SET007 Axioms: SET007+97.ax


%------------------------------------------------------------------------------
% File     : SET007+97 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Pigeon Hole Principle
% 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    : finseq_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  102 (  15 unit)
%            Number of atoms       :  536 (  85 equality)
%            Maximal formula depth :   15 (   6 average)
%            Number of connectives :  473 (  39 ~  ;   0  |; 223  &)
%                                         (  20 <=>; 191 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-3 arity)
%            Number of functors    :   38 (   6 constant; 0-4 arity)
%            Number of variables   :  226 (   0 singleton; 222 !;   4 ?)
%            Maximal term depth    :    5 (   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(d1_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r1_finseq_4(A,B)
        <=> k10_relat_1(A,k9_relat_1(A,k1_tarski(B))) = k1_tarski(B) ) ) )).

fof(t1_finseq_4,axiom,(
    $true )).

fof(t2_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r1_finseq_4(A,B)
         => r2_hidden(B,k1_relat_1(A)) ) ) )).

fof(t3_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r1_finseq_4(A,B)
        <=> ( r2_hidden(B,k1_relat_1(A))
            & k10_relat_1(A,k1_tarski(k1_funct_1(A,B))) = k1_tarski(B) ) ) ) )).

fof(t4_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r1_finseq_4(A,B)
        <=> ( r2_hidden(B,k1_relat_1(A))
            & ! [C] :
                ~ ( r2_hidden(C,k1_relat_1(A))
                  & B != C
                  & k1_funct_1(A,B) = k1_funct_1(A,C) ) ) ) ) )).

fof(t5_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ! [B] :
            ( r2_hidden(B,k1_relat_1(A))
           => r1_finseq_4(A,B) )
      <=> v2_funct_1(A) ) ) )).

fof(d2_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> ? [C] :
              ( v1_finset_1(C)
              & C = k10_relat_1(A,k1_tarski(B))
              & k4_card_1(C) = np__1 ) ) ) )).

fof(t6_finseq_4,axiom,(
    $true )).

fof(t7_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => r2_hidden(B,k2_relat_1(A)) ) ) )).

fof(t8_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> ? [C] : k1_tarski(C) = k10_relat_1(A,k1_tarski(B)) ) ) )).

fof(t9_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> ? [C] :
              ( r2_hidden(C,k1_relat_1(A))
              & B = k1_funct_1(A,C)
              & ! [D] :
                  ~ ( r2_hidden(D,k1_relat_1(A))
                    & D != C
                    & k1_funct_1(A,D) = B ) ) ) ) )).

fof(t10_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k2_relat_1(A))
           => r2_finseq_4(A,B) ) ) ) )).

fof(t11_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r1_finseq_4(A,B)
        <=> ( r2_hidden(B,k1_relat_1(A))
            & r2_finseq_4(A,k1_funct_1(A,B)) ) ) ) )).

fof(d3_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => ! [C] :
              ( C = k1_finseq_4(A,B)
            <=> ( r2_hidden(C,k1_relat_1(A))
                & k1_funct_1(A,C) = B ) ) ) ) )).

fof(t12_finseq_4,axiom,(
    $true )).

fof(t13_finseq_4,axiom,(
    $true )).

fof(t14_finseq_4,axiom,(
    $true )).

fof(t15_finseq_4,axiom,(
    $true )).

fof(t16_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => k9_relat_1(A,k1_tarski(k1_finseq_4(A,B))) = k1_tarski(B) ) ) )).

fof(t17_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => k10_relat_1(A,k1_tarski(B)) = k1_tarski(k1_finseq_4(A,B)) ) ) )).

fof(t18_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v2_funct_1(A)
            & r2_hidden(B,k2_relat_1(A)) )
         => k1_funct_1(k2_funct_1(A),B) = k1_finseq_4(A,B) ) ) )).

fof(t19_finseq_4,axiom,(
    $true )).

fof(t20_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r1_finseq_4(A,B)
         => k1_finseq_4(A,k1_funct_1(A,B)) = B ) ) )).

fof(t21_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => r1_finseq_4(A,k1_finseq_4(A,B)) ) ) )).

fof(d4_finseq_4,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ! [D] :
          ( r2_hidden(D,k1_relat_1(C))
         => k4_finseq_4(A,B,C,D) = k1_funct_1(C,D) ) ) )).

fof(t22_finseq_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => k4_finseq_4(A,B,C,D) = k8_funct_2(A,B,C,D) ) ) ) ) )).

fof(t23_finseq_4,axiom,(
    $true )).

fof(t24_finseq_4,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,C)
              & r1_xreal_0(C,k3_finseq_1(B)) )
           => k4_finseq_4(k5_numbers,A,B,C) = k1_funct_1(B,C) ) ) ) )).

fof(t25_finseq_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => k4_finseq_4(k5_numbers,A,k12_finseq_1(A,B),np__1) = B ) ) )).

fof(t26_finseq_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( k4_finseq_4(k5_numbers,A,k2_finseq_4(A,B,C),np__1) = B
                & k4_finseq_4(k5_numbers,A,k2_finseq_4(A,B,C),np__2) = C ) ) ) ) )).

fof(t27_finseq_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( k4_finseq_4(k5_numbers,A,k3_finseq_4(A,B,C,D),np__1) = B
                    & k4_finseq_4(k5_numbers,A,k3_finseq_4(A,B,C,D),np__2) = C
                    & k4_finseq_4(k5_numbers,A,k3_finseq_4(A,B,C,D),np__3) = D ) ) ) ) ) )).

fof(d5_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : k5_finseq_4(A,B) = k1_funct_1(k14_finseq_1(k10_relat_1(A,k1_tarski(B))),np__1) ) )).

fof(t28_finseq_4,axiom,(
    $true )).

fof(t29_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => k1_funct_1(A,k5_finseq_4(A,B)) = B ) ) )).

fof(t30_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => r2_hidden(k5_finseq_4(A,B),k4_finseq_1(A)) ) ) )).

fof(t31_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => ( r1_xreal_0(np__1,k5_finseq_4(A,B))
            & r1_xreal_0(k5_finseq_4(A,B),k3_finseq_1(A)) ) ) ) )).

fof(t32_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => ( m2_subset_1(k6_xcmplx_0(k5_finseq_4(A,B),np__1),k1_numbers,k5_numbers)
            & m2_subset_1(k6_xcmplx_0(k3_finseq_1(A),k5_finseq_4(A,B)),k1_numbers,k5_numbers) ) ) ) )).

fof(t33_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => r2_hidden(k5_finseq_4(A,B),k10_relat_1(A,k1_tarski(B))) ) ) )).

fof(t34_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ~ ( r2_hidden(C,k4_finseq_1(A))
              & ~ r1_xreal_0(k5_finseq_4(A,B),C)
              & k1_funct_1(A,C) = B ) ) ) )).

fof(t35_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => k1_finseq_4(A,B) = k5_finseq_4(A,B) ) ) )).

fof(t36_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( r2_hidden(C,k4_finseq_1(A))
                  & C != k5_finseq_4(A,B)
                  & k1_funct_1(A,C) = B ) ) ) ) )).

fof(t37_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ ( r2_hidden(C,k4_finseq_1(A))
                    & C != k5_finseq_4(A,B)
                    & k1_funct_1(A,C) = B ) ) )
         => r2_finseq_4(A,B) ) ) )).

fof(t38_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> ( r2_hidden(B,k2_relat_1(A))
            & k1_tarski(k5_finseq_4(A,B)) = k10_relat_1(A,k1_tarski(B)) ) ) ) )).

fof(t39_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v2_funct_1(A)
            & r2_hidden(B,k2_relat_1(A)) )
         => k1_tarski(k5_finseq_4(A,B)) = k10_relat_1(A,k1_tarski(B)) ) ) )).

fof(t40_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> k3_finseq_1(k1_finseq_3(A,k1_tarski(B))) = k6_xcmplx_0(k3_finseq_1(A),np__1) ) ) )).

fof(t41_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(k1_finseq_3(A,k1_tarski(B))))
               => ( ( ~ r1_xreal_0(k5_finseq_4(A,B),C)
                   => k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,C) )
                  & ( r1_xreal_0(k5_finseq_4(A,B),C)
                   => k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,k1_nat_1(C,np__1)) ) ) ) ) ) ) )).

fof(t42_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v2_funct_1(A)
            & r2_hidden(B,k2_relat_1(A)) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k4_finseq_1(k1_finseq_3(A,k1_tarski(B))))
               => ( ~ ( k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,C)
                      & r1_xreal_0(k5_finseq_4(A,B),C) )
                  & ( ~ r1_xreal_0(k5_finseq_4(A,B),C)
                   => k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,C) )
                  & ( k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,k1_nat_1(C,np__1))
                   => r1_xreal_0(k5_finseq_4(A,B),C) )
                  & ( r1_xreal_0(k5_finseq_4(A,B),C)
                   => k1_funct_1(k1_finseq_3(A,k1_tarski(B)),C) = k1_funct_1(A,k1_nat_1(C,np__1)) ) ) ) ) ) ) )).

fof(d6_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( C = k6_finseq_4(A,B)
              <=> ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & D = k6_xcmplx_0(k5_finseq_4(A,B),np__1)
                    & C = k7_relat_1(A,k2_finseq_1(D)) ) ) ) ) ) )).

fof(t43_finseq_4,axiom,(
    $true )).

fof(t44_finseq_4,axiom,(
    $true )).

fof(t45_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ( r2_hidden(B,k2_relat_1(A))
              & C = k6_xcmplx_0(k5_finseq_4(A,B),np__1) )
           => k7_relat_1(A,k2_finseq_1(C)) = k6_finseq_4(A,B) ) ) ) )).

fof(t46_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => k3_finseq_1(k6_finseq_4(A,B)) = k6_xcmplx_0(k5_finseq_4(A,B),np__1) ) ) )).

fof(t47_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ( r2_hidden(B,k2_relat_1(A))
              & C = k6_xcmplx_0(k5_finseq_4(A,B),np__1) )
           => k4_finseq_1(k6_finseq_4(A,B)) = k2_finseq_1(C) ) ) ) )).

fof(t48_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ( r2_hidden(B,k2_relat_1(A))
              & r2_hidden(C,k4_finseq_1(k6_finseq_4(A,B))) )
           => k1_funct_1(A,C) = k1_funct_1(k6_finseq_4(A,B),C) ) ) ) )).

fof(t49_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ~ ( r2_hidden(B,k2_relat_1(A))
            & r2_hidden(B,k2_relat_1(k6_finseq_4(A,B))) ) ) )).

fof(t50_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => r1_xboole_0(k2_relat_1(k6_finseq_4(A,B)),k1_tarski(B)) ) ) )).

fof(t51_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => r1_tarski(k2_relat_1(k6_finseq_4(A,B)),k2_relat_1(A)) ) ) )).

fof(t52_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => ( k5_finseq_4(A,B) = np__1
          <=> k6_finseq_4(A,B) = k1_xboole_0 ) ) ) )).

fof(t53_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( ~ v1_xboole_0(C)
         => ( ( r2_hidden(B,k2_relat_1(A))
              & m2_finseq_1(A,C) )
           => m2_finseq_1(k6_finseq_4(A,B),C) ) ) ) )).

fof(d7_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( C = k7_finseq_4(A,B)
              <=> ( k3_finseq_1(C) = k6_xcmplx_0(k3_finseq_1(A),k5_finseq_4(A,B))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(C))
                       => k1_funct_1(C,D) = k1_funct_1(A,k1_nat_1(D,k5_finseq_4(A,B))) ) ) ) ) ) ) ) )).

fof(t54_finseq_4,axiom,(
    $true )).

fof(t55_finseq_4,axiom,(
    $true )).

fof(t56_finseq_4,axiom,(
    $true )).

fof(t57_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ( r2_hidden(B,k2_relat_1(A))
              & C = k6_xcmplx_0(k3_finseq_1(A),k5_finseq_4(A,B)) )
           => k4_finseq_1(k7_finseq_4(A,B)) = k2_finseq_1(C) ) ) ) )).

fof(t58_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( ( r2_hidden(B,k2_relat_1(A))
              & r2_hidden(C,k4_finseq_1(k7_finseq_4(A,B))) )
           => r2_hidden(k1_nat_1(C,k5_finseq_4(A,B)),k4_finseq_1(A)) ) ) ) )).

fof(t59_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => r1_tarski(k2_relat_1(k7_finseq_4(A,B)),k2_relat_1(A)) ) ) )).

fof(t60_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> ( r2_hidden(B,k2_relat_1(A))
            & ~ r2_hidden(B,k2_relat_1(k7_finseq_4(A,B))) ) ) ) )).

fof(t61_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ~ ( r2_hidden(B,k2_relat_1(A))
            & v2_funct_1(A)
            & r2_hidden(B,k2_relat_1(k7_finseq_4(A,B))) ) ) )).

fof(t62_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> ( r2_hidden(B,k2_relat_1(A))
            & r1_xboole_0(k2_relat_1(k7_finseq_4(A,B)),k1_tarski(B)) ) ) ) )).

fof(t63_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & v2_funct_1(A) )
         => r1_xboole_0(k2_relat_1(k7_finseq_4(A,B)),k1_tarski(B)) ) ) )).

fof(t64_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => ( k5_finseq_4(A,B) = k3_finseq_1(A)
          <=> k7_finseq_4(A,B) = k1_xboole_0 ) ) ) )).

fof(t65_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( ~ v1_xboole_0(C)
         => ( ( r2_hidden(B,k2_relat_1(A))
              & m2_finseq_1(A,C) )
           => m2_finseq_1(k7_finseq_4(A,B),C) ) ) ) )).

fof(t66_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_hidden(B,k2_relat_1(A))
         => A = k7_finseq_1(k7_finseq_1(k6_finseq_4(A,B),k9_finseq_1(B)),k7_finseq_4(A,B)) ) ) )).

fof(t67_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & v2_funct_1(A) )
         => v2_funct_1(k6_finseq_4(A,B)) ) ) )).

fof(t68_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & v2_funct_1(A) )
         => v2_funct_1(k7_finseq_4(A,B)) ) ) )).

fof(t69_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( r2_finseq_4(A,B)
        <=> ( r2_hidden(B,k2_relat_1(A))
            & k1_finseq_3(A,k1_tarski(B)) = k7_finseq_1(k6_finseq_4(A,B),k7_finseq_4(A,B)) ) ) ) )).

fof(t70_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & v2_funct_1(A) )
         => k1_finseq_3(A,k1_tarski(B)) = k7_finseq_1(k6_finseq_4(A,B),k7_finseq_4(A,B)) ) ) )).

fof(t71_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & v2_funct_1(k1_finseq_3(A,k1_tarski(B)))
            & k1_finseq_3(A,k1_tarski(B)) = k7_finseq_1(k6_finseq_4(A,B),k7_finseq_4(A,B)) )
         => v2_funct_1(A) ) ) )).

fof(t72_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & v2_funct_1(A) )
         => r1_xboole_0(k2_relat_1(k6_finseq_4(A,B)),k2_relat_1(k7_finseq_4(A,B))) ) ) )).

fof(t73_finseq_4,axiom,(
    ! [A] :
      ~ ( v1_finset_1(A)
        & ! [B] :
            ( ( v1_relat_1(B)
              & v1_funct_1(B)
              & v1_finseq_1(B) )
           => ~ ( k2_relat_1(B) = A
                & v2_funct_1(B) ) ) ) )).

fof(t74_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( ( r1_tarski(k2_relat_1(A),k4_finseq_1(A))
          & v2_funct_1(A) )
       => k2_relat_1(A) = k4_finseq_1(A) ) ) )).

fof(t75_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( k2_relat_1(A) = k4_finseq_1(A)
       => v2_funct_1(A) ) ) )).

fof(t76_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( k2_relat_1(A) = k2_relat_1(B)
              & k3_finseq_1(A) = k3_finseq_1(B)
              & v2_funct_1(B) )
           => v2_funct_1(A) ) ) ) )).

fof(t77_finseq_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v2_funct_1(A)
      <=> k4_card_1(k2_relat_1(A)) = k3_finseq_1(A) ) ) )).

fof(t78_finseq_4,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ( ( k4_card_1(A) = k4_card_1(B)
                  & v2_funct_1(C) )
               => k2_relat_1(C) = B ) ) ) ) )).

fof(t79_finseq_4,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ( ( k4_card_1(A) = k4_card_1(B)
                  & k2_relat_1(C) = B )
               => v2_funct_1(C) ) ) ) ) )).

fof(t80_finseq_4,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,B,A)
        & m2_relset_1(C,B,A) )
     => ~ ( r2_hidden(k1_card_1(A),k1_card_1(B))
          & A != k1_xboole_0
          & ! [D,E] :
              ~ ( r2_hidden(D,B)
                & r2_hidden(E,B)
                & D != E
                & k1_funct_1(C,D) = k1_funct_1(C,E) ) ) ) )).

fof(t81_finseq_4,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ~ ( r2_hidden(k1_card_1(A),k1_card_1(B))
          & ! [D] :
              ~ ( r2_hidden(D,B)
                & ! [E] :
                    ~ ( r2_hidden(E,A)
                      & k1_funct_1(C,E) = D ) ) ) ) )).

fof(t82_finseq_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => k4_finseq_4(k5_numbers,A,k8_finseq_1(A,B,k12_finseq_1(A,C)),k1_nat_1(k3_finseq_1(B),np__1)) = C ) ) ) )).

fof(t83_finseq_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( r2_hidden(A,k4_finseq_1(C))
                   => k4_finseq_4(k5_numbers,B,k8_finseq_1(B,C,D),A) = k4_finseq_4(k5_numbers,B,C,A) ) ) ) ) ) )).

fof(t84_finseq_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( r2_hidden(A,k4_finseq_1(D))
                   => k4_finseq_4(k5_numbers,B,k8_finseq_1(B,C,D),k1_nat_1(k3_finseq_1(C),A)) = k4_finseq_4(k5_numbers,B,D,A) ) ) ) ) ) )).

fof(t85_finseq_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C,D] :
              ( m2_finseq_1(D,C)
             => ( r2_hidden(B,k4_finseq_1(k16_finseq_1(C,D,A)))
               => k4_finseq_4(k5_numbers,C,k16_finseq_1(C,D,A),B) = k4_finseq_4(k5_numbers,C,D,B) ) ) ) ) )).

fof(t86_finseq_4,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ! [D] :
              ( m2_subset_1(D,k1_numbers,k5_numbers)
             => ( ( r2_hidden(C,k4_finseq_1(B))
                  & r2_hidden(D,k2_finseq_1(C)) )
               => ( r2_hidden(D,k4_finseq_1(B))
                  & k4_finseq_4(k5_numbers,A,k16_finseq_1(A,B,C),D) = k4_finseq_4(k5_numbers,A,B,D) ) ) ) ) ) )).

fof(dt_k1_finseq_4,axiom,(
    $true )).

fof(dt_k2_finseq_4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m2_finseq_1(k2_finseq_4(A,B,C),A) ) )).

fof(redefinition_k2_finseq_4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k2_finseq_4(A,B,C) = k10_finseq_1(B,C) ) )).

fof(dt_k3_finseq_4,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A) )
     => m2_finseq_1(k3_finseq_4(A,B,C,D),A) ) )).

fof(redefinition_k3_finseq_4,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A) )
     => k3_finseq_4(A,B,C,D) = k11_finseq_1(B,C,D) ) )).

fof(dt_k4_finseq_4,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => m1_subset_1(k4_finseq_4(A,B,C,D),B) ) )).

fof(dt_k5_finseq_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => m2_subset_1(k5_finseq_4(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k6_finseq_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_relat_1(k6_finseq_4(A,B))
        & v1_funct_1(k6_finseq_4(A,B))
        & v1_finseq_1(k6_finseq_4(A,B)) ) ) )).

fof(dt_k7_finseq_4,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_relat_1(k7_finseq_4(A,B))
        & v1_funct_1(k7_finseq_4(A,B))
        & v1_finseq_1(k7_finseq_4(A,B)) ) ) )).
%------------------------------------------------------------------------------