SET007 Axioms: SET007+179.ax


%------------------------------------------------------------------------------
% File     : SET007+179 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Properties of Restrictions of Finite Sequences
% 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_5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  105 (   5 unit)
%            Number of atoms       :  562 (  85 equality)
%            Maximal formula depth :   13 (   8 average)
%            Number of connectives :  538 (  81 ~  ;   2  |; 156  &)
%                                         (   2 <=>; 297 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   1 propositional; 0-3 arity)
%            Number of functors    :   37 (   5 constant; 0-4 arity)
%            Number of variables   :  293 (   0 singleton; 289 !;   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(cc1_finseq_5,axiom,(
    ! [A] :
      ( v1_xboole_0(A)
     => v1_realset1(A) ) )).

fof(fc1_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k5_finseq_1(A))
      & v1_relat_1(k5_finseq_1(A))
      & v1_funct_1(k5_finseq_1(A))
      & v1_realset1(k5_finseq_1(A))
      & v1_finset_1(k5_finseq_1(A))
      & v1_finseq_1(k5_finseq_1(A)) ) )).

fof(fc2_finseq_5,axiom,(
    ! [A,B] :
      ( v1_relat_1(k10_finseq_1(A,B))
      & v1_funct_1(k10_finseq_1(A,B))
      & ~ v1_realset1(k10_finseq_1(A,B))
      & v1_finset_1(k10_finseq_1(A,B))
      & v1_finseq_1(k10_finseq_1(A,B)) ) )).

fof(rc1_finseq_5,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) )).

fof(rc2_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_finseq_1(B,A)
          & ~ v1_xboole_0(B)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B) ) ) )).

fof(fc3_finseq_5,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v2_funct_1(C)
        & m1_finseq_1(C,B) )
     => ( v1_relat_1(k16_finseq_1(B,C,A))
        & v1_funct_1(k16_finseq_1(B,C,A))
        & v2_funct_1(k16_finseq_1(B,C,A))
        & v1_finset_1(k16_finseq_1(B,C,A))
        & v1_finseq_1(k16_finseq_1(B,C,A)) ) ) )).

fof(fc4_finseq_5,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & v2_funct_1(C)
        & m1_finseq_1(C,B) )
     => ( v1_relat_1(k1_rfinseq(B,C,A))
        & v1_funct_1(k1_rfinseq(B,C,A))
        & v2_funct_1(k1_rfinseq(B,C,A))
        & v1_finset_1(k1_rfinseq(B,C,A))
        & v1_finseq_1(k1_rfinseq(B,C,A)) ) ) )).

fof(fc5_finseq_5,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & v2_funct_1(C)
        & m1_finseq_1(C,A) )
     => ( v1_relat_1(k1_finseq_5(A,C,B))
        & v1_funct_1(k1_finseq_5(A,C,B))
        & v2_funct_1(k1_finseq_5(A,C,B))
        & v1_finset_1(k1_finseq_5(A,C,B))
        & v1_finseq_1(k1_finseq_5(A,C,B)) ) ) )).

fof(fc6_finseq_5,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_finseq_1(C,A) )
     => ( ~ v1_xboole_0(k2_finseq_5(A,C,B))
        & v1_relat_1(k2_finseq_5(A,C,B))
        & v1_funct_1(k2_finseq_5(A,C,B))
        & v1_finset_1(k2_finseq_5(A,C,B))
        & v1_finseq_1(k2_finseq_5(A,C,B)) ) ) )).

fof(fc7_finseq_5,axiom,(
    ! [A] :
      ( ( v1_xboole_0(A)
        & v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_xboole_0(k3_finseq_5(A))
        & v1_relat_1(k3_finseq_5(A))
        & v1_funct_1(k3_finseq_5(A))
        & v2_funct_1(k3_finseq_5(A))
        & v1_realset1(k3_finseq_5(A))
        & v1_finset_1(k3_finseq_5(A))
        & v1_finseq_1(k3_finseq_5(A)) ) ) )).

fof(fc8_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_relat_1(k3_finseq_5(A))
        & v1_funct_1(k3_finseq_5(A))
        & v2_funct_1(k3_finseq_5(A))
        & v1_finset_1(k3_finseq_5(A))
        & v1_finseq_1(k3_finseq_5(A)) ) ) )).

fof(fc9_finseq_5,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,A) )
     => ( ~ v1_xboole_0(k5_finseq_5(A,B,D,C))
        & v1_relat_1(k5_finseq_5(A,B,D,C))
        & v1_funct_1(k5_finseq_5(A,B,D,C))
        & v1_finset_1(k5_finseq_5(A,B,D,C))
        & v1_finseq_1(k5_finseq_5(A,B,D,C)) ) ) )).

fof(t1_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
           => m2_subset_1(k2_xcmplx_0(k6_xcmplx_0(B,A),np__1),k1_numbers,k5_numbers) ) ) ) )).

fof(t2_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(A,k2_finseq_1(B))
           => r2_hidden(k2_xcmplx_0(k6_xcmplx_0(B,A),np__1),k2_finseq_1(B)) ) ) ) )).

fof(t3_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( k10_relat_1(A,k1_tarski(C)) = k1_tarski(B)
         => ( r2_hidden(B,k1_relat_1(A))
            & r2_hidden(C,k2_relat_1(A))
            & k1_funct_1(A,B) = C ) ) ) )).

fof(t4_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B] :
            ( r2_hidden(B,k1_relat_1(A))
           => k10_relat_1(A,k1_tarski(k1_funct_1(A,B))) = k1_tarski(B) ) ) ) )).

fof(t5_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( ( v2_funct_1(A)
            & r2_hidden(B,k2_relat_1(A))
            & r2_hidden(C,k2_relat_1(A))
            & k10_relat_1(A,k1_tarski(B)) = k10_relat_1(A,k1_tarski(C)) )
         => B = C ) ) )).

fof(t6_finseq_5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( r2_hidden(np__1,k4_finseq_1(A))
        & r2_hidden(k3_finseq_1(A),k4_finseq_1(A)) ) ) )).

fof(t7_finseq_5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & k1_nat_1(B,np__1) = k3_finseq_1(A) ) ) )).

fof(t8_finseq_5,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => k3_finseq_1(k7_finseq_1(k9_finseq_1(A),B)) = k1_nat_1(np__1,k3_finseq_1(B)) ) )).

fof(t9_finseq_5,axiom,(
    $true )).

fof(t10_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B,C] :
          ( ( r2_hidden(B,k2_relat_1(A))
            & r2_hidden(C,k2_relat_1(A))
            & k5_finseq_4(A,B) = k5_finseq_4(A,C) )
         => B = C ) ) )).

fof(t11_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( ( r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(B))
                  & C = k7_relat_1(B,k2_finseq_1(A)) )
               => k7_relat_1(B,k2_finseq_1(k1_nat_1(A,np__1))) = k7_finseq_1(C,k9_finseq_1(k1_funct_1(B,k1_nat_1(A,np__1)))) ) ) ) ) )).

fof(t12_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v2_funct_1(B)
            & v1_finseq_1(B) )
         => ( r2_hidden(A,k4_finseq_1(B))
           => k5_finseq_4(B,k1_funct_1(B,A)) = A ) ) ) )).

fof(t13_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( ( k4_finseq_1(B) = k4_finseq_1(C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(B))
                       => k4_finseq_4(k5_numbers,A,B,D) = k4_finseq_4(k5_numbers,A,C,D) ) ) )
               => B = C ) ) ) ) )).

fof(t14_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k3_finseq_1(B)) )
                       => k4_finseq_4(k5_numbers,A,B,D) = k4_finseq_4(k5_numbers,A,C,D) ) ) )
               => B = C ) ) ) ) )).

fof(t15_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ( k3_finseq_1(B) = np__1
           => B = k12_finseq_1(A,k4_finseq_4(k5_numbers,A,B,np__1)) ) ) ) )).

fof(t16_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => k4_finseq_4(k5_numbers,A,k8_finseq_1(A,k12_finseq_1(A,B),C),np__1) = B ) ) ) )).

fof(t17_finseq_5,axiom,(
    $true )).

fof(t18_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] :
          ( m2_finseq_1(C,B)
         => r1_xreal_0(k3_finseq_1(k16_finseq_1(B,C,A)),k3_finseq_1(C)) ) ) )).

fof(t19_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] :
          ( m2_finseq_1(C,B)
         => r1_xreal_0(k3_finseq_1(k16_finseq_1(B,C,A)),A) ) ) )).

fof(t20_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] :
          ( m2_finseq_1(C,B)
         => r1_tarski(k4_finseq_1(k16_finseq_1(B,C,A)),k4_finseq_1(C)) ) ) )).

fof(t21_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => r1_tarski(k2_relat_1(k16_finseq_1(B,C,A)),k2_relat_1(C)) ) ) ) )).

fof(t22_finseq_5,axiom,(
    $true )).

fof(t23_finseq_5,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ( ~ v1_xboole_0(B)
       => k16_finseq_1(A,B,np__1) = k9_finseq_1(k4_finseq_4(k5_numbers,A,B,np__1)) ) ) )).

fof(t24_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ( k1_nat_1(A,np__1) = k3_finseq_1(C)
               => C = k8_finseq_1(B,k16_finseq_1(B,C,A),k12_finseq_1(B,k4_finseq_4(k5_numbers,B,C,k3_finseq_1(C)))) ) ) ) ) )).

fof(t25_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] :
          ( m2_finseq_1(C,B)
         => ! [D] :
              ( m2_finseq_1(D,B)
             => ( r1_xreal_0(A,k3_finseq_1(C))
               => k16_finseq_1(B,k8_finseq_1(B,C,D),A) = k16_finseq_1(B,C,A) ) ) ) ) )).

fof(t26_finseq_5,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_finseq_1(C,A)
         => k16_finseq_1(A,k8_finseq_1(A,B,C),k3_finseq_1(B)) = B ) ) )).

fof(t27_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C,D] :
              ( m2_finseq_1(D,C)
             => ( r2_hidden(B,k2_relat_1(D))
               => k7_finseq_1(k6_finseq_4(D,B),k12_finseq_1(A,B)) = k16_finseq_1(C,D,k5_finseq_4(D,B)) ) ) ) ) )).

fof(t28_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => r1_xreal_0(k3_finseq_1(k1_rfinseq(B,C,A)),k3_finseq_1(C)) ) ) ) )).

fof(t29_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m2_finseq_1(D,C)
                 => ( r2_hidden(A,k4_finseq_1(k1_rfinseq(C,D,B)))
                   => r2_hidden(k1_nat_1(B,A),k4_finseq_1(D)) ) ) ) ) ) )).

fof(t30_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m2_finseq_1(D,C)
                 => ( r2_hidden(A,k4_finseq_1(k1_rfinseq(C,D,B)))
                   => k4_finseq_4(k5_numbers,C,k1_rfinseq(C,D,B),A) = k4_finseq_4(k5_numbers,C,D,k1_nat_1(B,A)) ) ) ) ) ) )).

fof(t31_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => k1_rfinseq(A,B,np__0) = B ) ) )).

fof(t32_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ( ~ v1_xboole_0(B)
           => B = k8_finseq_1(A,k12_finseq_1(A,k4_finseq_4(k5_numbers,A,B,np__1)),k1_rfinseq(A,B,np__1)) ) ) ) )).

fof(t33_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ( k1_nat_1(A,np__1) = k3_finseq_1(C)
               => k1_rfinseq(B,C,A) = k12_finseq_1(B,k4_finseq_4(k5_numbers,B,C,k3_finseq_1(C))) ) ) ) ) )).

fof(t34_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m2_finseq_1(D,C)
                 => ( ( k1_nat_1(A,np__1) = B
                      & r2_hidden(B,k4_finseq_1(D)) )
                   => k8_finseq_1(C,k12_finseq_1(C,k4_finseq_4(k5_numbers,C,D,B)),k1_rfinseq(C,D,B)) = k1_rfinseq(C,D,A) ) ) ) ) ) )).

fof(t35_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] :
          ( m2_finseq_1(C,B)
         => ( r1_xreal_0(k3_finseq_1(C),A)
           => v1_xboole_0(k1_rfinseq(B,C,A)) ) ) ) )).

fof(t36_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => r1_tarski(k2_relat_1(k1_rfinseq(B,C,A)),k2_relat_1(C)) ) ) ) )).

fof(t37_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ( v2_funct_1(C)
               => r1_xboole_0(k2_relat_1(k16_finseq_1(B,C,A)),k2_relat_1(k1_rfinseq(B,C,A))) ) ) ) ) )).

fof(t38_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => k7_finseq_4(C,B) = k1_rfinseq(A,C,k5_finseq_4(C,B)) ) ) ) ) )).

fof(t39_finseq_5,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)
                 => k1_rfinseq(B,k8_finseq_1(B,C,D),k1_nat_1(k3_finseq_1(C),A)) = k1_rfinseq(B,D,A) ) ) ) ) )).

fof(t40_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => k1_rfinseq(A,k8_finseq_1(A,B,C),k3_finseq_1(B)) = C ) ) ) )).

fof(t41_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => k4_finseq_4(k5_numbers,A,C,k5_finseq_4(C,B)) = B ) ) ) ) )).

fof(t42_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ( r2_hidden(A,k4_finseq_1(C))
               => r1_xreal_0(k5_finseq_4(C,k4_finseq_4(k5_numbers,B,C,A)),A) ) ) ) ) )).

fof(t43_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( r2_hidden(C,k2_relat_1(k16_finseq_1(B,D,A)))
                   => k5_finseq_4(k16_finseq_1(B,D,A),C) = k5_finseq_4(D,C) ) ) ) ) ) )).

fof(t44_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ( ( r2_hidden(A,k4_finseq_1(C))
                  & v2_funct_1(C) )
               => k5_finseq_4(C,k4_finseq_4(k5_numbers,B,C,A)) = A ) ) ) ) )).

fof(d1_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] : k1_finseq_5(A,B,C) = k16_finseq_1(A,B,k5_finseq_4(B,C)) ) ) )).

fof(t45_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => k3_finseq_1(k1_finseq_5(A,C,B)) = k5_finseq_4(C,B) ) ) ) ) )).

fof(t46_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( ( r2_hidden(C,k2_relat_1(D))
                      & r2_hidden(A,k2_finseq_1(k5_finseq_4(D,C))) )
                   => k4_finseq_4(k5_numbers,B,k1_finseq_5(B,D,C),A) = k4_finseq_4(k5_numbers,B,D,A) ) ) ) ) ) )).

fof(t47_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => k4_finseq_4(k5_numbers,A,k1_finseq_5(A,C,B),np__1) = k4_finseq_4(k5_numbers,A,C,np__1) ) ) ) ) )).

fof(t48_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => k4_finseq_4(k5_numbers,A,k1_finseq_5(A,C,B),k5_finseq_4(C,B)) = B ) ) ) ) )).

fof(t49_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ! [D] :
                  ( ( r2_hidden(D,k2_relat_1(C))
                    & r2_hidden(B,k2_relat_1(C))
                    & r1_xreal_0(k5_finseq_4(C,D),k5_finseq_4(C,B)) )
                 => r2_hidden(D,k2_relat_1(k1_finseq_5(A,C,B))) ) ) ) ) )).

fof(t50_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ~ ( r2_hidden(B,k2_relat_1(C))
                  & v1_xboole_0(k1_finseq_5(A,C,B)) ) ) ) ) )).

fof(t51_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => r1_tarski(k2_relat_1(k1_finseq_5(A,C,B)),k2_relat_1(C)) ) ) ) )).

fof(d2_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => k2_finseq_5(A,B,C) = k8_finseq_1(A,k12_finseq_1(A,C),k1_rfinseq(A,B,k5_finseq_4(B,C))) ) ) ) )).

fof(t52_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ~ ( r2_hidden(B,k2_relat_1(C))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( k1_nat_1(D,np__1) = k5_finseq_4(C,B)
                          & k2_finseq_5(A,C,B) = k1_rfinseq(A,C,D) ) ) ) ) ) ) )).

fof(t53_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => k3_finseq_1(k2_finseq_5(A,C,B)) = k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(C),k5_finseq_4(C,B)),np__1) ) ) ) ) )).

fof(t54_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( ( r2_hidden(C,k2_relat_1(D))
                      & r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(k2_finseq_5(B,D,C))) )
                   => r2_hidden(k1_nat_1(A,k5_finseq_4(D,C)),k4_finseq_1(D)) ) ) ) ) ) )).

fof(t55_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( ( r2_hidden(C,k2_relat_1(D))
                      & r2_hidden(k1_nat_1(A,np__1),k4_finseq_1(k2_finseq_5(B,D,C))) )
                   => k4_finseq_4(k5_numbers,B,k2_finseq_5(B,D,C),k1_nat_1(A,np__1)) = k4_finseq_4(k5_numbers,B,D,k1_nat_1(A,k5_finseq_4(D,C))) ) ) ) ) ) )).

fof(t56_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => k4_finseq_4(k5_numbers,A,k2_finseq_5(A,C,B),np__1) = B ) ) ) )).

fof(t57_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => k4_finseq_4(k5_numbers,A,k2_finseq_5(A,C,B),k3_finseq_1(k2_finseq_5(A,C,B))) = k4_finseq_4(k5_numbers,A,C,k3_finseq_1(C)) ) ) ) ) )).

fof(t58_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( r2_hidden(B,k2_relat_1(C))
               => r1_tarski(k2_relat_1(k2_finseq_5(A,C,B)),k2_relat_1(C)) ) ) ) ) )).

fof(t59_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => ( ( r2_hidden(B,k2_relat_1(C))
                  & v2_funct_1(C) )
               => v2_funct_1(k2_finseq_5(A,C,B)) ) ) ) ) )).

fof(d3_finseq_5,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) )
         => ( B = k3_finseq_5(A)
          <=> ( k3_finseq_1(B) = k3_finseq_1(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k4_finseq_1(B))
                   => k1_funct_1(B,C) = k1_funct_1(A,k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(A),C),np__1)) ) ) ) ) ) ) )).

fof(t60_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( k4_finseq_1(A) = k4_finseq_1(k3_finseq_5(A))
        & k2_relat_1(A) = k2_relat_1(k3_finseq_5(A)) ) ) )).

fof(t61_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( r2_hidden(A,k4_finseq_1(B))
           => k1_funct_1(k3_finseq_5(B),A) = k1_funct_1(B,k2_xcmplx_0(k6_xcmplx_0(k3_finseq_1(B),A),np__1)) ) ) ) )).

fof(t62_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r2_hidden(B,k4_finseq_1(A))
                  & k1_nat_1(B,C) = k1_nat_1(k3_finseq_1(A),np__1) )
               => r2_hidden(C,k4_finseq_1(k3_finseq_5(A))) ) ) ) ) )).

fof(t63_finseq_5,axiom,(
    ! [A] : k3_finseq_5(k9_finseq_1(A)) = k9_finseq_1(A) )).

fof(t64_finseq_5,axiom,(
    ! [A,B] : k3_finseq_5(k10_finseq_1(A,B)) = k10_finseq_1(B,A) )).

fof(t65_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( k1_funct_1(A,np__1) = k1_funct_1(k3_finseq_5(A),k3_finseq_1(A))
        & k1_funct_1(A,k3_finseq_1(A)) = k1_funct_1(k3_finseq_5(A),np__1) ) ) )).

fof(t66_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] : k3_finseq_5(k7_finseq_1(A,k9_finseq_1(B))) = k7_finseq_1(k9_finseq_1(B),k3_finseq_5(A)) ) )).

fof(t67_finseq_5,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) )
         => k3_finseq_5(k7_finseq_1(A,B)) = k7_finseq_1(k3_finseq_5(B),k3_finseq_5(A)) ) ) )).

fof(t68_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ( ~ v1_xboole_0(B)
           => ( k4_finseq_4(k5_numbers,A,B,np__1) = k4_finseq_4(k5_numbers,A,k4_finseq_5(A,B),k3_finseq_1(B))
              & k4_finseq_4(k5_numbers,A,B,k3_finseq_1(B)) = k4_finseq_4(k5_numbers,A,k4_finseq_5(A,B),np__1) ) ) ) ) )).

fof(t69_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m2_finseq_1(D,C)
                 => ( ( r2_hidden(A,k4_finseq_1(D))
                      & k1_nat_1(A,B) = k1_nat_1(k3_finseq_1(D),np__1) )
                   => k4_finseq_4(k5_numbers,C,D,A) = k4_finseq_4(k5_numbers,C,k4_finseq_5(C,D),B) ) ) ) ) ) )).

fof(d4_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_finseq_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => k5_finseq_5(A,B,C,D) = k8_finseq_1(A,k8_finseq_1(A,k16_finseq_1(A,B,D),k12_finseq_1(A,C)),k1_rfinseq(A,B,D)) ) ) ) ) )).

fof(t70_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( m2_finseq_1(C,A)
             => k5_finseq_5(A,C,B,np__0) = k8_finseq_1(A,k12_finseq_1(A,B),C) ) ) ) )).

fof(t71_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( r1_xreal_0(k3_finseq_1(D),A)
                   => k5_finseq_5(B,D,C,A) = k8_finseq_1(B,D,k12_finseq_1(B,C)) ) ) ) ) ) )).

fof(t72_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => k3_finseq_1(k5_finseq_5(B,D,C,A)) = k1_nat_1(k3_finseq_1(D),np__1) ) ) ) ) )).

fof(t73_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => k2_relat_1(k5_finseq_5(B,D,C,A)) = k2_xboole_0(k1_tarski(C),k2_relat_1(D)) ) ) ) ) )).

fof(t74_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => r2_hidden(C,k2_relat_1(k5_finseq_5(B,D,C,A))) ) ) ) ) )).

fof(t75_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,C)
                 => ! [E] :
                      ( m2_finseq_1(E,C)
                     => ( r2_hidden(A,k4_finseq_1(k16_finseq_1(C,E,B)))
                       => k4_finseq_4(k5_numbers,C,k5_finseq_5(C,E,D,B),A) = k4_finseq_4(k5_numbers,C,E,A) ) ) ) ) ) ) )).

fof(t76_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( r1_xreal_0(A,k3_finseq_1(D))
                   => k4_finseq_4(k5_numbers,B,k5_finseq_5(B,D,C,A),k1_nat_1(A,np__1)) = C ) ) ) ) ) )).

fof(t77_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,C)
                 => ! [E] :
                      ( m2_finseq_1(E,C)
                     => ( ( r1_xreal_0(k1_nat_1(A,np__1),B)
                          & r1_xreal_0(B,k3_finseq_1(E)) )
                       => k4_finseq_4(k5_numbers,C,k5_finseq_5(C,E,D,A),k1_nat_1(B,np__1)) = k4_finseq_4(k5_numbers,C,E,B) ) ) ) ) ) ) )).

fof(t78_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( r1_xreal_0(np__1,A)
                   => ( v1_xboole_0(D)
                      | k4_finseq_4(k5_numbers,B,k5_finseq_5(B,D,C,A),np__1) = k4_finseq_4(k5_numbers,B,D,np__1) ) ) ) ) ) ) )).

fof(t79_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_1(D,B)
                 => ( v2_funct_1(D)
                   => ( r2_hidden(C,k2_relat_1(D))
                      | v2_funct_1(k5_finseq_5(B,D,C,A)) ) ) ) ) ) ) )).

fof(t80_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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)
                 => ( r1_xreal_0(C,D)
                   => ( k16_finseq_1(A,k16_finseq_1(A,B,C),D) = k16_finseq_1(A,B,C)
                      & k16_finseq_1(A,k16_finseq_1(A,B,D),C) = k16_finseq_1(A,B,C) ) ) ) ) ) ) )).

fof(t81_finseq_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => k16_finseq_1(B,k6_finseq_1(B),A) = k6_finseq_1(B) ) ) )).

fof(t82_finseq_5,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_finseq_5(A,k6_finseq_1(A)) = k6_finseq_1(A) ) )).

fof(s1_finseq_5,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finseq_1(A)
      & k3_finseq_1(A) = f1_s1_finseq_5
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k4_finseq_1(A))
           => k1_funct_1(A,B) = f2_s1_finseq_5(B) ) ) ) )).

fof(dt_k1_finseq_5,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A) )
     => m2_finseq_1(k1_finseq_5(A,B,C),A) ) )).

fof(dt_k2_finseq_5,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A)
        & m1_subset_1(C,A) )
     => m2_finseq_1(k2_finseq_5(A,B,C),A) ) )).

fof(dt_k3_finseq_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_relat_1(k3_finseq_5(A))
        & v1_funct_1(k3_finseq_5(A))
        & v1_finseq_1(k3_finseq_5(A)) ) ) )).

fof(dt_k4_finseq_5,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A) )
     => m2_finseq_1(k4_finseq_5(A,B),A) ) )).

fof(redefinition_k4_finseq_5,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A) )
     => k4_finseq_5(A,B) = k3_finseq_5(B) ) )).

fof(dt_k5_finseq_5,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_finseq_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,k5_numbers) )
     => m2_finseq_1(k5_finseq_5(A,B,C,D),A) ) )).
%------------------------------------------------------------------------------