## SET007 Axioms: SET007+644.ax

```%------------------------------------------------------------------------------
% File     : SET007+644 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Definition of Integrability for Partial Functions from R to R
% Version  : [Urb08] axioms.
% English  : Definition of Integrability for Partial Functions from R to R and
%            Integrability for Continuous Functions

% 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    : integra5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   30 (   0 unit)
%            Number of atoms       :  221 (  30 equality)
%            Maximal formula depth :   16 (   8 average)
%            Number of connectives :  198 (   7 ~  ;   2  |;  93  &)
%                                         (   1 <=>;  95 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   0 propositional; 1-3 arity)
%            Number of functors    :   32 (   2 constant; 0-4 arity)
%            Number of variables   :   81 (   0 singleton;  81 !;   0 ?)
%            Maximal term depth    :    6 (   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_integra5,axiom,(
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ! [C] :
( m2_finseq_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ~ ( ( B = k8_finseq_1(k1_numbers,k12_finseq_1(k1_numbers,D),A)
| B = k8_finseq_1(k1_numbers,A,k12_finseq_1(k1_numbers,D)) )
& ( C = k8_finseq_1(k1_numbers,k12_finseq_1(k1_numbers,E),A)
| C = k8_finseq_1(k1_numbers,A,k12_finseq_1(k1_numbers,E)) )
& k15_rvsum_1(k7_rvsum_1(B,C)) != k10_binop_2(D,E) ) ) ) ) ) ) )).

fof(t2_integra5,axiom,(
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( k3_finseq_1(A) = k3_finseq_1(B)
=> ( k3_finseq_1(k3_rvsum_1(A,B)) = k3_finseq_1(A)
& k3_finseq_1(k7_rvsum_1(A,B)) = k3_finseq_1(A)
& k15_rvsum_1(k3_rvsum_1(A,B)) = k9_binop_2(k15_rvsum_1(A),k15_rvsum_1(B))
& k15_rvsum_1(k7_rvsum_1(A,B)) = k10_binop_2(k15_rvsum_1(A),k15_rvsum_1(B)) ) ) ) ) )).

fof(t3_integra5,axiom,(
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( k3_finseq_1(A) = k3_finseq_1(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k4_finseq_1(A))
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) )
=> r1_xreal_0(k15_rvsum_1(A),k15_rvsum_1(B)) ) ) ) )).

fof(d1_integra5,axiom,(
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> k1_integra5(A,B) = k2_partfun1(k1_numbers,k1_numbers,B,A) ) ) )).

fof(t4_integra5,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] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> k8_seq_1(C,k1_numbers,k1_integra5(C,A),k1_integra5(C,B)) = k1_integra5(C,k8_seq_1(k1_numbers,k1_numbers,A,B)) ) ) ) )).

fof(t5_integra5,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] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> k1_integra5(C,k6_seq_1(k1_numbers,k1_numbers,A,B)) = k6_seq_1(C,k1_numbers,k1_integra5(C,A),k1_integra5(C,B)) ) ) ) )).

fof(d2_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_integra5(A,B)
<=> r3_integra1(A,k1_integra5(A,B)) ) ) ) )).

fof(d3_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> k2_integra5(A,B) = k13_integra1(A,k1_integra5(A,B)) ) ) )).

fof(t6_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
=> v1_partfun1(k1_integra5(A,B),A,k1_numbers) ) ) ) )).

fof(t7_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r1_rfunct_1(B,A)
=> r1_rfunct_1(k1_integra5(A,B),A) ) ) ) )).

fof(t8_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_rfunct_1(B,A)
=> r2_rfunct_1(k1_integra5(A,B),A) ) ) ) )).

fof(t9_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r3_rfunct_1(B,A)
=> r3_rfunct_1(k1_integra5(A,B),A) ) ) ) )).

fof(t10_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,A)
=> r3_rfunct_1(B,A) ) ) ) )).

fof(t11_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( r2_fcont_1(B,A)
=> r1_integra5(A,B) ) ) ) )).

fof(t12_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ! [D] :
( m3_integra1(D,A,k8_integra1(A))
=> ( ( r1_tarski(A,B)
& r2_fdiff_1(C,B)
& r3_rfunct_1(k2_fdiff_1(C,B),A) )
=> ( r1_xreal_0(k7_integra1(A,k1_integra5(A,k2_fdiff_1(C,B)),k8_integra1(A),D),k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,C,k3_pscomp_1(A)),k2_seq_1(k1_numbers,k1_numbers,C,k4_pscomp_1(A))))
& r1_xreal_0(k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,C,k3_pscomp_1(A)),k2_seq_1(k1_numbers,k1_numbers,C,k4_pscomp_1(A))),k6_integra1(A,k1_integra5(A,k2_fdiff_1(C,B)),k8_integra1(A),D)) ) ) ) ) ) )).

fof(t13_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r1_tarski(A,B)
& r2_fdiff_1(C,B)
& r1_integra5(A,k2_fdiff_1(C,B))
& r3_rfunct_1(k2_fdiff_1(C,B),A) )
=> k2_integra5(A,k2_fdiff_1(C,B)) = k10_binop_2(k2_seq_1(k1_numbers,k1_numbers,C,k3_pscomp_1(A)),k2_seq_1(k1_numbers,k1_numbers,C,k4_pscomp_1(A))) ) ) ) )).

fof(t14_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r3_rfunct_2(B,A)
& r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B)) )
=> v3_seq_4(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A))) ) ) ) )).

fof(t15_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r3_rfunct_2(B,A)
& r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B)) )
=> ( k4_pscomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A))) = k2_seq_1(k1_numbers,k1_numbers,B,k4_pscomp_1(A))
& k3_pscomp_1(k5_relset_1(k1_numbers,k1_numbers,k2_partfun1(k1_numbers,k1_numbers,B,A))) = k2_seq_1(k1_numbers,k1_numbers,B,k3_pscomp_1(A)) ) ) ) ) )).

fof(t16_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ( ( r5_rfunct_2(B,A)
& r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B)) )
=> r1_integra5(A,B) ) ) ) )).

fof(t17_integra5,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_integra1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ( ( r2_fcont_1(A,B)
& r1_tarski(C,B) )
=> r1_integra5(C,A) ) ) ) ) )).

fof(t18_integra5,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_integra1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ! [D] :
( ( v1_integra1(D)
& m1_subset_1(D,k1_zfmisc_1(k1_numbers)) )
=> ! [E] :
( ( r1_tarski(B,E)
& r2_fdiff_1(A,E)
& r2_fcont_1(k2_fdiff_1(A,E),B)
& k4_pscomp_1(B) = k4_pscomp_1(C)
& k3_pscomp_1(C) = k4_pscomp_1(D)
& k3_pscomp_1(D) = k3_pscomp_1(B) )
=> ( r1_tarski(C,B)
& r1_tarski(D,B)
& k2_integra5(B,k2_fdiff_1(A,E)) = k9_binop_2(k2_integra5(C,k2_fdiff_1(A,E)),k2_integra5(D,k2_fdiff_1(A,E))) ) ) ) ) ) ) )).

fof(d4_integra5,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k3_integra5(A,B) = k1_rcomp_1(A,B) ) ) ) )).

fof(d5_integra5,axiom,(
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ( ( r1_xreal_0(A,B)
=> k4_integra5(A,B,C) = k2_integra5(k3_integra5(A,B),C) )
& ( ~ r1_xreal_0(A,B)
=> k4_integra5(A,B,C) = k7_binop_2(k2_integra5(k3_integra5(B,A),C)) ) ) ) ) ) )).

fof(t19_integra5,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( B = k1_rcomp_1(C,D)
=> k2_integra5(B,A) = k4_integra5(C,D,A) ) ) ) ) ) )).

fof(t20_integra5,axiom,(
! [A] :
( ( v1_funct_1(A)
& m2_relset_1(A,k1_numbers,k1_numbers) )
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ( B = k1_rcomp_1(D,C)
=> k7_binop_2(k2_integra5(B,A)) = k4_integra5(C,D,A) ) ) ) ) ) )).

fof(t21_integra5,axiom,(
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,k1_numbers,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,k1_numbers,k1_numbers) )
=> ! [D] :
( ( v3_rcomp_1(D)
& m1_subset_1(D,k1_zfmisc_1(k1_numbers)) )
=> ( ( r2_fdiff_1(B,D)
& r2_fdiff_1(C,D)
& r1_tarski(A,D)
& r1_integra5(A,k2_fdiff_1(B,D))
& r3_rfunct_1(k2_fdiff_1(B,D),A)
& r1_integra5(A,k2_fdiff_1(C,D))
& r3_rfunct_1(k2_fdiff_1(C,D),A) )
=> k2_integra5(A,k8_seq_1(k1_numbers,k1_numbers,k2_fdiff_1(B,D),C)) = k10_binop_2(k10_binop_2(k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,B,k3_pscomp_1(A)),k2_seq_1(k1_numbers,k1_numbers,C,k3_pscomp_1(A))),k11_binop_2(k2_seq_1(k1_numbers,k1_numbers,B,k4_pscomp_1(A)),k2_seq_1(k1_numbers,k1_numbers,C,k4_pscomp_1(A)))),k2_integra5(A,k8_seq_1(k1_numbers,k1_numbers,B,k2_fdiff_1(C,D)))) ) ) ) ) ) )).

fof(dt_k1_integra5,axiom,(
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& v1_funct_1(B)
& m1_relset_1(B,k1_numbers,k1_numbers) )
=> ( v1_funct_1(k1_integra5(A,B))
& m2_relset_1(k1_integra5(A,B),A,k1_numbers) ) ) )).

fof(dt_k2_integra5,axiom,(
! [A,B] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& v1_funct_1(B)
& m1_relset_1(B,k1_numbers,k1_numbers) )
=> m1_subset_1(k2_integra5(A,B),k1_numbers) ) )).

fof(dt_k3_integra5,axiom,(
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_integra1(k3_integra5(A,B))
& m1_subset_1(k3_integra5(A,B),k1_zfmisc_1(k1_numbers)) ) ) )).

fof(dt_k4_integra5,axiom,(
! [A,B,C] :
( ( v1_xreal_0(A)
& v1_xreal_0(B)
& v1_funct_1(C)
& m1_relset_1(C,k1_numbers,k1_numbers) )
=> m1_subset_1(k4_integra5(A,B,C),k1_numbers) ) )).
%------------------------------------------------------------------------------
```