_funct_1(E),D) )
                           => ( r2_hidden(C,k4_relset_1(A,B,E))
                              & D = k1_funct_1(E,C) ) )
                          & ( ( r2_hidden(C,k4_relset_1(A,B,E))
                              & D = k1_funct_1(E,C) )
                           => ( r2_hidden(D,k2_relat_1(E))
                              & C = k1_funct_1(k2_funct_1(E),D) ) ) ) ) ) ) ) ) ) )).

fof(s1_partfun2,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & m2_relset_1(A,f1_s1_partfun2,f2_s1_partfun2)
      & ! [B] :
          ( m1_subset_1(B,f1_s1_partfun2)
         => ( r2_hidden(B,k4_relset_1(f1_s1_partfun2,f2_s1_partfun2,A))
          <=> ? [C] :
                ( m1_subset_1(C,f2_s1_partfun2)
                & p1_s1_partfun2(B,C) ) ) )
      & ! [B] :
          ( m1_subset_1(B,f1_s1_partfun2)
         => ( r2_hidden(B,k4_relset_1(f1_s1_partfun2,f2_s1_partfun2,A))
           => p1_s1_partfun2(B,k4_finseq_4(f1_s1_partfun2,f2_s1_partfun2,A,B)) ) ) ) )).

fof(s2_partfun2,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & m2_relset_1(A,f1_s2_partfun2,f2_s2_partfun2)
      & ! [B] :
          ( m1_subset_1(B,f1_s2_partfun2)
         => ( r2_hidden(B,k4_relset_1(f1_s2_partfun2,f2_s2_partfun2,A))
          <=> p1_s2_partfun2(B) ) )
      & ! [B] :
          ( m1_subset_1(B,f1_s2_partfun2)
         => ( r2_hidden(B,k4_relset_1(f1_s2_partfun2,f2_s2_partfun2,A))
           => k4_finseq_4(f1_s2_partfun2,f2_s2_partfun2,A,B) = f3_s2_partfun2(B) ) ) ) )).

fof(s3_partfun2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,f1_s3_partfun2,f2_s3_partfun2) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,f1_s3_partfun2,f2_s3_partfun2) )
         => ( ( k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,A) = f3_s3_partfun2
              & ! [C] :
                  ( m1_subset_1(C,f1_s3_partfun2)
                 => ( r2_hidden(C,k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,A))
                   => k4_finseq_4(f1_s3_partfun2,f2_s3_partfun2,A,C) = f4_s3_partfun2(C) ) )
              & k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,B) = f3_s3_partfun2
              & ! [C] :
                  ( m1_subset_1(C,f1_s3_partfun2)
                 => ( r2_hidden(C,k4_relset_1(f1_s3_partfun2,f2_s3_partfun2,B))
                   => k4_finseq_4(f1_s3_partfun2,f2_s3_partfun2,B,C) = f4_s3_partfun2(C) ) ) )
           => A = B ) ) ) )).

fof(dt_k1_partfun2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ( v1_funct_1(k1_partfun2(A,B))
        & m2_relset_1(k1_partfun2(A,B),A,A) ) ) )).

fof(redefinition_k1_partfun2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => k1_partfun2(A,B) = k6_relat_1(B) ) )).

fof(dt_k2_partfun2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v2_funct_1(C)
        & m1_relset_1(C,A,B) )
     => ( v1_funct_1(k2_partfun2(A,B,C))
        & m2_relset_1(k2_partfun2(A,B,C),B,A) ) ) )).

fof(redefinition_k2_partfun2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v2_funct_1(C)
        & m1_relset_1(C,A,B) )
     => k2_partfun2(A,B,C) = k2_funct_1(C) ) )).

fof(dt_k3_partfun2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => ( v1_funct_1(k3_partfun2(A,B,C,D))
        & m2_relset_1(k3_partfun2(A,B,C,D),A,B) ) ) )).

fof(redefinition_k3_partfun2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,B) )
     => k3_partfun2(A,B,C,D) = k8_relat_1(C,D) ) )).

fof(dt_k4_partfun2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,B) )
     => ( v1_funct_1(k4_partfun2(A,B,C,D))
        & m2_relset_1(k4_partfun2(A,B,C,D),A,B) ) ) )).

fof(redefinition_k4_partfun2,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,B) )
     => k4_partfun2(A,B,C,D) = k2_funcop_1(C,D) ) )).
%------------------------------------------------------------------------------