SET007 Axioms: SET007+127.ax
%------------------------------------------------------------------------------
% File : SET007+127 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Functional Sequence from a Domain to a Domain
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : seqfunc [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 74 ( 1 unt; 0 def)
% Number of atoms : 450 ( 85 equ)
% Maximal formula atoms : 15 ( 6 avg)
% Number of connectives : 450 ( 74 ~; 3 |; 104 &)
% ( 19 <=>; 250 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-4 aty)
% Number of functors : 47 ( 47 usr; 7 con; 0-4 aty)
% Number of variables : 287 ( 276 !; 11 ?)
% 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_seqfunc,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( m1_seqfunc(C,A,B)
<=> ( k1_relat_1(C) = k5_numbers
& r1_tarski(k2_relat_1(C),k4_partfun1(A,B)) ) ) ) ).
fof(t1_seqfunc,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( m1_seqfunc(C,A,B)
<=> ( k1_relat_1(C) = k5_numbers
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( v1_funct_1(k1_funct_1(C,D))
& m2_relset_1(k1_funct_1(C,D),A,B) ) ) ) ) ) ).
fof(t2_seqfunc,axiom,
! [A,B,C] :
( m1_seqfunc(C,B,A)
=> ! [D] :
( m1_seqfunc(D,B,A)
=> ( ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_seqfunc(B,A,C,E) = k1_seqfunc(B,A,D,E) )
=> C = D ) ) ) ).
fof(d2_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( D = k2_seqfunc(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,D,E) = k13_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,E),C) ) ) ) ) ) ) ).
fof(d3_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ( C = k3_seqfunc(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,C,D) = k4_rfunct_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,D)) ) ) ) ) ) ).
fof(d4_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ( C = k4_seqfunc(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,C,D) = k16_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,D)) ) ) ) ) ) ).
fof(d5_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ( C = k5_seqfunc(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,C,D) = k21_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,D)) ) ) ) ) ) ).
fof(d6_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( D = k6_seqfunc(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,D,E) = k6_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,E),k1_seqfunc(A,k1_numbers,C,E)) ) ) ) ) ) ) ).
fof(d7_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> k7_seqfunc(A,B,C) = k6_seqfunc(A,B,k4_seqfunc(A,C)) ) ) ) ).
fof(d8_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( D = k8_seqfunc(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,D,E) = k8_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,E),k1_seqfunc(A,k1_numbers,C,E)) ) ) ) ) ) ) ).
fof(d9_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> k9_seqfunc(A,B,C) = k8_seqfunc(A,C,k3_seqfunc(A,B)) ) ) ) ).
fof(t3_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( B = k9_seqfunc(A,D,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,B,E) = k2_rfunct_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,C,E),k1_seqfunc(A,k1_numbers,D,E)) ) ) ) ) ) ) ).
fof(t4_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( B = k7_seqfunc(A,C,D)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k1_seqfunc(A,k1_numbers,B,E) = k7_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,C,E),k1_seqfunc(A,k1_numbers,D,E)) ) ) ) ) ) ) ).
fof(t5_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( k6_seqfunc(A,B,C) = k6_seqfunc(A,C,B)
& k6_seqfunc(A,k6_seqfunc(A,B,C),D) = k6_seqfunc(A,B,k6_seqfunc(A,C,D)) ) ) ) ) ) ).
fof(t6_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( k8_seqfunc(A,B,C) = k8_seqfunc(A,C,B)
& k8_seqfunc(A,k8_seqfunc(A,B,C),D) = k8_seqfunc(A,B,k8_seqfunc(A,C,D)) ) ) ) ) ) ).
fof(t7_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( k8_seqfunc(A,k6_seqfunc(A,B,C),D) = k6_seqfunc(A,k8_seqfunc(A,B,D),k8_seqfunc(A,C,D))
& k8_seqfunc(A,D,k6_seqfunc(A,B,C)) = k6_seqfunc(A,k8_seqfunc(A,D,B),k8_seqfunc(A,D,C)) ) ) ) ) ) ).
fof(t8_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> k4_seqfunc(A,B) = k2_seqfunc(A,B,k1_real_1(np__1)) ) ) ).
fof(t9_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( k8_seqfunc(A,k7_seqfunc(A,B,C),D) = k7_seqfunc(A,k8_seqfunc(A,B,D),k8_seqfunc(A,C,D))
& k7_seqfunc(A,k8_seqfunc(A,D,B),k8_seqfunc(A,D,C)) = k8_seqfunc(A,D,k7_seqfunc(A,B,C)) ) ) ) ) ) ).
fof(t10_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> ( k2_seqfunc(A,k6_seqfunc(A,C,D),B) = k6_seqfunc(A,k2_seqfunc(A,C,B),k2_seqfunc(A,D,B))
& k2_seqfunc(A,k7_seqfunc(A,C,D),B) = k7_seqfunc(A,k2_seqfunc(A,C,B),k2_seqfunc(A,D,B)) ) ) ) ) ) ).
fof(t11_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_seqfunc(D,A,k1_numbers)
=> k2_seqfunc(A,D,k4_real_1(B,C)) = k2_seqfunc(A,k2_seqfunc(A,D,C),B) ) ) ) ) ).
fof(t12_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> k2_seqfunc(A,B,np__1) = B ) ) ).
fof(t13_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> k4_seqfunc(A,k4_seqfunc(A,B)) = B ) ) ).
fof(t14_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> k8_seqfunc(A,k3_seqfunc(A,B),k3_seqfunc(A,C)) = k3_seqfunc(A,k8_seqfunc(A,B,C)) ) ) ) ).
fof(t15_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ( B != np__0
=> k3_seqfunc(A,k2_seqfunc(A,C,B)) = k2_seqfunc(A,k3_seqfunc(A,C),k2_real_1(B)) ) ) ) ) ).
fof(t16_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> k3_seqfunc(A,k5_seqfunc(A,B)) = k5_seqfunc(A,k3_seqfunc(A,B)) ) ) ).
fof(t17_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> k5_seqfunc(A,k8_seqfunc(A,B,C)) = k8_seqfunc(A,k5_seqfunc(A,B),k5_seqfunc(A,C)) ) ) ) ).
fof(t18_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> k5_seqfunc(A,k9_seqfunc(A,C,B)) = k9_seqfunc(A,k5_seqfunc(A,C),k5_seqfunc(A,B)) ) ) ) ).
fof(t19_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> k5_seqfunc(A,k2_seqfunc(A,C,B)) = k2_seqfunc(A,k5_seqfunc(A,C),k18_complex1(B)) ) ) ) ).
fof(d10_seqfunc,axiom,
! [A,B,C] :
( m1_seqfunc(C,A,B)
=> ! [D] :
( r1_seqfunc(A,B,C,D)
<=> ( D != k1_xboole_0
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> r1_tarski(D,k4_relset_1(A,B,k1_seqfunc(A,B,C,E))) ) ) ) ) ).
fof(d11_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( D = k10_seqfunc(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,E),C) ) ) ) ) ) ) ).
fof(d12_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r2_seqfunc(A,B,C)
<=> ( r1_seqfunc(A,k1_numbers,B,C)
& ? [D] :
( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers)
& C = k4_relset_1(A,k1_numbers,D)
& ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,C)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,np__0)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ? [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
& r1_xreal_0(G,H)
& r1_xreal_0(F,k18_complex1(k5_real_1(k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,H),E),k2_seq_1(A,k1_numbers,D,E)))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t20_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r2_seqfunc(A,B,C)
<=> ( r1_seqfunc(A,k1_numbers,B,C)
& ? [D] :
( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers)
& C = k4_relset_1(A,k1_numbers,D)
& ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,C)
=> ( v4_seq_2(k10_seqfunc(A,B,E))
& k2_seq_2(k10_seqfunc(A,B,E)) = k2_seq_1(A,k1_numbers,D,E) ) ) ) ) ) ) ) ) ).
fof(t21_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r2_seqfunc(A,B,C)
<=> ( r1_seqfunc(A,k1_numbers,B,C)
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,C)
=> v4_seq_2(k10_seqfunc(A,B,D)) ) ) ) ) ) ) ).
fof(d13_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r3_seqfunc(A,B,C)
<=> ( r1_seqfunc(A,k1_numbers,B,C)
& ? [D] :
( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers)
& C = k4_relset_1(A,k1_numbers,D)
& ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ~ r1_xreal_0(E,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& ? [H] :
( m1_subset_1(H,A)
& r1_xreal_0(F,G)
& r2_hidden(H,C)
& r1_xreal_0(E,k18_complex1(k5_real_1(k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,G),H),k2_seq_1(A,k1_numbers,D,H)))) ) ) ) ) ) ) ) ) ) ) ).
fof(d14_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r2_seqfunc(A,B,C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( D = k11_seqfunc(A,B,C)
<=> ( k4_relset_1(A,k1_numbers,D) = C
& ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,k4_relset_1(A,k1_numbers,D))
=> k2_seq_1(A,k1_numbers,D,E) = k2_seq_2(k10_seqfunc(A,B,E)) ) ) ) ) ) ) ) ) ).
fof(t22_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( r2_seqfunc(A,B,C)
=> ( D = k11_seqfunc(A,B,C)
<=> ( k4_relset_1(A,k1_numbers,D) = C
& ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,C)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( ~ r1_xreal_0(F,np__0)
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ? [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
& r1_xreal_0(G,H)
& r1_xreal_0(F,k18_complex1(k5_real_1(k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,H),E),k2_seq_1(A,k1_numbers,D,E)))) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t23_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r3_seqfunc(A,B,C)
=> r2_seqfunc(A,B,C) ) ) ) ).
fof(t24_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C,D] :
( ( r1_tarski(C,D)
& r1_seqfunc(A,k1_numbers,B,D) )
=> ( C = k1_xboole_0
| r1_seqfunc(A,k1_numbers,B,C) ) ) ) ) ).
fof(t25_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C,D] :
( ( r1_tarski(C,D)
& r2_seqfunc(A,B,D) )
=> ( C = k1_xboole_0
| ( r2_seqfunc(A,B,C)
& k2_partfun1(A,k1_numbers,k11_seqfunc(A,B,D),C) = k11_seqfunc(A,B,C) ) ) ) ) ) ).
fof(t26_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C,D] :
( ( r1_tarski(C,D)
& r3_seqfunc(A,B,D) )
=> ( C = k1_xboole_0
| r3_seqfunc(A,B,C) ) ) ) ) ).
fof(t27_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r1_seqfunc(A,k1_numbers,B,C)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,C)
=> r1_seqfunc(A,k1_numbers,B,k1_tarski(D)) ) ) ) ) ) ).
fof(t28_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r2_seqfunc(A,B,C)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,C)
=> r1_seqfunc(A,k1_numbers,B,k1_tarski(D)) ) ) ) ) ) ).
fof(t29_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r1_seqfunc(A,k1_numbers,B,k1_tarski(D))
& r1_seqfunc(A,k1_numbers,C,k1_tarski(D)) )
=> ( k9_seq_1(k10_seqfunc(A,B,D),k10_seqfunc(A,C,D)) = k10_seqfunc(A,k6_seqfunc(A,B,C),D)
& k10_seq_1(k10_seqfunc(A,B,D),k10_seqfunc(A,C,D)) = k10_seqfunc(A,k7_seqfunc(A,B,C),D)
& k11_seq_1(k10_seqfunc(A,B,D),k10_seqfunc(A,C,D)) = k10_seqfunc(A,k8_seqfunc(A,B,C),D) ) ) ) ) ) ) ).
fof(t30_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_subset_1(C,A)
=> ( r1_seqfunc(A,k1_numbers,B,k1_tarski(C))
=> ( k10_seqfunc(A,k5_seqfunc(A,B),C) = k22_seq_1(k10_seqfunc(A,B,C))
& k10_seqfunc(A,k4_seqfunc(A,B),C) = k17_seq_1(k10_seqfunc(A,B,C)) ) ) ) ) ) ).
fof(t31_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r1_seqfunc(A,k1_numbers,C,k1_tarski(D))
=> k10_seqfunc(A,k2_seqfunc(A,C,B),D) = k14_seq_1(k10_seqfunc(A,C,D),B) ) ) ) ) ) ).
fof(t32_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( ( r1_seqfunc(A,k1_numbers,B,D)
& r1_seqfunc(A,k1_numbers,C,D) )
=> ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,D)
=> ( k9_seq_1(k10_seqfunc(A,B,E),k10_seqfunc(A,C,E)) = k10_seqfunc(A,k6_seqfunc(A,B,C),E)
& k10_seq_1(k10_seqfunc(A,B,E),k10_seqfunc(A,C,E)) = k10_seqfunc(A,k7_seqfunc(A,B,C),E)
& k11_seq_1(k10_seqfunc(A,B,E),k10_seqfunc(A,C,E)) = k10_seqfunc(A,k8_seqfunc(A,B,C),E) ) ) ) ) ) ) ) ).
fof(t33_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r1_seqfunc(A,k1_numbers,B,C)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,C)
=> ( k10_seqfunc(A,k5_seqfunc(A,B),D) = k22_seq_1(k10_seqfunc(A,B,D))
& k10_seqfunc(A,k4_seqfunc(A,B),D) = k17_seq_1(k10_seqfunc(A,B,D)) ) ) ) ) ) ) ).
fof(t34_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( r1_seqfunc(A,k1_numbers,C,D)
=> ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,D)
=> k10_seqfunc(A,k2_seqfunc(A,C,B),E) = k14_seq_1(k10_seqfunc(A,C,E),B) ) ) ) ) ) ) ).
fof(t35_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( ( r2_seqfunc(A,B,D)
& r2_seqfunc(A,C,D) )
=> ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,D)
=> ( k9_seq_1(k10_seqfunc(A,B,E),k10_seqfunc(A,C,E)) = k10_seqfunc(A,k6_seqfunc(A,B,C),E)
& k10_seq_1(k10_seqfunc(A,B,E),k10_seqfunc(A,C,E)) = k10_seqfunc(A,k7_seqfunc(A,B,C),E)
& k11_seq_1(k10_seqfunc(A,B,E),k10_seqfunc(A,C,E)) = k10_seqfunc(A,k8_seqfunc(A,B,C),E) ) ) ) ) ) ) ) ).
fof(t36_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r2_seqfunc(A,B,C)
=> ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,C)
=> ( k10_seqfunc(A,k5_seqfunc(A,B),D) = k22_seq_1(k10_seqfunc(A,B,D))
& k10_seqfunc(A,k4_seqfunc(A,B),D) = k17_seq_1(k10_seqfunc(A,B,D)) ) ) ) ) ) ) ).
fof(t37_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( r2_seqfunc(A,C,D)
=> ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,D)
=> k10_seqfunc(A,k2_seqfunc(A,C,B),E) = k14_seq_1(k10_seqfunc(A,C,E),B) ) ) ) ) ) ) ).
fof(t38_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( ( r1_seqfunc(A,k1_numbers,B,D)
& r1_seqfunc(A,k1_numbers,C,D) )
=> ( r1_seqfunc(A,k1_numbers,k6_seqfunc(A,B,C),D)
& r1_seqfunc(A,k1_numbers,k7_seqfunc(A,B,C),D)
& r1_seqfunc(A,k1_numbers,k8_seqfunc(A,B,C),D) ) ) ) ) ) ).
fof(t39_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r1_seqfunc(A,k1_numbers,B,C)
=> ( r1_seqfunc(A,k1_numbers,k5_seqfunc(A,B),C)
& r1_seqfunc(A,k1_numbers,k4_seqfunc(A,B),C) ) ) ) ) ).
fof(t40_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( r1_seqfunc(A,k1_numbers,C,D)
=> r1_seqfunc(A,k1_numbers,k2_seqfunc(A,C,B),D) ) ) ) ) ).
fof(t41_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( ( r2_seqfunc(A,B,D)
& r2_seqfunc(A,C,D) )
=> ( r2_seqfunc(A,k6_seqfunc(A,B,C),D)
& k11_seqfunc(A,k6_seqfunc(A,B,C),D) = k6_seq_1(A,k1_numbers,k11_seqfunc(A,B,D),k11_seqfunc(A,C,D))
& r2_seqfunc(A,k7_seqfunc(A,B,C),D)
& k11_seqfunc(A,k7_seqfunc(A,B,C),D) = k7_seq_1(A,k1_numbers,k11_seqfunc(A,B,D),k11_seqfunc(A,C,D))
& r2_seqfunc(A,k8_seqfunc(A,B,C),D)
& k11_seqfunc(A,k8_seqfunc(A,B,C),D) = k8_seq_1(A,k1_numbers,k11_seqfunc(A,B,D),k11_seqfunc(A,C,D)) ) ) ) ) ) ).
fof(t42_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r2_seqfunc(A,B,C)
=> ( r2_seqfunc(A,k5_seqfunc(A,B),C)
& k11_seqfunc(A,k5_seqfunc(A,B),C) = k21_seq_1(A,k1_numbers,k11_seqfunc(A,B,C))
& r2_seqfunc(A,k4_seqfunc(A,B),C)
& k11_seqfunc(A,k4_seqfunc(A,B),C) = k16_seq_1(A,k1_numbers,k11_seqfunc(A,B,C)) ) ) ) ) ).
fof(t43_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_seqfunc(C,A,k1_numbers)
=> ! [D] :
( r2_seqfunc(A,C,D)
=> ( r2_seqfunc(A,k2_seqfunc(A,C,B),D)
& k11_seqfunc(A,k2_seqfunc(A,C,B),D) = k13_seq_1(A,k1_numbers,k11_seqfunc(A,C,D),B) ) ) ) ) ) ).
fof(t44_seqfunc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_seqfunc(B,A,k1_numbers)
=> ! [C] :
( r3_seqfunc(A,B,C)
<=> ( r1_seqfunc(A,k1_numbers,B,C)
& r2_seqfunc(A,B,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& ? [G] :
( m1_subset_1(G,A)
& r1_xreal_0(E,F)
& r2_hidden(G,C)
& r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,F),G),k2_seq_1(A,k1_numbers,k11_seqfunc(A,B,C),G)))) ) ) ) ) ) ) ) ) ) ).
fof(t45_seqfunc,axiom,
! [A,B] :
( m1_seqfunc(B,k1_numbers,k1_numbers)
=> ( ( r3_seqfunc(k1_numbers,B,A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r2_fcont_1(k1_seqfunc(k1_numbers,k1_numbers,B,C),A) ) )
=> r2_fcont_1(k11_seqfunc(k1_numbers,B,A),A) ) ) ).
fof(s1_seqfunc,axiom,
? [A] :
( m1_seqfunc(A,f1_s1_seqfunc,f2_s1_seqfunc)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k1_seqfunc(f1_s1_seqfunc,f2_s1_seqfunc,A,B) = f3_s1_seqfunc(B) ) ) ).
fof(dt_m1_seqfunc,axiom,
! [A,B,C] :
( m1_seqfunc(C,A,B)
=> ( v1_relat_1(C)
& v1_funct_1(C) ) ) ).
fof(existence_m1_seqfunc,axiom,
! [A,B] :
? [C] : m1_seqfunc(C,A,B) ).
fof(dt_k1_seqfunc,axiom,
! [A,B,C,D] :
( ( m1_seqfunc(C,A,B)
& v4_ordinal2(D) )
=> ( v1_funct_1(k1_seqfunc(A,B,C,D))
& m2_relset_1(k1_seqfunc(A,B,C,D),A,B) ) ) ).
fof(redefinition_k1_seqfunc,axiom,
! [A,B,C,D] :
( ( m1_seqfunc(C,A,B)
& v4_ordinal2(D) )
=> k1_seqfunc(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k2_seqfunc,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers)
& v1_xreal_0(C) )
=> m1_seqfunc(k2_seqfunc(A,B,C),A,k1_numbers) ) ).
fof(dt_k3_seqfunc,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers) )
=> m1_seqfunc(k3_seqfunc(A,B),A,k1_numbers) ) ).
fof(dt_k4_seqfunc,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers) )
=> m1_seqfunc(k4_seqfunc(A,B),A,k1_numbers) ) ).
fof(dt_k5_seqfunc,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers) )
=> m1_seqfunc(k5_seqfunc(A,B),A,k1_numbers) ) ).
fof(dt_k6_seqfunc,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers)
& m1_seqfunc(C,A,k1_numbers) )
=> m1_seqfunc(k6_seqfunc(A,B,C),A,k1_numbers) ) ).
fof(dt_k7_seqfunc,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers)
& m1_seqfunc(C,A,k1_numbers) )
=> m1_seqfunc(k7_seqfunc(A,B,C),A,k1_numbers) ) ).
fof(dt_k8_seqfunc,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers)
& m1_seqfunc(C,A,k1_numbers) )
=> m1_seqfunc(k8_seqfunc(A,B,C),A,k1_numbers) ) ).
fof(dt_k9_seqfunc,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers)
& m1_seqfunc(C,A,k1_numbers) )
=> m1_seqfunc(k9_seqfunc(A,B,C),A,k1_numbers) ) ).
fof(dt_k10_seqfunc,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers)
& m1_subset_1(C,A) )
=> ( v1_funct_1(k10_seqfunc(A,B,C))
& v1_funct_2(k10_seqfunc(A,B,C),k5_numbers,k1_numbers)
& m2_relset_1(k10_seqfunc(A,B,C),k5_numbers,k1_numbers) ) ) ).
fof(dt_k11_seqfunc,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_seqfunc(B,A,k1_numbers) )
=> ( v1_funct_1(k11_seqfunc(A,B,C))
& m2_relset_1(k11_seqfunc(A,B,C),A,k1_numbers) ) ) ).
%------------------------------------------------------------------------------