SET007 Axioms: SET007+105.ax


%------------------------------------------------------------------------------
% File     : SET007+105 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of Real Functions
% 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    : rfunct_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   94 (   7 unit)
%            Number of atoms       :  622 (  43 equality)
%            Maximal formula depth :   14 (   7 average)
%            Number of connectives :  534 (   6 ~  ;   9  |; 269  &)
%                                         (  14 <=>; 236 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-4 arity)
%            Number of functors    :   49 (   4 constant; 0-6 arity)
%            Number of variables   :  237 (   0 singleton; 236 !;   1 ?)
%            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_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v2_funct_1(B) )
     => ( v1_relat_1(k7_relat_1(B,A))
        & v1_funct_1(k7_relat_1(B,A))
        & v2_funct_1(k7_relat_1(B,A)) ) ) )).

fof(t1_rfunct_2,axiom,(
    $true )).

fof(t2_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : k5_relat_1(k7_relat_1(A,C),k7_relat_1(B,k9_relat_1(A,C))) = k7_relat_1(k5_relat_1(A,B),C) ) ) )).

fof(t3_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C,D] : k5_relat_1(k7_relat_1(A,C),k7_relat_1(B,D)) = k7_relat_1(k5_relat_1(A,B),k3_xboole_0(C,k10_relat_1(A,D))) ) ) )).

fof(t4_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] :
              ( r1_tarski(C,k1_relat_1(k5_relat_1(A,B)))
            <=> ( r1_tarski(C,k1_relat_1(A))
                & r1_tarski(k9_relat_1(A,C),k1_relat_1(B)) ) ) ) ) )).

fof(t5_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : k9_relat_1(k7_relat_1(A,B),B) = k9_relat_1(A,B) ) )).

fof(t6_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( A = k10_seq_1(B,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,A,D) = k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) )).

fof(t7_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r1_tarski(k1_rfunct_2(k1_seqm_3(A,B)),k1_rfunct_2(A)) ) ) )).

fof(t8_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,C))
               => r2_hidden(k2_seq_1(k5_numbers,k1_numbers,A,B),k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )).

fof(t9_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,k1_rfunct_2(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k2_seq_1(k5_numbers,k1_numbers,B,C) ) ) ) )).

fof(t10_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => r2_hidden(k2_seq_1(k5_numbers,k1_numbers,A,B),k1_rfunct_2(A)) ) ) )).

fof(t11_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( m1_seqm_3(A,B)
           => r1_tarski(k1_rfunct_2(A),k1_rfunct_2(B)) ) ) ) )).

fof(t12_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( m1_seqm_3(A,B)
              & v2_relat_1(B) )
           => v2_relat_1(A) ) ) ) )).

fof(t13_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v1_seqm_3(C)
                & v7_seqm_3(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( k3_seqm_3(C,k9_seq_1(A,B)) = k9_seq_1(k3_seqm_3(C,A),k3_seqm_3(C,B))
                & k3_seqm_3(C,k10_seq_1(A,B)) = k10_seq_1(k3_seqm_3(C,A),k3_seqm_3(C,B))
                & k3_seqm_3(C,k11_seq_1(A,B)) = k11_seq_1(k3_seqm_3(C,A),k3_seqm_3(C,B)) ) ) ) ) )).

fof(t14_rfunct_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v1_seqm_3(C)
                & v7_seqm_3(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k3_seqm_3(C,k14_seq_1(B,A)) = k14_seq_1(k3_seqm_3(C,B),A) ) ) ) )).

fof(t15_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_seqm_3(B)
            & v7_seqm_3(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( k3_seqm_3(B,k17_seq_1(A)) = k17_seq_1(k3_seqm_3(B,A))
            & k3_seqm_3(B,k22_seq_1(A)) = k22_seq_1(k3_seqm_3(B,A)) ) ) ) )).

fof(t16_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_seqm_3(B)
            & v7_seqm_3(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k18_seq_1(k3_seqm_3(B,A)) = k3_seqm_3(B,k18_seq_1(A)) ) ) )).

fof(t17_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v1_seqm_3(C)
                & v7_seqm_3(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => k3_seqm_3(C,k19_seq_1(A,B)) = k19_seq_1(k3_seqm_3(C,A),k3_seqm_3(C,B)) ) ) ) )).

fof(t18_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ( v4_seq_2(A)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0) ) )
       => r1_xreal_0(k2_seq_2(A),np__0) ) ) )).

fof(t19_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,C),A) )
       => r1_tarski(k1_rfunct_2(B),A) ) ) )).

fof(d1_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,A))
           => k2_rfunct_2(A,B) = k5_relat_1(B,A) ) ) ) )).

fof(t20_rfunct_2,axiom,(
    $true )).

fof(t21_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,C))
               => k2_seq_1(k5_numbers,k1_numbers,k2_rfunct_2(C,A),B) = k2_seq_1(k1_numbers,k1_numbers,C,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) )).

fof(t22_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,C))
               => k1_seqm_3(k2_rfunct_2(C,A),B) = k2_rfunct_2(C,k1_seqm_3(A,B)) ) ) ) ) )).

fof(t23_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k1_rfunct_2(A),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C)))
               => ( k2_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k9_seq_1(k2_rfunct_2(B,A),k2_rfunct_2(C,A))
                  & k2_rfunct_2(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k10_seq_1(k2_rfunct_2(B,A),k2_rfunct_2(C,A))
                  & k2_rfunct_2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k11_seq_1(k2_rfunct_2(B,A),k2_rfunct_2(C,A)) ) ) ) ) ) )).

fof(t24_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,B))
               => k2_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,B,C),A) = k14_seq_1(k2_rfunct_2(B,A),C) ) ) ) ) )).

fof(t25_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,B))
           => ( k22_seq_1(k2_rfunct_2(B,A)) = k2_rfunct_2(k21_seq_1(k1_numbers,k1_numbers,B),A)
              & k17_seq_1(k2_rfunct_2(B,A)) = k2_rfunct_2(k16_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t26_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B)))
           => v2_relat_1(k2_rfunct_2(B,A)) ) ) ) )).

fof(t27_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,B)))
           => k2_rfunct_2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k18_seq_1(k2_rfunct_2(B,A)) ) ) ) )).

fof(t28_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v1_seqm_3(B)
            & v7_seqm_3(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,C))
               => k3_seqm_3(B,k2_rfunct_2(C,A)) = k2_rfunct_2(C,k3_seqm_3(B,A)) ) ) ) ) )).

fof(t29_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,C))
                  & m1_seqm_3(B,A) )
               => m1_seqm_3(k2_rfunct_2(C,B),k2_rfunct_2(C,A)) ) ) ) ) )).

fof(t30_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( v1_partfun1(C,k1_numbers,k1_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,k2_rfunct_2(C,A),B) = k2_seq_1(k1_numbers,k1_numbers,C,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ) )).

fof(t31_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( v1_partfun1(C,k1_numbers,k1_numbers)
               => k2_rfunct_2(C,k1_seqm_3(A,B)) = k1_seqm_3(k2_rfunct_2(C,A),B) ) ) ) ) )).

fof(t32_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( v1_partfun1(B,k1_numbers,k1_numbers)
                  & v1_partfun1(C,k1_numbers,k1_numbers) )
               => ( k2_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k9_seq_1(k2_rfunct_2(B,A),k2_rfunct_2(C,A))
                  & k2_rfunct_2(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k10_seq_1(k2_rfunct_2(B,A),k2_rfunct_2(C,A))
                  & k2_rfunct_2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k11_seq_1(k2_rfunct_2(B,A),k2_rfunct_2(C,A)) ) ) ) ) ) )).

fof(t33_rfunct_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( v1_partfun1(C,k1_numbers,k1_numbers)
               => k2_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,A),B) = k14_seq_1(k2_rfunct_2(C,B),A) ) ) ) ) )).

fof(t34_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,A)))
           => k2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,C,A),B) = k2_rfunct_2(C,B) ) ) ) )).

fof(t35_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,k5_numbers,k1_numbers)
        & m2_relset_1(C,k5_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,D,A)))
           => ( ( ~ r1_tarski(k1_rfunct_2(C),k4_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,D,B)))
                & ~ r1_tarski(A,B) )
              | k2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,D,A),C) = k2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,D,B),C) ) ) ) ) )).

fof(t36_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,A)))
           => k22_seq_1(k2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,C,A),B)) = k2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,k21_seq_1(k1_numbers,k1_numbers,C),A),B) ) ) ) )).

fof(t37_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( ( r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,A)))
              & k11_relset_1(k1_numbers,k1_numbers,C,k1_seq_4(np__0)) = k1_xboole_0 )
           => k2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,k4_rfunct_1(k1_numbers,k1_numbers,C),A),B) = k18_seq_1(k2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,C,A),B)) ) ) ) )).

fof(t38_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,B))
           => k10_relset_1(k1_numbers,k1_numbers,B,k1_rfunct_2(A)) = k1_rfunct_2(k2_rfunct_2(B,A)) ) ) ) )).

fof(t39_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k1_rfunct_2(A),k4_relset_1(k1_numbers,k1_numbers,k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,B)))
               => k2_rfunct_2(B,k2_rfunct_2(C,A)) = k2_rfunct_2(k1_partfun1(k1_numbers,k1_numbers,k1_numbers,k1_numbers,C,B),A) ) ) ) ) )).

fof(t40_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v2_funct_1(B) )
     => k2_funct_1(k7_relat_1(B,A)) = k7_relat_1(k2_funct_1(B),k9_relat_1(B,A)) ) )).

fof(t41_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ( ( v3_seq_4(k5_relset_1(k1_numbers,k1_numbers,A))
          & k4_seq_4(k5_relset_1(k1_numbers,k1_numbers,A)) = k5_seq_4(k5_relset_1(k1_numbers,k1_numbers,A)) )
       => r1_partfun2(k1_numbers,k1_numbers,A,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) )).

fof(t42_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & v3_seq_4(k10_relset_1(k1_numbers,k1_numbers,B,A))
          & k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,B,A)) = k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,B,A)) )
       => r1_partfun2(k1_numbers,k1_numbers,B,A) ) ) )).

fof(d2_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( r1_rfunct_2(A,B)
        <=> ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( r2_hidden(C,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A)))
                      & r2_hidden(D,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A)))
                      & ~ r1_xreal_0(D,C)
                      & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,A,C)) ) ) ) ) ) )).

fof(d3_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( r2_rfunct_2(A,B)
        <=> ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ~ ( r2_hidden(C,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A)))
                      & r2_hidden(D,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A)))
                      & ~ r1_xreal_0(D,C)
                      & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,C),k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) ) ) )).

fof(d4_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( r3_rfunct_2(A,B)
        <=> ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r2_hidden(C,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A)))
                      & r2_hidden(D,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A))) )
                   => ( r1_xreal_0(D,C)
                      | r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,C),k2_seq_1(k1_numbers,k1_numbers,A,D)) ) ) ) ) ) ) )).

fof(d5_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( r4_rfunct_2(A,B)
        <=> ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r2_hidden(C,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A)))
                      & r2_hidden(D,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,A))) )
                   => ( r1_xreal_0(D,C)
                      | r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,D),k2_seq_1(k1_numbers,k1_numbers,A,C)) ) ) ) ) ) ) )).

fof(d6_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( r5_rfunct_2(A,B)
        <=> ( r3_rfunct_2(A,B)
            | r4_rfunct_2(A,B) ) ) ) )).

fof(t43_rfunct_2,axiom,(
    $true )).

fof(t44_rfunct_2,axiom,(
    $true )).

fof(t45_rfunct_2,axiom,(
    $true )).

fof(t46_rfunct_2,axiom,(
    $true )).

fof(t47_rfunct_2,axiom,(
    $true )).

fof(t48_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r3_rfunct_2(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,k1_numbers)
           => ! [D] :
                ( m1_subset_1(D,k1_numbers)
               => ( ( r2_hidden(C,k3_xboole_0(A,k4_relset_1(k1_numbers,k1_numbers,B)))
                    & r2_hidden(D,k3_xboole_0(A,k4_relset_1(k1_numbers,k1_numbers,B)))
                    & r1_xreal_0(C,D) )
                 => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,C),k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) ) ) )).

fof(t49_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r4_rfunct_2(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,k1_numbers)
           => ! [D] :
                ( m1_subset_1(D,k1_numbers)
               => ( ( r2_hidden(C,k3_xboole_0(A,k4_relset_1(k1_numbers,k1_numbers,B)))
                    & r2_hidden(D,k3_xboole_0(A,k4_relset_1(k1_numbers,k1_numbers,B)))
                    & r1_xreal_0(C,D) )
                 => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),k2_seq_1(k1_numbers,k1_numbers,B,C)) ) ) ) ) ) )).

fof(t50_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_rfunct_2(B,A)
      <=> r1_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,B,A),A) ) ) )).

fof(t51_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r2_rfunct_2(B,A)
      <=> r2_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,B,A),A) ) ) )).

fof(t52_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r3_rfunct_2(B,A)
      <=> r3_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,B,A),A) ) ) )).

fof(t53_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r4_rfunct_2(B,A)
      <=> r4_rfunct_2(k2_partfun1(k1_numbers,k1_numbers,B,A),A) ) ) )).

fof(t54_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_xboole_0(A,k4_relset_1(k1_numbers,k1_numbers,B))
       => ( r1_rfunct_2(B,A)
          & r2_rfunct_2(B,A)
          & r3_rfunct_2(B,A)
          & r4_rfunct_2(B,A)
          & r5_rfunct_2(B,A) ) ) ) )).

fof(t55_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_rfunct_2(B,A)
       => r3_rfunct_2(B,A) ) ) )).

fof(t56_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r2_rfunct_2(B,A)
       => r4_rfunct_2(B,A) ) ) )).

fof(t57_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_partfun2(k1_numbers,k1_numbers,B,A)
       => r3_rfunct_2(B,A) ) ) )).

fof(t58_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_partfun2(k1_numbers,k1_numbers,B,A)
       => r4_rfunct_2(B,A) ) ) )).

fof(t59_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ( ( r3_rfunct_2(C,A)
          & r4_rfunct_2(C,B) )
       => r1_partfun2(k1_numbers,k1_numbers,C,k3_xboole_0(A,B)) ) ) )).

fof(t60_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,B)
          & r1_rfunct_2(C,B) )
       => r1_rfunct_2(C,A) ) ) )).

fof(t61_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,B)
          & r2_rfunct_2(C,B) )
       => r2_rfunct_2(C,A) ) ) )).

fof(t62_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,B)
          & r3_rfunct_2(C,B) )
       => r3_rfunct_2(C,A) ) ) )).

fof(t63_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,B)
          & r4_rfunct_2(C,B) )
       => r4_rfunct_2(C,A) ) ) )).

fof(t64_rfunct_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( ( r1_rfunct_2(C,A)
             => ( r1_xreal_0(B,np__0)
                | r1_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
            & ( B = np__0
             => r1_partfun2(k1_numbers,k1_numbers,k13_seq_1(k1_numbers,k1_numbers,C,B),A) )
            & ( r1_rfunct_2(C,A)
             => ( r1_xreal_0(np__0,B)
                | r2_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) )).

fof(t65_rfunct_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( ( r2_rfunct_2(C,A)
             => ( r1_xreal_0(B,np__0)
                | r2_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
            & ( r2_rfunct_2(C,A)
             => ( r1_xreal_0(np__0,B)
                | r1_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) )).

fof(t66_rfunct_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( ( ( r3_rfunct_2(C,A)
                & r1_xreal_0(np__0,B) )
             => r3_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) )
            & ( ( r3_rfunct_2(C,A)
                & r1_xreal_0(B,np__0) )
             => r4_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) )).

fof(t67_rfunct_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( ( ( r4_rfunct_2(C,A)
                & r1_xreal_0(np__0,B) )
             => r4_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) )
            & ( ( r4_rfunct_2(C,A)
                & r1_xreal_0(B,np__0) )
             => r3_rfunct_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) )).

fof(t68_rfunct_2,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_numbers)
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ! [E] :
              ( ( v1_funct_1(E)
                & m2_relset_1(E,k1_numbers,k1_numbers) )
             => ( r2_hidden(C,k3_xboole_0(k3_xboole_0(A,B),k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,D,E))))
               => ( r2_hidden(C,k3_xboole_0(A,k4_relset_1(k1_numbers,k1_numbers,D)))
                  & r2_hidden(C,k3_xboole_0(B,k4_relset_1(k1_numbers,k1_numbers,E))) ) ) ) ) ) )).

fof(t69_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( ( r1_rfunct_2(C,A)
                & r1_rfunct_2(D,B) )
             => r1_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) )
            & ( ( r2_rfunct_2(C,A)
                & r2_rfunct_2(D,B) )
             => r2_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) )
            & ( ( r3_rfunct_2(C,A)
                & r3_rfunct_2(D,B) )
             => r3_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) )
            & ( ( r4_rfunct_2(C,A)
                & r4_rfunct_2(D,B) )
             => r4_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) ) )).

fof(t70_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( ( r1_rfunct_2(C,A)
                & r1_partfun2(k1_numbers,k1_numbers,D,B) )
             => r1_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) )
            & ( ( r2_rfunct_2(C,A)
                & r1_partfun2(k1_numbers,k1_numbers,D,B) )
             => r2_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) ) )).

fof(t71_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( r1_rfunct_2(C,A)
              & r3_rfunct_2(D,B) )
           => r1_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) )).

fof(t72_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( r4_rfunct_2(C,A)
              & r1_partfun2(k1_numbers,k1_numbers,D,B) )
           => r4_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) )).

fof(t73_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( r2_rfunct_2(C,A)
              & r4_rfunct_2(D,B) )
           => r2_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) )).

fof(t74_rfunct_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( r3_rfunct_2(C,A)
              & r1_partfun2(k1_numbers,k1_numbers,D,B) )
           => r3_rfunct_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) )).

fof(t75_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => r1_rfunct_2(B,k1_tarski(A)) ) )).

fof(t76_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => r2_rfunct_2(B,k1_tarski(A)) ) )).

fof(t77_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => r3_rfunct_2(B,k1_tarski(A)) ) )).

fof(t78_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => r4_rfunct_2(B,k1_tarski(A)) ) )).

fof(t79_rfunct_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => r1_rfunct_2(k1_partfun2(k1_numbers,A),A) ) )).

fof(t80_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_rfunct_2(B,A)
       => r2_rfunct_2(k16_seq_1(k1_numbers,k1_numbers,B),A) ) ) )).

fof(t81_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r3_rfunct_2(B,A)
       => r4_rfunct_2(k16_seq_1(k1_numbers,k1_numbers,B),A) ) ) )).

fof(t82_rfunct_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_rfunct_2(C,k1_rcomp_1(A,B))
                  | r2_rfunct_2(C,k1_rcomp_1(A,B)) )
               => v2_funct_1(k2_partfun1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) ) ) ) ) )).

fof(t83_rfunct_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v2_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_rfunct_2(C,k1_rcomp_1(A,B))
               => r1_rfunct_2(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))),k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) ) ) ) ) )).

fof(t84_rfunct_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v2_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r2_rfunct_2(C,k1_rcomp_1(A,B))
               => r2_rfunct_2(k2_partfun2(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))),k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) ) ) ) ) )).

fof(dt_k1_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_subset_1(k1_rfunct_2(A),k1_zfmisc_1(k1_numbers)) ) )).

fof(redefinition_k1_rfunct_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => k1_rfunct_2(A) = k2_relat_1(A) ) )).

fof(dt_k2_rfunct_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & m1_relset_1(A,k1_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k2_rfunct_2(A,B))
        & v1_funct_2(k2_rfunct_2(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k2_rfunct_2(A,B),k5_numbers,k1_numbers) ) ) )).
%------------------------------------------------------------------------------