SET007 Axioms: SET007+631.ax
%------------------------------------------------------------------------------
% File : SET007+631 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Scalar Multiple of Riemann Definite Integral
% 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 : integra2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 0 unt; 0 def)
% Number of atoms : 454 ( 27 equ)
% Maximal formula atoms : 16 ( 8 avg)
% Number of connectives : 434 ( 32 ~; 1 |; 195 &)
% ( 8 <=>; 198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-4 aty)
% Number of functors : 40 ( 40 usr; 4 con; 0-4 aty)
% Number of variables : 173 ( 168 !; 5 ?)
% 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_integra2,axiom,
? [A] :
( m1_finseq_1(A,k1_numbers)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_seq_1(A)
& v1_integra2(A) ) ).
fof(t1_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( v1_xreal_0(B)
=> ( r2_hidden(B,A)
<=> ( r1_xreal_0(k4_pscomp_1(A),B)
& r1_xreal_0(B,k3_pscomp_1(A)) ) ) ) ) ).
fof(d1_integra2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ( v1_integra2(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( r2_hidden(B,k4_finseq_1(A))
& r2_hidden(k1_nat_1(B,np__1),k4_finseq_1(A)) )
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,np__1))) ) ) ) ) ).
fof(t2_integra2,axiom,
! [A] :
( ( v1_integra2(A)
& m2_finseq_1(A,k1_numbers) )
=> ! [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))
& r2_hidden(C,k4_finseq_1(A))
& r1_xreal_0(B,C) )
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ).
fof(t3_integra2,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ? [B] :
( v1_integra2(B)
& m2_finseq_1(B,k1_numbers)
& r1_rfinseq(A,B) ) ) ).
fof(t4_integra2,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)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(E,k3_finseq_1(B))
& r1_xreal_0(C,D) )
=> ( r1_xreal_0(E,D)
| k8_finseq_1(A,k1_jordan3(A,B,C,D),k1_jordan3(A,B,k1_nat_1(D,np__1),E)) = k1_jordan3(A,B,C,E) ) ) ) ) ) ) ) ).
fof(d2_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_numbers))
=> ( C = k1_integra2(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,C)
<=> ? [E] :
( m1_subset_1(E,k1_numbers)
& r2_hidden(E,A)
& D = k3_xcmplx_0(B,E) ) ) ) ) ) ) ) ).
fof(t5_integra2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( ( r1_rfunct_1(C,A)
& r1_tarski(B,A) )
=> r1_rfunct_1(k2_partfun1(A,k1_numbers,C,B),B) ) ) ) ) ).
fof(t6_integra2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( ( r2_rfunct_1(C,A)
& r1_tarski(B,A) )
=> r2_rfunct_1(k2_partfun1(A,k1_numbers,C,B),B) ) ) ) ) ).
fof(t7_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ~ v1_xboole_0(k1_integra2(B,A)) ) ) ).
fof(t9_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v1_seq_4(B)
& r1_xreal_0(np__0,A) )
=> v1_seq_4(k1_integra2(B,A)) ) ) ) ).
fof(t10_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v1_seq_4(B)
& r1_xreal_0(A,np__0) )
=> v2_seq_4(k1_integra2(B,A)) ) ) ) ).
fof(t11_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v2_seq_4(B)
& r1_xreal_0(np__0,A) )
=> v2_seq_4(k1_integra2(B,A)) ) ) ) ).
fof(t12_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v2_seq_4(B)
& r1_xreal_0(A,np__0) )
=> v1_seq_4(k1_integra2(B,A)) ) ) ) ).
fof(t13_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v1_seq_4(B)
& r1_xreal_0(np__0,A) )
=> k3_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k3_pscomp_1(B)) ) ) ) ).
fof(t14_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v1_seq_4(B)
& r1_xreal_0(A,np__0) )
=> k4_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k3_pscomp_1(B)) ) ) ) ).
fof(t15_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v2_seq_4(B)
& r1_xreal_0(np__0,A) )
=> k4_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k4_pscomp_1(B)) ) ) ) ).
fof(t16_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ( v2_seq_4(B)
& r1_xreal_0(A,np__0) )
=> k3_pscomp_1(k1_integra2(B,A)) = k4_real_1(A,k4_pscomp_1(B)) ) ) ) ).
fof(t17_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> k5_relset_1(B,k1_numbers,k13_seq_1(B,k1_numbers,C,A)) = k1_integra2(k1_pscomp_1(B,k1_numbers,C),A) ) ) ) ).
fof(t18_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,B,k1_numbers) )
=> k5_relset_1(B,k1_numbers,k13_seq_1(B,k1_numbers,k2_partfun1(B,k1_numbers,D,C),A)) = k1_integra2(k5_relset_1(B,k1_numbers,k2_partfun1(B,k1_numbers,D,C)),A) ) ) ) ) ).
fof(t19_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ( ( r3_rfunct_1(C,B)
& r1_xreal_0(np__0,A) )
=> r1_xreal_0(k4_real_1(k4_real_1(A,k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B)),k2_seq_1(k8_integra1(B),k1_numbers,k9_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D)) ) ) ) ) ) ).
fof(t20_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ( ( r3_rfunct_1(C,B)
& r1_xreal_0(A,np__0) )
=> r1_xreal_0(k4_real_1(k4_real_1(A,k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B)),k2_seq_1(k8_integra1(B),k1_numbers,k9_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D)) ) ) ) ) ) ).
fof(t21_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ( ( r3_rfunct_1(C,B)
& r1_xreal_0(np__0,A) )
=> r1_xreal_0(k2_seq_1(k8_integra1(B),k1_numbers,k10_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D),k4_real_1(k4_real_1(A,k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B))) ) ) ) ) ) ).
fof(t22_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ( ( r3_rfunct_1(C,B)
& r1_xreal_0(A,np__0) )
=> r1_xreal_0(k2_seq_1(k8_integra1(B),k1_numbers,k10_integra1(B,k13_seq_1(B,k1_numbers,C,A)),D),k4_real_1(k4_real_1(A,k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k3_integra1(B))) ) ) ) ) ) ).
fof(t23_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
& r1_rfunct_1(C,B)
& r1_xreal_0(np__0,A) )
=> k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) ).
fof(t24_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
& r1_rfunct_1(C,B)
& r1_xreal_0(A,np__0) )
=> k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) ).
fof(t25_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
& r2_rfunct_1(C,B)
& r1_xreal_0(np__0,A) )
=> k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) ).
fof(t26_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k2_finseq_1(k3_finseq_1(E)))
& r2_rfunct_1(C,B)
& r1_xreal_0(A,np__0) )
=> k2_seq_1(k5_numbers,k1_numbers,k14_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E),F) = k4_real_1(A,k2_seq_1(k5_numbers,k1_numbers,k15_integra1(B,C,D,E),F)) ) ) ) ) ) ) ) ).
fof(t27_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ( ( r1_rfunct_1(C,B)
& r1_xreal_0(np__0,A) )
=> k6_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k6_integra1(B,C,D,E)) ) ) ) ) ) ) ).
fof(t28_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ( ( r1_rfunct_1(C,B)
& r1_xreal_0(A,np__0) )
=> k7_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k6_integra1(B,C,D,E)) ) ) ) ) ) ) ).
fof(t29_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ( ( r2_rfunct_1(C,B)
& r1_xreal_0(np__0,A) )
=> k7_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k7_integra1(B,C,D,E)) ) ) ) ) ) ) ).
fof(t30_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m2_integra1(D,B) )
=> ! [E] :
( m3_integra1(E,B,D)
=> ( ( r2_rfunct_1(C,B)
& r1_xreal_0(A,np__0) )
=> k6_integra1(B,k13_seq_1(B,k1_numbers,C,A),D,E) = k4_real_1(A,k7_integra1(B,C,D,E)) ) ) ) ) ) ) ).
fof(t31_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ( ( r3_rfunct_1(C,B)
& r3_integra1(B,C) )
=> ( r3_integra1(B,k13_seq_1(B,k1_numbers,C,A))
& k13_integra1(B,k13_seq_1(B,k1_numbers,C,A)) = k4_real_1(A,k13_integra1(B,C)) ) ) ) ) ) ).
fof(t32_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( r2_hidden(C,A)
=> r1_xreal_0(np__0,k2_seq_1(A,k1_numbers,B,C)) ) ) )
=> r1_xreal_0(np__0,k13_integra1(A,B)) ) ) ) ).
fof(t33_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m2_relset_1(C,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B)
& r3_rfunct_1(C,A)
& r3_integra1(A,C) )
=> ( r3_integra1(A,k7_seq_1(A,k1_numbers,B,C))
& k13_integra1(A,k7_seq_1(A,k1_numbers,B,C)) = k5_real_1(k13_integra1(A,B),k13_integra1(A,C)) ) ) ) ) ) ).
fof(t34_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m2_relset_1(C,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B)
& r3_rfunct_1(C,A)
& r3_integra1(A,C)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( r2_hidden(D,A)
=> r1_xreal_0(k2_seq_1(A,k1_numbers,C,D),k2_seq_1(A,k1_numbers,B,D)) ) ) )
=> r1_xreal_0(k13_integra1(A,C),k13_integra1(A,B)) ) ) ) ) ).
fof(t35_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( r3_rfunct_1(B,A)
=> v2_seq_4(k5_relset_1(k8_integra1(A),k1_numbers,k9_integra1(A,B))) ) ) ) ).
fof(t36_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( r3_rfunct_1(B,A)
=> v1_seq_4(k5_relset_1(k8_integra1(A),k1_numbers,k10_integra1(A,B))) ) ) ) ).
fof(d3_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k8_integra1(A))
& m2_relset_1(B,k5_numbers,k8_integra1(A)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( C = k2_integra2(A,B)
<=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,C,D) = k17_integra1(A,k8_funct_2(k5_numbers,k8_integra1(A),B,D)) ) ) ) ) ) ).
fof(d4_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m2_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( D = k3_integra2(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k6_integra1(A,B,k8_integra1(A),k8_funct_2(k5_numbers,k8_integra1(A),C,E)) ) ) ) ) ) ) ).
fof(d5_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m2_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,k1_numbers)
& m2_relset_1(D,k5_numbers,k1_numbers) )
=> ( D = k4_integra2(A,B,C)
<=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k2_seq_1(k5_numbers,k1_numbers,D,E) = k7_integra1(A,B,k8_integra1(A),k8_funct_2(k5_numbers,k8_integra1(A),C,E)) ) ) ) ) ) ) ).
fof(t37_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ! [C] :
( m3_integra1(C,A,k8_integra1(A))
=> ( r4_integra1(A,k8_integra1(A),B,C)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(C))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(E,k4_finseq_1(B))
& r1_tarski(k2_integra1(A,k8_integra1(A),C,D),k2_integra1(A,k8_integra1(A),B,E)) ) ) ) ) ) ) ) ) ).
fof(t38_integra2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( r1_tarski(A,B)
=> r1_xreal_0(k1_pre_circ(A),k1_pre_circ(B)) ) ) ) ).
fof(t39_integra2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( ? [C] :
( m1_subset_1(C,k1_numbers)
& r2_hidden(C,B)
& r1_xreal_0(k1_pre_circ(A),C) )
=> r1_xreal_0(k1_pre_circ(A),k1_pre_circ(B)) ) ) ) ).
fof(t40_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ( r1_tarski(A,B)
=> r1_xreal_0(k3_integra1(A),k3_integra1(B)) ) ) ) ).
fof(t41_integra2,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ! [C] :
( m3_integra1(C,A,k8_integra1(A))
=> ( r4_integra1(A,k8_integra1(A),B,C)
=> r1_xreal_0(k17_integra1(A,C),k17_integra1(A,B)) ) ) ) ) ).
fof(dt_k1_integra2,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& v1_xreal_0(B) )
=> m1_subset_1(k1_integra2(A,B),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k2_integra2,axiom,
! [A,B] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k8_integra1(A))
& m1_relset_1(B,k5_numbers,k8_integra1(A)) )
=> ( v1_funct_1(k2_integra2(A,B))
& v1_funct_2(k2_integra2(A,B),k5_numbers,k1_numbers)
& m2_relset_1(k2_integra2(A,B),k5_numbers,k1_numbers) ) ) ).
fof(dt_k3_integra2,axiom,
! [A,B,C] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m1_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ( v1_funct_1(k3_integra2(A,B,C))
& v1_funct_2(k3_integra2(A,B,C),k5_numbers,k1_numbers)
& m2_relset_1(k3_integra2(A,B,C),k5_numbers,k1_numbers) ) ) ).
fof(dt_k4_integra2,axiom,
! [A,B,C] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& v1_funct_1(B)
& m1_relset_1(B,A,k1_numbers)
& v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m1_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ( v1_funct_1(k4_integra2(A,B,C))
& v1_funct_2(k4_integra2(A,B,C),k5_numbers,k1_numbers)
& m2_relset_1(k4_integra2(A,B,C),k5_numbers,k1_numbers) ) ) ).
fof(t8_integra2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> k1_integra2(B,A) = a_2_0_integra2(A,B) ) ) ).
fof(fraenkel_a_2_0_integra2,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_numbers)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ( r2_hidden(A,a_2_0_integra2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = k4_real_1(B,D)
& r2_hidden(D,C) ) ) ) ).
%------------------------------------------------------------------------------