SET007 Axioms: SET007+166.ax
%------------------------------------------------------------------------------
% File : SET007+166 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Partial Functions from a Domain to the Real Numbers
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : rfunct_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 151 ( 5 unt; 0 def)
% Number of atoms : 820 ( 112 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 807 ( 138 ~; 3 |; 241 &)
% ( 21 <=>; 404 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 31 ( 29 usr; 1 prp; 0-4 aty)
% Number of functors : 93 ( 93 usr; 6 con; 0-6 aty)
% Number of variables : 476 ( 472 !; 4 ?)
% 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(rc1_rfunct_3,axiom,
! [A,B] :
? [C] :
( m1_rfunct_3(C,A,B)
& ~ v1_xboole_0(C) ) ).
fof(rc2_rfunct_3,axiom,
! [A,B] :
? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finset_1(C) ) ).
fof(fc1_rfunct_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_finset_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k19_rfunct_3(A,B))
& v1_funct_1(k19_rfunct_3(A,B))
& v1_finset_1(k19_rfunct_3(A,B))
& v1_seq_1(k19_rfunct_3(A,B)) ) ) ).
fof(fc2_rfunct_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_finset_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_relat_1(k20_rfunct_3(A,B))
& v1_funct_1(k20_rfunct_3(A,B))
& v1_finset_1(k20_rfunct_3(A,B))
& v1_seq_1(k20_rfunct_3(A,B)) ) ) ).
fof(d1_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k2_rfunct_3(A) = k2_square_1(A,np__0) ) ).
fof(d2_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k3_rfunct_3(A) = k2_square_1(k4_xcmplx_0(A),np__0) ) ).
fof(t1_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> A = k5_real_1(k2_rfunct_3(A),k3_rfunct_3(A)) ) ).
fof(t2_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k18_complex1(A) = k3_real_1(k2_rfunct_3(A),k3_rfunct_3(A)) ) ).
fof(t3_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> k4_real_1(np__2,k2_rfunct_3(A)) = k2_xcmplx_0(A,k18_complex1(A)) ) ).
fof(t4_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(np__0,A)
=> k2_rfunct_3(k3_xcmplx_0(A,B)) = k3_xcmplx_0(A,k2_rfunct_3(B)) ) ) ) ).
fof(t5_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_xreal_0(k2_rfunct_3(k2_xcmplx_0(A,B)),k3_real_1(k2_rfunct_3(A),k2_rfunct_3(B))) ) ) ).
fof(t6_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ( r1_xreal_0(np__0,k2_rfunct_3(A))
& r1_xreal_0(np__0,k3_rfunct_3(A)) ) ) ).
fof(t7_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r1_xreal_0(A,C)
& r1_xreal_0(B,D) )
=> r1_xreal_0(k2_square_1(A,B),k2_square_1(C,D)) ) ) ) ) ) ).
fof(t8_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( C != np__0
=> k11_relset_1(A,k1_numbers,B,k1_tarski(k7_xcmplx_0(D,C))) = k11_relset_1(A,k1_numbers,k13_seq_1(A,k1_numbers,B,C),k1_tarski(D)) ) ) ) ) ) ).
fof(t9_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k11_relset_1(A,k1_numbers,k13_seq_1(A,k1_numbers,B,np__0),k2_setwiseo(k1_numbers,np__0)) = k4_relset_1(A,k1_numbers,B) ) ) ).
fof(t10_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k11_relset_1(A,k1_numbers,k21_seq_1(A,k1_numbers,B),k2_setwiseo(k1_numbers,C)) = k11_relset_1(A,k1_numbers,B,k3_setwiseo(k1_numbers,k1_real_1(C),C)) ) ) ) ) ).
fof(t11_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k11_relset_1(A,k1_numbers,k21_seq_1(A,k1_numbers,B),k2_setwiseo(k1_numbers,np__0)) = k11_relset_1(A,k1_numbers,B,k2_setwiseo(k1_numbers,np__0)) ) ) ).
fof(t12_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ r1_xreal_0(np__0,C)
=> k11_relset_1(A,k1_numbers,k21_seq_1(A,k1_numbers,B),k2_setwiseo(k1_numbers,C)) = k1_xboole_0 ) ) ) ) ).
fof(t13_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( E != np__0
=> ( r1_rfinseq(C,D)
<=> r1_rfinseq(k13_seq_1(A,k1_numbers,C,E),k13_seq_1(B,k1_numbers,D,E)) ) ) ) ) ) ) ) ).
fof(t14_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ( r1_rfinseq(C,D)
<=> r1_rfinseq(k16_seq_1(A,k1_numbers,C),k16_seq_1(B,k1_numbers,D)) ) ) ) ) ) ).
fof(t15_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ( r1_rfinseq(C,D)
=> r1_rfinseq(k21_seq_1(A,k1_numbers,C),k21_seq_1(B,k1_numbers,D)) ) ) ) ) ) ).
fof(d3_rfunct_3,axiom,
! [A,B,C] :
( m1_rfunct_3(C,A,B)
<=> ! [D] :
( m1_subset_1(D,C)
=> ( v1_funct_1(D)
& m2_relset_1(D,A,B) ) ) ) ).
fof(d4_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k4_rfunct_3(A,k1_numbers),k4_rfunct_3(A,k1_numbers)),k4_rfunct_3(A,k1_numbers))
& m2_relset_1(B,k2_zfmisc_1(k4_rfunct_3(A,k1_numbers),k4_rfunct_3(A,k1_numbers)),k4_rfunct_3(A,k1_numbers)) )
=> ( B = k14_rfunct_3(A)
<=> ! [C] :
( m2_rfunct_3(C,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> ! [D] :
( m2_rfunct_3(D,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> k2_binop_1(k4_rfunct_3(A,k1_numbers),k4_rfunct_3(A,k1_numbers),k4_rfunct_3(A,k1_numbers),B,C,D) = k6_rfunct_3(A,k1_numbers,C,D) ) ) ) ) ) ).
fof(t16_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> v1_binop_1(k14_rfunct_3(A),k4_rfunct_3(A,k1_numbers)) ) ).
fof(t17_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> v2_binop_1(k14_rfunct_3(A),k4_rfunct_3(A,k1_numbers)) ) ).
fof(t18_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> r3_binop_1(k4_rfunct_3(A,k1_numbers),k5_rfunct_3(A,k1_numbers,k2_subset_1(A),np__0),k14_rfunct_3(A)) ) ).
fof(t19_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k3_binop_1(k4_rfunct_3(A,k1_numbers),k14_rfunct_3(A)) = k5_rfunct_3(A,k1_numbers,k2_subset_1(A),np__0) ) ).
fof(t20_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> v1_setwiseo(k14_rfunct_3(A),k4_rfunct_3(A,k1_numbers)) ) ).
fof(d5_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> k15_rfunct_3(A,B) = k2_finsop_1(k4_rfunct_3(A,k1_numbers),B,k14_rfunct_3(A)) ) ) ).
fof(t21_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k15_rfunct_3(A,k6_finseq_1(k4_rfunct_3(A,k1_numbers))) = k5_rfunct_3(A,k1_numbers,k2_subset_1(A),np__0) ) ).
fof(t22_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_rfunct_3(B,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> k15_rfunct_3(A,k12_finseq_1(k4_rfunct_3(A,k1_numbers),B)) = B ) ) ).
fof(t23_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m2_rfunct_3(C,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> k15_rfunct_3(A,k8_finseq_1(k4_rfunct_3(A,k1_numbers),B,k12_finseq_1(k4_rfunct_3(A,k1_numbers),C))) = k6_rfunct_3(A,k1_numbers,k15_rfunct_3(A,B),C) ) ) ) ).
fof(t24_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,k1_numbers))
=> k15_rfunct_3(A,k8_finseq_1(k4_rfunct_3(A,k1_numbers),B,C)) = k6_rfunct_3(A,k1_numbers,k15_rfunct_3(A,B),k15_rfunct_3(A,C)) ) ) ) ).
fof(t25_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m2_rfunct_3(C,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> k15_rfunct_3(A,k8_finseq_1(k4_rfunct_3(A,k1_numbers),k12_finseq_1(k4_rfunct_3(A,k1_numbers),C),B)) = k6_rfunct_3(A,k1_numbers,C,k15_rfunct_3(A,B)) ) ) ) ).
fof(t26_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_rfunct_3(B,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m2_rfunct_3(C,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> k15_rfunct_3(A,k2_finseq_4(k4_rfunct_3(A,k1_numbers),B,C)) = k6_rfunct_3(A,k1_numbers,B,C) ) ) ) ).
fof(t27_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_rfunct_3(B,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m2_rfunct_3(C,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> ! [D] :
( m2_rfunct_3(D,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> k15_rfunct_3(A,k3_finseq_4(k4_rfunct_3(A,k1_numbers),B,C,D)) = k6_rfunct_3(A,k1_numbers,k6_rfunct_3(A,k1_numbers,B,C),D) ) ) ) ) ).
fof(t28_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,k1_numbers))
=> ( r1_rfinseq(B,C)
=> k15_rfunct_3(A,B) = k15_rfunct_3(A,C) ) ) ) ) ).
fof(d6_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,k1_numbers))
=> ( C = k16_rfunct_3(A,B)
<=> ( k3_finseq_1(C) = k3_finseq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(C))
=> k1_funct_1(C,D) = k5_rfunct_1(k1_funct_1(B,D),A) ) ) ) ) ) ) ) ).
fof(d7_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m2_finseq_1(D,k4_rfunct_3(A,k1_numbers))
=> ( D = k17_rfunct_3(A,B,C)
<=> ( k3_finseq_1(D) = k1_rfunct_3(k3_finseq_1(C),k3_finseq_1(B))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k4_finseq_1(D))
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,A,k1_numbers) )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( G = k2_seq_1(k5_numbers,k1_numbers,C,E)
& F = k1_funct_1(B,E) )
=> k1_funct_1(D,E) = k13_seq_1(A,k1_numbers,F,G) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d8_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( D = k18_rfunct_3(A,B,C)
<=> ( k3_finseq_1(D) = k3_finseq_1(B)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_rfunct_3(F,A,k1_numbers,k4_rfunct_3(A,k1_numbers))
=> ( ( r2_hidden(E,k4_finseq_1(D))
& k1_funct_1(B,E) = F )
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k2_seq_1(A,k1_numbers,F,C) ) ) ) ) ) ) ) ) ) ).
fof(d9_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ( r1_rfunct_3(A,B,C,D)
<=> ! [E] :
( m2_rfunct_3(E,A,B,k4_rfunct_3(A,B))
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k4_finseq_1(C))
& k1_funct_1(C,F) = E )
=> r2_hidden(D,k4_relset_1(A,B,E)) ) ) ) ) ) ) ) ) ).
fof(t29_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_rfunct_3(A,B,C,D)
=> ( E = np__0
| r1_rfunct_3(A,B,k16_finseq_1(k4_rfunct_3(A,B),C,E),D) ) ) ) ) ) ) ) ).
fof(t30_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_rfunct_3(A,B,C,D)
=> r1_rfunct_3(A,B,k1_rfinseq(k4_rfunct_3(A,B),C,E),D) ) ) ) ) ) ) ).
fof(t31_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,k1_numbers))
=> ( k3_finseq_1(C) != np__0
=> ( r1_rfunct_3(A,k1_numbers,C,B)
<=> r2_hidden(B,k4_relset_1(A,k1_numbers,k15_rfunct_3(A,C))) ) ) ) ) ) ).
fof(t32_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,k4_rfunct_3(A,k1_numbers))
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k18_rfunct_3(A,k16_finseq_1(k4_rfunct_3(A,k1_numbers),B,D),C) = k16_finseq_1(k1_numbers,k18_rfunct_3(A,B,C),D) ) ) ) ) ).
fof(t33_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m1_subset_1(C,A)
=> r1_rfunct_3(A,k1_numbers,k16_rfunct_3(A,B),C) ) ) ) ).
fof(t34_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,k1_numbers))
=> ! [D] :
( m2_finseq_1(D,k1_numbers)
=> ( r1_rfunct_3(A,k1_numbers,C,B)
=> r1_rfunct_3(A,k1_numbers,k17_rfunct_3(A,C,D),B) ) ) ) ) ) ).
fof(t35_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,A)
=> r1_rfunct_3(A,k1_numbers,k17_rfunct_3(A,k16_rfunct_3(A,B),C),D) ) ) ) ) ).
fof(t36_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m2_finseq_1(C,k4_rfunct_3(A,k1_numbers))
=> ( r1_rfunct_3(A,k1_numbers,C,B)
=> k2_seq_1(A,k1_numbers,k15_rfunct_3(A,C),B) = k15_rvsum_1(k18_rfunct_3(A,C,B)) ) ) ) ) ).
fof(d10_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( C = k19_rfunct_3(A,B)
<=> ( k4_relset_1(A,k1_numbers,C) = k4_relset_1(A,k1_numbers,B)
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k4_relset_1(A,k1_numbers,C))
=> k2_seq_1(A,k1_numbers,C,D) = k2_rfunct_3(k2_seq_1(A,k1_numbers,B,D)) ) ) ) ) ) ) ) ).
fof(d11_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( C = k20_rfunct_3(A,B)
<=> ( k4_relset_1(A,k1_numbers,C) = k4_relset_1(A,k1_numbers,B)
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k4_relset_1(A,k1_numbers,C))
=> k2_seq_1(A,k1_numbers,C,D) = k3_rfunct_3(k2_seq_1(A,k1_numbers,B,D)) ) ) ) ) ) ) ) ).
fof(t37_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ( B = k7_seq_1(A,k1_numbers,k19_rfunct_3(A,B),k20_rfunct_3(A,B))
& k21_seq_1(A,k1_numbers,B) = k6_seq_1(A,k1_numbers,k19_rfunct_3(A,B),k20_rfunct_3(A,B))
& k13_seq_1(A,k1_numbers,k19_rfunct_3(A,B),np__2) = k6_seq_1(A,k1_numbers,B,k21_seq_1(A,k1_numbers,B)) ) ) ) ).
fof(t38_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k11_relset_1(A,k1_numbers,B,k2_setwiseo(k1_numbers,C)) = k11_relset_1(A,k1_numbers,k19_rfunct_3(A,B),k2_setwiseo(k1_numbers,C)) ) ) ) ) ).
fof(t39_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k11_relset_1(A,k1_numbers,B,k2_limfunc1(np__0)) = k11_relset_1(A,k1_numbers,k19_rfunct_3(A,B),k2_setwiseo(k1_numbers,np__0)) ) ) ).
fof(t40_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> r1_xreal_0(np__0,k2_seq_1(A,k1_numbers,k19_rfunct_3(A,B),C)) ) ) ) ) ).
fof(t41_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> k11_relset_1(A,k1_numbers,B,k2_setwiseo(k1_numbers,k1_real_1(C))) = k11_relset_1(A,k1_numbers,k20_rfunct_3(A,B),k2_setwiseo(k1_numbers,C)) ) ) ) ) ).
fof(t42_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k11_relset_1(A,k1_numbers,B,k3_limfunc1(np__0)) = k11_relset_1(A,k1_numbers,k20_rfunct_3(A,B),k2_setwiseo(k1_numbers,np__0)) ) ) ).
fof(t43_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> r1_xreal_0(np__0,k2_seq_1(A,k1_numbers,k20_rfunct_3(A,B),C)) ) ) ) ) ).
fof(t44_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ( r1_rfinseq(C,D)
=> r1_rfinseq(k19_rfunct_3(A,C),k19_rfunct_3(B,D)) ) ) ) ) ) ).
fof(t45_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ( r1_rfinseq(C,D)
=> r1_rfinseq(k20_rfunct_3(A,C),k20_rfunct_3(B,D)) ) ) ) ) ) ).
fof(t46_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_finset_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_finset_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ( ( r1_rfinseq(k19_rfunct_3(A,C),k19_rfunct_3(B,D))
& r1_rfinseq(k20_rfunct_3(A,C),k20_rfunct_3(B,D)) )
=> r1_rfinseq(C,D) ) ) ) ) ) ).
fof(t47_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] : k2_partfun1(A,k1_numbers,k19_rfunct_3(A,B),C) = k19_rfunct_3(A,k2_partfun1(A,k1_numbers,B,C)) ) ) ).
fof(t48_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] : k2_partfun1(A,k1_numbers,k20_rfunct_3(A,B),C) = k20_rfunct_3(A,k2_partfun1(A,k1_numbers,B,C)) ) ) ).
fof(t49_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> r1_xreal_0(np__0,k2_seq_1(A,k1_numbers,B,C)) ) )
=> k19_rfunct_3(A,B) = B ) ) ) ).
fof(t50_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> r1_xreal_0(k2_seq_1(A,k1_numbers,B,C),np__0) ) )
=> k20_rfunct_3(A,B) = k16_seq_1(A,k1_numbers,B) ) ) ) ).
fof(d12_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( D = k21_rfunct_3(A,B,C)
<=> ( k4_relset_1(A,k1_numbers,D) = k4_relset_1(A,k1_numbers,B)
& ! [E] :
( m1_subset_1(E,A)
=> ( r2_hidden(E,k4_relset_1(A,k1_numbers,D))
=> k2_seq_1(A,k1_numbers,D,E) = k5_real_1(k2_seq_1(A,k1_numbers,B,E),C) ) ) ) ) ) ) ) ) ).
fof(t51_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k21_rfunct_3(A,B,np__0) = B ) ) ).
fof(t52_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] : k21_rfunct_3(A,k2_partfun1(A,k1_numbers,B,D),C) = k2_partfun1(A,k1_numbers,k21_rfunct_3(A,B,C),D) ) ) ) ).
fof(t53_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> k11_relset_1(A,k1_numbers,B,k2_setwiseo(k1_numbers,k3_real_1(D,C))) = k11_relset_1(A,k1_numbers,k21_rfunct_3(A,B,C),k2_setwiseo(k1_numbers,D)) ) ) ) ) ).
fof(t54_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( r1_rfinseq(C,D)
<=> r1_rfinseq(k21_rfunct_3(A,C,E),k21_rfunct_3(B,D,E)) ) ) ) ) ) ) ).
fof(d13_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( r2_rfunct_3(A,B)
<=> ( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( r1_xreal_0(np__0,C)
& r1_xreal_0(C,np__1) )
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(D,B)
& r2_hidden(E,B)
& r2_hidden(k3_real_1(k4_real_1(C,D),k4_real_1(k5_real_1(np__1,C),E)),B) )
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,A,k3_real_1(k4_real_1(C,D),k4_real_1(k5_real_1(np__1,C),E))),k3_real_1(k4_real_1(C,k2_seq_1(k1_numbers,k1_numbers,A,D)),k4_real_1(k5_real_1(np__1,C),k2_seq_1(k1_numbers,k1_numbers,A,E)))) ) ) ) ) ) ) ) ) ).
fof(t55_rfunct_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r2_rfunct_3(C,k1_rcomp_1(A,B))
<=> ( r1_tarski(k1_rcomp_1(A,B),k4_relset_1(k1_numbers,k1_numbers,C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( ( r1_xreal_0(np__0,D)
& r1_xreal_0(D,np__1) )
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r2_hidden(E,k1_rcomp_1(A,B))
& r2_hidden(F,k1_rcomp_1(A,B)) )
=> r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,k3_real_1(k4_real_1(D,E),k4_real_1(k5_real_1(np__1,D),F))),k3_real_1(k4_real_1(D,k2_seq_1(k1_numbers,k1_numbers,C,E)),k4_real_1(k5_real_1(np__1,D),k2_seq_1(k1_numbers,k1_numbers,C,F)))) ) ) ) ) ) ) ) ) ) ) ).
fof(t56_rfunct_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( r2_rfunct_3(C,k1_rcomp_1(A,B))
<=> ( r1_tarski(k1_rcomp_1(A,B),k4_relset_1(k1_numbers,k1_numbers,C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ( ( r2_hidden(D,k1_rcomp_1(A,B))
& r2_hidden(E,k1_rcomp_1(A,B))
& r2_hidden(F,k1_rcomp_1(A,B)) )
=> ( r1_xreal_0(E,D)
| r1_xreal_0(F,E)
| r1_xreal_0(k6_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,D),k2_seq_1(k1_numbers,k1_numbers,C,E)),k5_real_1(D,E)),k6_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,C,F)),k5_real_1(E,F))) ) ) ) ) ) ) ) ) ) ) ).
fof(t57_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B,C] :
( ( r2_rfunct_3(A,B)
& r1_tarski(C,B) )
=> r2_rfunct_3(A,C) ) ) ).
fof(t58_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B,C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_rfunct_3(A,B)
<=> r2_rfunct_3(k21_rfunct_3(k1_numbers,A,C),B) ) ) ) ).
fof(t59_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B,C] :
( m1_subset_1(C,k1_numbers)
=> ( ~ r1_xreal_0(C,np__0)
=> ( r2_rfunct_3(A,B)
<=> r2_rfunct_3(k13_seq_1(k1_numbers,k1_numbers,A,C),B) ) ) ) ) ).
fof(t60_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
=> r2_rfunct_3(k13_seq_1(k1_numbers,k1_numbers,A,np__0),B) ) ) ).
fof(t61_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( r2_rfunct_3(A,C)
& r2_rfunct_3(B,C) )
=> r2_rfunct_3(k6_seq_1(k1_numbers,k1_numbers,A,B),C) ) ) ) ).
fof(t62_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B,C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_rfunct_3(A,B)
=> r2_rfunct_3(k19_rfunct_3(k1_numbers,k21_rfunct_3(k1_numbers,A,C)),B) ) ) ) ).
fof(t63_rfunct_3,axiom,
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( r2_rfunct_3(A,B)
=> r2_rfunct_3(k19_rfunct_3(k1_numbers,A),B) ) ) ).
fof(t64_rfunct_3,axiom,
r2_rfunct_3(k1_partfun2(k1_numbers,k2_subset_1(k1_numbers)),k1_numbers) ).
fof(t65_rfunct_3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> r2_rfunct_3(k19_rfunct_3(k1_numbers,k21_rfunct_3(k1_numbers,k1_partfun2(k1_numbers,k2_subset_1(k1_numbers)),A)),k1_numbers) ) ).
fof(d14_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
=> ! [D] :
( ( v1_rfinseq(D)
& m2_finseq_1(D,k1_numbers) )
=> ( D = k22_rfunct_3(A,B,C)
<=> r1_rfinseq(k2_partfun1(A,k1_numbers,B,C),D) ) ) ) ) ) ).
fof(t66_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
=> k22_rfunct_3(A,B,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C))) = k22_rfunct_3(A,B,C) ) ) ) ).
fof(t67_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
=> k22_rfunct_3(A,k2_partfun1(A,k1_numbers,B,C),C) = k22_rfunct_3(A,B,C) ) ) ) ).
fof(t68_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( ( v1_finset_1(C)
& r2_hidden(B,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,D,C))) )
=> r1_rfinseq(k8_finseq_1(k1_numbers,k22_rfunct_3(A,D,k4_xboole_0(C,k2_setwiseo(A,B))),k12_finseq_1(k1_numbers,k2_seq_1(A,k1_numbers,D,B))),k2_partfun1(A,k1_numbers,D,C)) ) ) ) ) ).
fof(t69_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C,D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k1_numbers) )
=> ( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,D,C)))
& r2_hidden(B,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,D,C))) )
=> r1_rfinseq(k8_finseq_1(k1_numbers,k22_rfunct_3(A,D,k4_xboole_0(C,k2_setwiseo(A,B))),k12_finseq_1(k1_numbers,k2_seq_1(A,k1_numbers,D,B))),k2_partfun1(A,k1_numbers,D,C)) ) ) ) ) ).
fof(t70_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( v1_finset_1(D)
=> ( D = k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C))
=> k3_finseq_1(k22_rfunct_3(A,B,C)) = k4_card_1(D) ) ) ) ) ).
fof(t71_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k22_rfunct_3(A,B,k1_xboole_0) = k6_finseq_1(k1_numbers) ) ) ).
fof(t72_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> k22_rfunct_3(A,B,k2_setwiseo(A,C)) = k12_finseq_1(k1_numbers,k2_seq_1(A,k1_numbers,B,C)) ) ) ) ) ).
fof(t73_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( m1_subset_1(D,A)
=> ( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
& r2_hidden(D,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
& k2_seq_1(k5_numbers,k1_numbers,k22_rfunct_3(A,B,C),k3_finseq_1(k22_rfunct_3(A,B,C))) = k2_seq_1(A,k1_numbers,B,D) )
=> k22_rfunct_3(A,B,C) = k8_finseq_1(k1_numbers,k22_rfunct_3(A,B,k4_xboole_0(C,k2_setwiseo(A,D))),k12_finseq_1(k1_numbers,k2_seq_1(A,k1_numbers,B,D))) ) ) ) ) ).
fof(t74_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
& r1_tarski(D,C)
& ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ( ( r2_hidden(E,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D)))
& r2_hidden(F,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k4_xboole_0(C,D)))) )
=> r1_xreal_0(k2_seq_1(A,k1_numbers,B,F),k2_seq_1(A,k1_numbers,B,E)) ) ) ) )
=> k22_rfunct_3(A,B,C) = k8_finseq_1(k1_numbers,k22_rfunct_3(A,B,D),k22_rfunct_3(A,B,k4_xboole_0(C,D))) ) ) ) ).
fof(t75_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D,E] :
( m1_subset_1(E,A)
=> ( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D)))
& r2_hidden(E,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D))) )
=> ( k2_seq_1(k5_numbers,k1_numbers,k22_rfunct_3(A,k21_rfunct_3(A,B,C),D),k3_finseq_1(k22_rfunct_3(A,k21_rfunct_3(A,B,C),D))) = k2_seq_1(A,k1_numbers,k21_rfunct_3(A,B,C),E)
<=> k2_seq_1(k5_numbers,k1_numbers,k22_rfunct_3(A,B,D),k3_finseq_1(k22_rfunct_3(A,B,D))) = k2_seq_1(A,k1_numbers,B,E) ) ) ) ) ) ) ).
fof(t76_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D,E] :
( v1_finset_1(E)
=> ( E = k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D))
=> k22_rfunct_3(A,k21_rfunct_3(A,B,C),D) = k7_rvsum_1(k22_rfunct_3(A,B,D),k4_finseqop(k1_numbers,k4_card_1(E),C)) ) ) ) ) ) ).
fof(t77_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
& ! [D] :
( m1_subset_1(D,A)
=> ( r2_hidden(D,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
=> r1_xreal_0(np__0,k2_seq_1(A,k1_numbers,B,D)) ) ) )
=> k22_rfunct_3(A,k19_rfunct_3(A,B),C) = k22_rfunct_3(A,B,C) ) ) ) ).
fof(t78_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( v1_finset_1(E)
=> ( ( E = k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C))
& k5_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)) = k2_setwiseo(k1_numbers,D) )
=> k22_rfunct_3(A,B,C) = k4_finseqop(k1_numbers,k4_card_1(E),D) ) ) ) ) ) ).
fof(t79_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k2_xboole_0(C,D))))
& r1_xboole_0(C,D) )
=> r1_rfinseq(k22_rfunct_3(A,B,k2_xboole_0(C,D)),k8_finseq_1(k1_numbers,k22_rfunct_3(A,B,C),k22_rfunct_3(A,B,D))) ) ) ) ).
fof(d15_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] : k24_rfunct_3(A,B,C) = k15_rvsum_1(k22_rfunct_3(A,B,C)) ) ) ).
fof(t80_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( m1_subset_1(D,k1_numbers)
=> ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)))
=> k24_rfunct_3(A,k13_seq_1(A,k1_numbers,B,D),C) = k4_real_1(D,k24_rfunct_3(A,B,C)) ) ) ) ) ).
fof(t81_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D,E] :
( v1_finset_1(E)
=> ( ( E = k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D))
& k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D)) = k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,C,D)) )
=> k24_rfunct_3(A,k6_seq_1(A,k1_numbers,B,C),D) = k3_real_1(k24_rfunct_3(A,B,D),k24_rfunct_3(A,C,D)) ) ) ) ) ) ).
fof(t82_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ! [D] :
( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D)))
& k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D)) = k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,C,D)) )
=> k24_rfunct_3(A,k7_seq_1(A,k1_numbers,B,C),D) = k5_real_1(k24_rfunct_3(A,B,D),k24_rfunct_3(A,C,D)) ) ) ) ) ).
fof(t83_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( v1_finset_1(E)
=> ( E = k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C))
=> k24_rfunct_3(A,k21_rfunct_3(A,B,D),C) = k5_real_1(k24_rfunct_3(A,B,C),k4_real_1(D,k4_card_1(E))) ) ) ) ) ) ).
fof(t84_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k24_rfunct_3(A,B,k1_xboole_0) = np__0 ) ) ).
fof(t85_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,k4_relset_1(A,k1_numbers,B))
=> k24_rfunct_3(A,B,k2_setwiseo(A,C)) = k2_seq_1(A,k1_numbers,B,C) ) ) ) ) ).
fof(t86_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k2_xboole_0(C,D))))
& r1_xboole_0(C,D) )
=> k24_rfunct_3(A,B,k2_xboole_0(C,D)) = k3_real_1(k24_rfunct_3(A,B,C),k24_rfunct_3(A,B,D)) ) ) ) ).
fof(t87_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C,D] :
( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k2_xboole_0(C,D))))
& r1_xboole_0(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)),k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D))) )
=> k24_rfunct_3(A,B,k2_xboole_0(C,D)) = k3_real_1(k24_rfunct_3(A,B,C),k24_rfunct_3(A,B,D)) ) ) ) ).
fof(dt_m1_rfunct_3,axiom,
$true ).
fof(existence_m1_rfunct_3,axiom,
! [A,B] :
? [C] : m1_rfunct_3(C,A,B) ).
fof(dt_m2_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& m1_rfunct_3(C,A,B) )
=> ! [D] :
( m2_rfunct_3(D,A,B,C)
=> ( v1_funct_1(D)
& m2_relset_1(D,A,B) ) ) ) ).
fof(existence_m2_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& m1_rfunct_3(C,A,B) )
=> ? [D] : m2_rfunct_3(D,A,B,C) ) ).
fof(redefinition_m2_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& m1_rfunct_3(C,A,B) )
=> ! [D] :
( m2_rfunct_3(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_k1_rfunct_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k1_rfunct_3(A,B),k1_numbers,k5_numbers) ) ).
fof(commutativity_k1_rfunct_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_rfunct_3(A,B) = k1_rfunct_3(B,A) ) ).
fof(idempotence_k1_rfunct_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_rfunct_3(A,A) = A ) ).
fof(redefinition_k1_rfunct_3,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_rfunct_3(A,B) = k1_square_1(A,B) ) ).
fof(dt_k2_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k2_rfunct_3(A),k1_numbers) ) ).
fof(dt_k3_rfunct_3,axiom,
! [A] :
( v1_xreal_0(A)
=> m1_subset_1(k3_rfunct_3(A),k1_numbers) ) ).
fof(dt_k4_rfunct_3,axiom,
! [A,B] : m1_rfunct_3(k4_rfunct_3(A,B),A,B) ).
fof(redefinition_k4_rfunct_3,axiom,
! [A,B] : k4_rfunct_3(A,B) = k4_partfun1(A,B) ).
fof(dt_k5_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,B) )
=> m2_rfunct_3(k5_rfunct_3(A,B,C,D),A,B,k4_rfunct_3(A,B)) ) ).
fof(redefinition_k5_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,B) )
=> k5_rfunct_3(A,B,C,D) = k2_funcop_1(C,D) ) ).
fof(dt_k6_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> m2_rfunct_3(k6_rfunct_3(A,B,C,D),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(commutativity_k6_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> k6_rfunct_3(A,B,C,D) = k6_rfunct_3(A,B,D,C) ) ).
fof(redefinition_k6_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> k6_rfunct_3(A,B,C,D) = k3_seq_1(C,D) ) ).
fof(dt_k7_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> m2_rfunct_3(k7_rfunct_3(A,B,C,D),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(redefinition_k7_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> k7_rfunct_3(A,B,C,D) = k4_seq_1(C,D) ) ).
fof(dt_k8_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> m2_rfunct_3(k8_rfunct_3(A,B,C,D),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(commutativity_k8_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> k8_rfunct_3(A,B,C,D) = k8_rfunct_3(A,B,D,C) ) ).
fof(redefinition_k8_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> k8_rfunct_3(A,B,C,D) = k5_seq_1(C,D) ) ).
fof(dt_k9_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> m2_rfunct_3(k9_rfunct_3(A,B,C,D),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(redefinition_k9_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& m1_subset_1(D,k4_rfunct_3(A,B)) )
=> k9_rfunct_3(A,B,C,D) = k1_rfunct_1(C,D) ) ).
fof(dt_k10_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B)) )
=> m2_rfunct_3(k10_rfunct_3(A,B,C),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(redefinition_k10_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B)) )
=> k10_rfunct_3(A,B,C) = k20_seq_1(C) ) ).
fof(dt_k11_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B)) )
=> m2_rfunct_3(k11_rfunct_3(A,B,C),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(redefinition_k11_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B)) )
=> k11_rfunct_3(A,B,C) = k15_seq_1(C) ) ).
fof(dt_k12_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B)) )
=> m2_rfunct_3(k12_rfunct_3(A,B,C),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(redefinition_k12_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B)) )
=> k12_rfunct_3(A,B,C) = k3_rfunct_1(C) ) ).
fof(dt_k13_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& v1_xreal_0(D) )
=> m2_rfunct_3(k13_rfunct_3(A,B,C,D),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(redefinition_k13_rfunct_3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v2_membered(B)
& m1_subset_1(C,k4_rfunct_3(A,B))
& v1_xreal_0(D) )
=> k13_rfunct_3(A,B,C,D) = k12_seq_1(C,D) ) ).
fof(dt_k14_rfunct_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_funct_1(k14_rfunct_3(A))
& v1_funct_2(k14_rfunct_3(A),k2_zfmisc_1(k4_rfunct_3(A,k1_numbers),k4_rfunct_3(A,k1_numbers)),k4_rfunct_3(A,k1_numbers))
& m2_relset_1(k14_rfunct_3(A),k2_zfmisc_1(k4_rfunct_3(A,k1_numbers),k4_rfunct_3(A,k1_numbers)),k4_rfunct_3(A,k1_numbers)) ) ) ).
fof(dt_k15_rfunct_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k4_rfunct_3(A,k1_numbers)) )
=> m2_rfunct_3(k15_rfunct_3(A,B),A,k1_numbers,k4_rfunct_3(A,k1_numbers)) ) ).
fof(dt_k16_rfunct_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> m2_finseq_1(k16_rfunct_3(A,B),k4_rfunct_3(A,k1_numbers)) ) ).
fof(dt_k17_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k4_rfunct_3(A,k1_numbers))
& m1_finseq_1(C,k1_numbers) )
=> m2_finseq_1(k17_rfunct_3(A,B,C),k4_rfunct_3(A,k1_numbers)) ) ).
fof(dt_k18_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_finseq_1(B,k4_rfunct_3(A,k1_numbers))
& m1_subset_1(C,A) )
=> m2_finseq_1(k18_rfunct_3(A,B,C),k1_numbers) ) ).
fof(dt_k19_rfunct_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_funct_1(k19_rfunct_3(A,B))
& m2_relset_1(k19_rfunct_3(A,B),A,k1_numbers) ) ) ).
fof(dt_k20_rfunct_3,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_funct_1(k20_rfunct_3(A,B))
& m2_relset_1(k20_rfunct_3(A,B),A,k1_numbers) ) ) ).
fof(dt_k21_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers)
& m1_subset_1(C,k1_numbers) )
=> ( v1_funct_1(k21_rfunct_3(A,B,C))
& m2_relset_1(k21_rfunct_3(A,B,C),A,k1_numbers) ) ) ).
fof(dt_k22_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> ( v1_rfinseq(k22_rfunct_3(A,B,C))
& m2_finseq_1(k22_rfunct_3(A,B,C),k1_numbers) ) ) ).
fof(dt_k23_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers)
& v1_finset_1(C) )
=> ( v1_funct_1(k23_rfunct_3(A,B,C))
& v1_finset_1(k23_rfunct_3(A,B,C))
& m2_relset_1(k23_rfunct_3(A,B,C),A,k1_numbers) ) ) ).
fof(redefinition_k23_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers)
& v1_finset_1(C) )
=> k23_rfunct_3(A,B,C) = k7_relat_1(B,C) ) ).
fof(dt_k24_rfunct_3,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers) )
=> m1_subset_1(k24_rfunct_3(A,B,C),k1_numbers) ) ).
%------------------------------------------------------------------------------