## SET007 Axioms: SET007+71.ax

```%------------------------------------------------------------------------------
% File     : SET007+71 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Non-contiguous Substrings and One-to-one 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  126 (  24 unit)
%            Number of atoms       :  601 ( 140 equality)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :  522 (  47 ~  ;  12  |; 216  &)
%                                         (  24 <=>; 223 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   1 propositional; 0-3 arity)
%            Number of functors    :   48 (  12 constant; 0-8 arity)
%            Number of variables   :  278 (   0 singleton; 276 !;   2 ?)
%            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(t1_finseq_3,axiom,(
k2_finseq_1(np__3) = k1_enumset1(np__1,np__2,np__3) )).

fof(t2_finseq_3,axiom,(
k2_finseq_1(np__4) = k2_enumset1(np__1,np__2,np__3,np__4) )).

fof(t3_finseq_3,axiom,(
k2_finseq_1(np__5) = k3_enumset1(np__1,np__2,np__3,np__4,np__5) )).

fof(t4_finseq_3,axiom,(
k2_finseq_1(np__6) = k4_enumset1(np__1,np__2,np__3,np__4,np__5,np__6) )).

fof(t5_finseq_3,axiom,(
k2_finseq_1(np__7) = k5_enumset1(np__1,np__2,np__3,np__4,np__5,np__6,np__7) )).

fof(t6_finseq_3,axiom,(
k2_finseq_1(np__8) = k6_enumset1(np__1,np__2,np__3,np__4,np__5,np__6,np__7,np__8) )).

fof(t7_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ( k2_finseq_1(A) = k1_xboole_0
<=> ~ r2_hidden(A,k2_finseq_1(A)) ) ) )).

fof(t8_finseq_3,axiom,(
\$true )).

fof(t9_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ~ r2_hidden(k2_xcmplx_0(A,np__1),k2_finseq_1(A)) ) )).

fof(t10_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( A != np__0
=> r2_hidden(A,k2_finseq_1(k2_xcmplx_0(A,B))) ) ) ) )).

fof(t11_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r2_hidden(k2_xcmplx_0(A,B),k2_finseq_1(A))
=> B = np__0 ) ) ) )).

fof(t12_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(B,A)
=> r2_hidden(k2_xcmplx_0(A,np__1),k2_finseq_1(B)) ) ) ) )).

fof(t13_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r2_hidden(A,k2_finseq_1(B))
=> ( r1_xreal_0(A,C)
| r2_hidden(k6_xcmplx_0(A,C),k2_finseq_1(B)) ) ) ) ) ) )).

fof(t14_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r2_hidden(k6_xcmplx_0(A,B),k2_finseq_1(A))
<=> ~ r1_xreal_0(A,B) ) ) ) )).

fof(t15_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> r1_xboole_0(k2_finseq_1(A),k1_tarski(k2_xcmplx_0(A,np__1))) ) )).

fof(t16_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> k6_subset_1(k5_numbers,k2_finseq_1(k2_xcmplx_0(A,np__1)),k2_finseq_1(A)) = k1_tarski(k2_xcmplx_0(A,np__1)) ) )).

fof(t17_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> k2_finseq_1(A) != k2_finseq_1(k2_xcmplx_0(A,np__1)) ) )).

fof(t18_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( k2_finseq_1(A) = k2_finseq_1(k2_xcmplx_0(A,B))
=> B = np__0 ) ) ) )).

fof(t19_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> r1_tarski(k2_finseq_1(A),k2_finseq_1(k2_xcmplx_0(A,B))) ) ) )).

fof(t20_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> r3_xboole_0(k2_finseq_1(A),k2_finseq_1(B)) ) ) )).

fof(t21_finseq_3,axiom,(
\$true )).

fof(t22_finseq_3,axiom,(
! [A,B] :
( v4_ordinal2(B)
=> ( k2_finseq_1(B) = k1_tarski(A)
=> ( B = np__1
& A = np__1 ) ) ) )).

fof(t23_finseq_3,axiom,(
! [A,B,C] :
( v4_ordinal2(C)
=> ( k2_finseq_1(C) = k2_tarski(A,B)
=> ( A = B
| ( C = np__2
& k2_tarski(A,B) = k2_tarski(np__1,np__2) ) ) ) ) )).

fof(t24_finseq_3,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) )
=> ! [C] :
( r2_hidden(C,k4_finseq_1(A))
=> r2_hidden(C,k4_finseq_1(k7_finseq_1(A,B))) ) ) ) )).

fof(t25_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k4_finseq_1(A))
=> m2_subset_1(B,k1_numbers,k5_numbers) ) ) )).

fof(t26_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
~ ( r2_hidden(B,k4_finseq_1(A))
& B = np__0 ) ) )).

fof(t27_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( v4_ordinal2(B)
=> ( r2_hidden(B,k4_finseq_1(A))
<=> ( r1_xreal_0(np__1,B)
& r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) )).

fof(t28_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( v4_ordinal2(B)
=> ( r2_hidden(B,k4_finseq_1(A))
<=> ( m2_subset_1(k6_xcmplx_0(B,np__1),k1_numbers,k5_numbers)
& m2_subset_1(k6_xcmplx_0(k3_finseq_1(A),B),k1_numbers,k5_numbers) ) ) ) ) )).

fof(t29_finseq_3,axiom,(
! [A,B] : k4_finseq_1(k10_finseq_1(A,B)) = k2_finseq_1(np__2) )).

fof(t30_finseq_3,axiom,(
! [A,B,C] : k4_finseq_1(k11_finseq_1(A,B,C)) = k2_finseq_1(np__3) )).

fof(t31_finseq_3,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_1(A) = k3_finseq_1(B)
<=> k4_finseq_1(A) = k4_finseq_1(B) ) ) ) )).

fof(t32_finseq_3,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) )
=> ( r1_xreal_0(k3_finseq_1(A),k3_finseq_1(B))
<=> r1_tarski(k4_finseq_1(A),k4_finseq_1(B)) ) ) ) )).

fof(t33_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> r2_hidden(np__1,k4_finseq_1(A)) ) ) )).

fof(t34_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( k2_relat_1(A) != k1_xboole_0
=> r2_hidden(np__1,k4_finseq_1(A)) ) ) )).

fof(t35_finseq_3,axiom,(
\$true )).

fof(t36_finseq_3,axiom,(
\$true )).

fof(t37_finseq_3,axiom,(
\$true )).

fof(t38_finseq_3,axiom,(
! [A,B] : k1_xboole_0 != k10_finseq_1(A,B) )).

fof(t39_finseq_3,axiom,(
! [A,B,C] : k1_xboole_0 != k11_finseq_1(A,B,C) )).

fof(t40_finseq_3,axiom,(
! [A,B,C] : k9_finseq_1(A) != k10_finseq_1(B,C) )).

fof(t41_finseq_3,axiom,(
! [A,B,C,D] : k9_finseq_1(A) != k11_finseq_1(B,C,D) )).

fof(t42_finseq_3,axiom,(
! [A,B,C,D,E] : k10_finseq_1(A,B) != k11_finseq_1(C,D,E) )).

fof(t43_finseq_3,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) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( k3_finseq_1(A) = k1_nat_1(k3_finseq_1(B),k3_finseq_1(C))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> k1_funct_1(A,D) = k1_funct_1(B,D) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> k1_funct_1(A,k1_nat_1(k3_finseq_1(B),D)) = k1_funct_1(C,D) ) ) )
=> A = k7_finseq_1(B,C) ) ) ) ) )).

fof(t44_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_finset_1(B)
=> ( r1_tarski(B,k2_finseq_1(A))
=> k3_finseq_1(k14_finseq_1(B)) = k4_card_1(B) ) ) ) )).

fof(t45_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v1_finset_1(B)
=> ( r1_tarski(B,k2_finseq_1(A))
=> k4_finseq_1(k14_finseq_1(B)) = k2_finseq_1(k4_card_1(B)) ) ) ) )).

fof(t46_finseq_3,axiom,(
! [A,B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ! [E] :
( v4_ordinal2(E)
=> ! [F] :
( v4_ordinal2(F)
=> ~ ( r1_tarski(A,k2_finseq_1(B))
& ~ r1_xreal_0(D,C)
& r1_xreal_0(np__1,E)
& r1_xreal_0(F,k3_finseq_1(k14_finseq_1(A)))
& k1_funct_1(k14_finseq_1(A),F) = C
& k1_funct_1(k14_finseq_1(A),E) = D
& r1_xreal_0(E,F) ) ) ) ) ) ) )).

fof(t47_finseq_3,axiom,(
\$true )).

fof(t48_finseq_3,axiom,(
! [A,B,C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( ( r1_tarski(A,k2_finseq_1(C))
& r1_tarski(B,k2_finseq_1(D)) )
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(E,A)
& r2_hidden(F,B)
& r1_xreal_0(F,E) ) ) )
<=> k14_finseq_1(k2_xboole_0(A,B)) = k8_finseq_1(k5_numbers,k14_finseq_1(A),k14_finseq_1(B)) ) ) ) ) )).

fof(t49_finseq_3,axiom,(
k14_finseq_1(k1_xboole_0) = k1_xboole_0 )).

fof(t50_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ( np__0 != A
=> k14_finseq_1(k1_tarski(A)) = k9_finseq_1(A) ) ) )).

fof(t51_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,A)
& k14_finseq_1(k2_tarski(A,B)) != k10_finseq_1(A,B) ) ) ) )).

fof(t52_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> k3_finseq_1(k14_finseq_1(k2_finseq_1(A))) = A ) )).

fof(t53_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k7_relat_1(k14_finseq_1(k2_finseq_1(k2_xcmplx_0(A,B))),k2_finseq_1(A)) = k14_finseq_1(k2_finseq_1(A)) ) ) )).

fof(t54_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> k14_finseq_1(k2_finseq_1(A)) = k1_finseq_2(A) ) )).

fof(t55_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( v4_ordinal2(B)
=> ( k7_relat_1(A,k2_finseq_1(B)) = A
<=> r1_xreal_0(k3_finseq_1(A),B) ) ) ) )).

fof(t56_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k7_relat_1(k1_finseq_2(k2_xcmplx_0(A,B)),k2_finseq_1(A)) = k1_finseq_2(A) ) ) )).

fof(t57_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( k7_relat_1(k1_finseq_2(A),k2_finseq_1(B)) = k1_finseq_2(B)
<=> r1_xreal_0(B,A) ) ) ) )).

fof(t58_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( k7_relat_1(k1_finseq_2(A),k2_finseq_1(B)) = k1_finseq_2(A)
<=> r1_xreal_0(A,B) ) ) ) )).

fof(t59_finseq_3,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) )
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( ( k3_finseq_1(A) = k2_xcmplx_0(C,D)
& B = k7_relat_1(A,k2_finseq_1(C)) )
=> k3_finseq_1(B) = C ) ) ) ) ) )).

fof(t60_finseq_3,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) )
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( ( k3_finseq_1(A) = k2_xcmplx_0(C,D)
& B = k7_relat_1(A,k2_finseq_1(C)) )
=> k4_finseq_1(B) = k2_finseq_1(C) ) ) ) ) ) )).

fof(t61_finseq_3,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) )
=> ! [C] :
( v4_ordinal2(C)
=> ( ( k3_finseq_1(A) = k2_xcmplx_0(C,np__1)
& B = k7_relat_1(A,k2_finseq_1(C)) )
=> A = k7_finseq_1(B,k9_finseq_1(k1_funct_1(A,k2_xcmplx_0(C,np__1)))) ) ) ) ) )).

fof(t62_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B))
& v1_finseq_1(k7_relat_1(A,B)) )
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& k3_xboole_0(B,k4_finseq_1(A)) = k2_finseq_1(C) ) ) ) )).

fof(t63_finseq_3,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) )
=> ! [C] : k4_card_1(k10_relat_1(k7_finseq_1(A,B),C)) = k1_nat_1(k4_card_1(k10_relat_1(A,C)),k4_card_1(k10_relat_1(B,C))) ) ) )).

fof(t64_finseq_3,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) )
=> ! [C] : r1_tarski(k10_relat_1(A,C),k10_relat_1(k7_finseq_1(A,B),C)) ) ) )).

fof(d1_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : k1_finseq_3(A,B) = k5_relat_1(k14_finseq_1(k4_xboole_0(k4_finseq_1(A),k10_relat_1(A,B))),A) ) )).

fof(t65_finseq_3,axiom,(
\$true )).

fof(t66_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : k3_finseq_1(k1_finseq_3(A,B)) = k6_xcmplx_0(k3_finseq_1(A),k4_card_1(k10_relat_1(A,B))) ) )).

fof(t67_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : r1_xreal_0(k3_finseq_1(k1_finseq_3(A,B)),k3_finseq_1(A)) ) )).

fof(t68_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( k3_finseq_1(k1_finseq_3(A,B)) = k3_finseq_1(A)
=> r1_xboole_0(B,k2_relat_1(A)) ) ) )).

fof(t69_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( v4_ordinal2(C)
=> ( C = k6_xcmplx_0(k3_finseq_1(A),k4_card_1(k10_relat_1(A,B)))
=> k4_finseq_1(k1_finseq_3(A,B)) = k2_finseq_1(C) ) ) ) )).

fof(t70_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : r1_tarski(k4_finseq_1(k1_finseq_3(A,B)),k4_finseq_1(A)) ) )).

fof(t71_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( k4_finseq_1(k1_finseq_3(A,B)) = k4_finseq_1(A)
=> r1_xboole_0(B,k2_relat_1(A)) ) ) )).

fof(t72_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : k2_relat_1(k1_finseq_3(A,B)) = k4_xboole_0(k2_relat_1(A),B) ) )).

fof(t73_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] : r1_tarski(k2_relat_1(k1_finseq_3(A,B)),k2_relat_1(A)) ) )).

fof(t74_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( k2_relat_1(k1_finseq_3(A,B)) = k2_relat_1(A)
=> r1_xboole_0(B,k2_relat_1(A)) ) ) )).

fof(t75_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( k1_finseq_3(A,B) = k1_xboole_0
<=> r1_tarski(k2_relat_1(A),B) ) ) )).

fof(t76_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( k1_finseq_3(A,B) = A
<=> r1_xboole_0(B,k2_relat_1(A)) ) ) )).

fof(t77_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( k1_finseq_3(A,k1_tarski(B)) = A
<=> ~ r2_hidden(B,k2_relat_1(A)) ) ) )).

fof(t78_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> k1_finseq_3(A,k1_xboole_0) = A ) )).

fof(t79_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> k1_finseq_3(A,k2_relat_1(A)) = k1_xboole_0 ) )).

fof(t80_finseq_3,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) )
=> ! [C] : k1_finseq_3(k7_finseq_1(A,B),C) = k7_finseq_1(k1_finseq_3(A,C),k1_finseq_3(B,C)) ) ) )).

fof(t81_finseq_3,axiom,(
! [A] : k1_finseq_3(k1_xboole_0,A) = k1_xboole_0 )).

fof(t82_finseq_3,axiom,(
! [A,B] :
( k1_finseq_3(k9_finseq_1(A),B) = k9_finseq_1(A)
<=> ~ r2_hidden(A,B) ) )).

fof(t83_finseq_3,axiom,(
! [A,B] :
( k1_finseq_3(k9_finseq_1(A),B) = k1_xboole_0
<=> r2_hidden(A,B) ) )).

fof(t84_finseq_3,axiom,(
! [A,B,C] :
( k1_finseq_3(k10_finseq_1(A,B),C) = k1_xboole_0
<=> ( r2_hidden(A,C)
& r2_hidden(B,C) ) ) )).

fof(t85_finseq_3,axiom,(
! [A,B,C] :
( r2_hidden(A,B)
=> ( r2_hidden(C,B)
| k1_finseq_3(k10_finseq_1(A,C),B) = k9_finseq_1(C) ) ) )).

fof(t86_finseq_3,axiom,(
! [A,B,C] :
( k1_finseq_3(k10_finseq_1(A,B),C) = k9_finseq_1(B)
=> ( A = B
| ( r2_hidden(A,C)
& ~ r2_hidden(B,C) ) ) ) )).

fof(t87_finseq_3,axiom,(
! [A,B,C] :
( r2_hidden(C,B)
=> ( r2_hidden(A,B)
| k1_finseq_3(k10_finseq_1(A,C),B) = k9_finseq_1(A) ) ) )).

fof(t88_finseq_3,axiom,(
! [A,B,C] :
( k1_finseq_3(k10_finseq_1(A,B),C) = k9_finseq_1(A)
=> ( A = B
| ( ~ r2_hidden(A,C)
& r2_hidden(B,C) ) ) ) )).

fof(t89_finseq_3,axiom,(
! [A,B,C] :
( k1_finseq_3(k10_finseq_1(A,B),C) = k10_finseq_1(A,B)
<=> ( ~ r2_hidden(A,C)
& ~ r2_hidden(B,C) ) ) )).

fof(t90_finseq_3,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) )
=> ! [C,D] :
( v4_ordinal2(D)
=> ( ( k3_finseq_1(A) = k2_xcmplx_0(D,np__1)
& B = k7_relat_1(A,k2_finseq_1(D)) )
=> ( r2_hidden(k1_funct_1(A,k2_xcmplx_0(D,np__1)),C)
<=> k1_finseq_3(A,C) = k1_finseq_3(B,C) ) ) ) ) ) )).

fof(t91_finseq_3,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) )
=> ! [C,D] :
( v4_ordinal2(D)
=> ( ( k3_finseq_1(A) = k2_xcmplx_0(D,np__1)
& B = k7_relat_1(A,k2_finseq_1(D)) )
=> ( ~ r2_hidden(k1_funct_1(A,k2_xcmplx_0(D,np__1)),C)
<=> k1_finseq_3(A,C) = k7_finseq_1(k1_finseq_3(B,C),k9_finseq_1(k1_funct_1(A,k2_xcmplx_0(D,np__1)))) ) ) ) ) ) )).

fof(t93_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( m2_finseq_1(A,B)
=> m2_finseq_1(k1_finseq_3(A,C),B) ) ) )).

fof(t94_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( v2_funct_1(A)
=> v2_funct_1(k1_finseq_3(A,B)) ) ) )).

fof(t95_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( v2_funct_1(A)
=> k3_finseq_1(k1_finseq_3(A,B)) = k6_xcmplx_0(k3_finseq_1(A),k4_card_1(k3_xboole_0(B,k2_relat_1(A)))) ) ) )).

fof(t96_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( v1_finset_1(B)
=> ( ( v2_funct_1(A)
& r1_tarski(B,k2_relat_1(A)) )
=> k3_finseq_1(k1_finseq_3(A,B)) = k6_xcmplx_0(k3_finseq_1(A),k4_card_1(B)) ) ) ) )).

fof(t97_finseq_3,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)) )
=> k3_finseq_1(k1_finseq_3(A,k1_tarski(B))) = k6_xcmplx_0(k3_finseq_1(A),np__1) ) ) )).

fof(t98_finseq_3,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) )
=> ( ( r1_xboole_0(k2_relat_1(A),k2_relat_1(B))
& v2_funct_1(A)
& v2_funct_1(B) )
<=> v2_funct_1(k7_finseq_1(A,B)) ) ) ) )).

fof(t99_finseq_3,axiom,(
! [A,B] :
( v4_ordinal2(B)
=> ( r1_tarski(A,k2_finseq_1(B))
=> v2_funct_1(k14_finseq_1(A)) ) ) )).

fof(t100_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> v2_funct_1(k1_finseq_2(A)) ) )).

fof(t101_finseq_3,axiom,(
\$true )).

fof(t102_finseq_3,axiom,(
! [A] : v2_funct_1(k9_finseq_1(A)) )).

fof(t103_finseq_3,axiom,(
! [A,B] :
( A != B
<=> v2_funct_1(k10_finseq_1(A,B)) ) )).

fof(t104_finseq_3,axiom,(
! [A,B,C] :
( ( A != B
& B != C
& C != A )
<=> v2_funct_1(k11_finseq_1(A,B,C)) ) )).

fof(t105_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v2_funct_1(A)
& k2_relat_1(A) = k1_tarski(B) )
=> k3_finseq_1(A) = np__1 ) ) )).

fof(t106_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( ( v2_funct_1(A)
& k2_relat_1(A) = k1_tarski(B) )
=> A = k9_finseq_1(B) ) ) )).

fof(t107_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( ( v2_funct_1(A)
& k2_relat_1(A) = k2_tarski(B,C) )
=> ( B = C
| k3_finseq_1(A) = np__2 ) ) ) )).

fof(t108_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
~ ( v2_funct_1(A)
& k2_relat_1(A) = k2_tarski(B,C)
& B != C
& A != k10_finseq_1(B,C)
& A != k10_finseq_1(C,B) ) ) )).

fof(t109_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C,D] :
( ( v2_funct_1(A)
& k2_relat_1(A) = k1_enumset1(B,C,D)
& v2_funct_1(k11_finseq_1(B,C,D)) )
=> k3_finseq_1(A) = np__3 ) ) )).

fof(t110_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C,D] :
( ( v2_funct_1(A)
& k2_relat_1(A) = k1_enumset1(B,C,D) )
=> ( B = C
| C = D
| B = D
| k3_finseq_1(A) = np__3 ) ) ) )).

fof(t111_finseq_3,axiom,(
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ~ ( ~ v1_xboole_0(B)
& ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_finseq_1(D,A)
=> ~ ( C = k1_funct_1(B,np__1)
& B = k8_finseq_1(A,k12_finseq_1(A,C),D) ) ) ) ) ) ) )).

fof(t112_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( r2_hidden(A,k4_finseq_1(B))
=> k1_funct_1(k7_finseq_1(k9_finseq_1(C),B),k2_xcmplx_0(A,np__1)) = k1_funct_1(B,A) ) ) ) )).

fof(d2_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> k2_finseq_3(A,B) = k5_relat_1(k14_finseq_1(k4_xboole_0(k4_finseq_1(B),k1_tarski(A))),B) ) ) )).

fof(t113_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( ~ ( r2_hidden(A,k4_finseq_1(B))
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( k3_finseq_1(B) = k1_nat_1(C,np__1)
& k3_finseq_1(k2_finseq_3(A,B)) = C ) ) )
& ( ~ r2_hidden(A,k4_finseq_1(B))
=> k2_finseq_3(A,B) = B ) ) ) ) )).

fof(t114_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,B)
=> m2_finseq_1(k2_finseq_3(A,C),B) ) ) ) )).

fof(t115_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> r1_tarski(k2_relat_1(k2_finseq_3(A,B)),k2_relat_1(B)) ) ) )).

fof(t116_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( A = k2_xcmplx_0(B,np__1)
& r2_hidden(C,k2_finseq_1(A)) )
=> k3_finseq_1(k14_finseq_1(k4_xboole_0(k2_finseq_1(A),k1_tarski(C)))) = B ) ) ) ) )).

fof(t117_finseq_3,axiom,(
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( D = k1_nat_1(C,np__1)
& r2_hidden(B,k2_finseq_1(D))
& r2_hidden(A,k2_finseq_1(C)) )
=> ( ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(B,A)
| k1_funct_1(k14_finseq_1(k4_xboole_0(k2_finseq_1(D),k1_tarski(B))),A) = A ) )
& ( ( r1_xreal_0(B,A)
& r1_xreal_0(A,C) )
=> k1_funct_1(k14_finseq_1(k4_xboole_0(k2_finseq_1(D),k1_tarski(B))),A) = k1_nat_1(A,np__1) ) ) ) ) ) ) ) )).

fof(t118_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( k3_finseq_1(A) = k2_xcmplx_0(C,np__1)
& r2_hidden(B,k4_finseq_1(A)) )
=> k3_finseq_1(k2_finseq_3(B,A)) = C ) ) ) ) )).

fof(t119_finseq_3,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k3_finseq_1(A) = k1_nat_1(C,np__1)
=> ( r1_xreal_0(B,D)
| k1_funct_1(k2_finseq_3(B,A),D) = k1_funct_1(A,D) ) ) ) ) ) ) )).

fof(t120_finseq_3,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( k3_finseq_1(A) = k1_nat_1(C,np__1)
& r2_hidden(B,k4_finseq_1(A))
& r1_xreal_0(B,D)
& r1_xreal_0(D,C) )
=> k1_funct_1(k2_finseq_3(B,A),D) = k1_funct_1(A,k1_nat_1(D,np__1)) ) ) ) ) ) )).

fof(t121_finseq_3,axiom,(
! [A] :
( v4_ordinal2(A)
=> ! [B,C] :
( m2_finseq_1(C,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(D,A)
=> k1_funct_1(k16_finseq_1(B,C,A),D) = k1_funct_1(C,D) ) ) ) ) )).

fof(dt_k1_finseq_3,axiom,(
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ( v1_relat_1(k1_finseq_3(A,B))
& v1_funct_1(k1_finseq_3(A,B))
& v1_finseq_1(k1_finseq_3(A,B)) ) ) )).

fof(dt_k2_finseq_3,axiom,(
! [A,B] :
( ( v4_ordinal2(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( v1_relat_1(k2_finseq_3(A,B))
& v1_funct_1(k2_finseq_3(A,B))
& v1_finseq_1(k2_finseq_3(A,B)) ) ) )).

fof(t92_finseq_3,axiom,(
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finseq_1(A) )
=> ! [B,C] :
( v4_ordinal2(C)
=> ( r2_hidden(C,k4_finseq_1(A))
=> ! [D] :
( v1_finset_1(D)
=> ~ ( D = a_3_0_finseq_3(A,B,C)
& ~ r2_hidden(k1_funct_1(A,C),B)
& k1_funct_1(k1_finseq_3(A,B),k6_xcmplx_0(C,k4_card_1(D))) != k1_funct_1(A,C) ) ) ) ) ) )).

fof(fraenkel_a_3_0_finseq_3,axiom,(
! [A,B,C,D] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B)
& v4_ordinal2(D) )
=> ( r2_hidden(A,a_3_0_finseq_3(B,C,D))
<=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& A = E
& r2_hidden(E,k4_finseq_1(B))
& r1_xreal_0(E,D)
& r2_hidden(k1_funct_1(B,E),C) ) ) ) )).
%------------------------------------------------------------------------------
```