environ
vocabularies HIDDEN, PRE_TOPC, NORMSP_1, FUNCT_1, ARYTM_1, NAT_1, SUBSET_1, NUMBERS, CARD_1, XXREAL_0, REAL_1, ARYTM_3, RELAT_1, ORDINAL2, SEQ_2, LOPBAN_1, STRUCT_0, COMPLEX1, VALUED_1, SUPINF_2, XBOOLE_0, INTEGRA1, INTEGRA2, FINSEQ_1, TARSKI, INTEGRA5, INTEGR15, INTEGRA4, MEASURE7, CARD_3, XXREAL_1, PARTFUN1, SEQ_4, XXREAL_2, MEASURE5, FUNCT_3, ORDINAL4, RFINSEQ;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0, NAT_1, VALUED_0, COMPLEX1, XXREAL_2, FINSEQ_1, FINSEQ_2, SEQ_2, RFUNCT_1, RVSUM_1, RFINSEQ, SEQ_4, RCOMP_1, MEASURE5, INTEGRA1, INTEGRA2, INTEGRA3, INTEGRA4, INTEGRA5, STRUCT_0, PRE_TOPC, RLVECT_1, VFUNCT_1, NORMSP_0, NORMSP_1, LOPBAN_1, INTEGR15, INTEGR18, NFCONT_3, INTEGR19;
definitions INTEGR19, TARSKI;
theorems TARSKI, ABSVALUE, RLVECT_1, NORMSP_0, SEQ_2, PARTFUN1, PARTFUN2, FUNCT_1, NAT_1, FUNCT_2, NORMSP_1, XREAL_1, LOPBAN_1, XXREAL_0, XREAL_0, ORDINAL1, RELAT_1, NFCONT_3, VFUNCT_1, XBOOLE_0, FINSEQ_1, COMSEQ_2, SEQM_3, INTEGR15, INTEGRA2, XXREAL_1, COMPLEX1, XBOOLE_1, INTEGR19, INTEGRA4, INTEGRA5, INTEGRA3, INTEGRA1, INTEGR18, INTEGRA6, FINSEQ_3, FINSEQ_5, RFINSEQ, XXREAL_2, INTEGR20, SUBSET_1;
schemes FUNCT_2, FINSEQ_1;
registrations STRUCT_0, NAT_1, XXREAL_0, NUMBERS, XBOOLE_0, FUNCT_2, XREAL_0, MEMBERED, NORMSP_1, RELAT_1, RELSET_1, INTEGRA1, ORDINAL1, NFCONT_3, XXREAL_2, RLVECT_1, VALUED_0, FINSEQ_1, FINSET_1, MEASURE5, SEQM_3, INT_1, CARD_1;
constructors HIDDEN, ABIAN, SEQ_2, RELSET_1, SEQ_4, VFUNCT_1, PDIFF_1, RLSUB_1, FCONT_1, NFCONT_1, INTEGRA3, INTEGRA4, NDIFF_3, INTEGR18, RVSUM_1, INTEGR19, INTEGR15, COMSEQ_2, VALUED_2, BINOM, FDIFF_1, NFCONT_2, C0SP2, ORDEQ_01, RFINSEQ, RFUNCT_1, RFINSEQ2, SQUARE_1, PDIFF_7, NDIFF_5, INTEGR20, NEWTON;
requirements HIDDEN, SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
equalities ALGSTR_0, FINSEQ_1, INTEGRA5, INTEGRA1, INTEGRA3, INTEGR15, INTEGR18, XCMPLX_0;
expansions INTEGRA5, INTEGRA4, INTEGR18, NORMSP_1, INTEGR19, TARSKI;
Lm2:
for a, b, c, d being Real st a <= b & c in ['a,b'] & d in ['a,b'] holds
['(min (c,d)),(max (c,d))'] c= ['a,b']
Lm3:
for X, Y, Z being non empty set
for f being PartFunc of X,Y st Z c= dom f holds
f | Z is Function of Z,Y
Lm62:
for a, b, c being Real
for X being RealBanachSpace
for f being continuous PartFunc of REAL, the carrier of X st a <= b & ['a,b'] c= dom f & c in ['a,b'] & b <> c holds
integral (f,a,b) = (integral (f,a,c)) + (integral (f,c,b))
theorem Th1908:
for
a,
b,
c being
Real for
Y being
RealBanachSpace for
f being
continuous PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
c in ['a,b'] holds
(
f is_integrable_on ['a,c'] &
f is_integrable_on ['c,b'] &
integral (
f,
a,
b)
= (integral (f,a,c)) + (integral (f,c,b)) )
Lm8:
for A being non empty closed_interval Subset of REAL
for Y being RealBanachSpace
for h being Function of A, the carrier of Y
for f being Function of A,REAL st h is bounded & h is integrable & f = ||.h.|| & f is integrable holds
||.(integral h).|| <= integral f
theorem Th1921:
for
a,
b being
Real for
Y being
RealBanachSpace for
f being
PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
f is_integrable_on ['a,b'] &
||.f.|| is_integrable_on ['a,b'] &
f | ['a,b'] is
bounded holds
||.(integral (f,a,b)).|| <= integral (
||.f.||,
a,
b)
Lm10:
for a, b, c, d being Real
for Y being RealBanachSpace
for f being continuous PartFunc of REAL, the carrier of Y st a <= b & c <= d & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( f is_integrable_on ['c,d'] & ||.f.|| is_integrable_on ['c,d'] & ||.f.|| | ['c,d'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,c,d) )
theorem Th1922:
for
a,
b,
c,
d being
Real for
Y being
RealBanachSpace for
f being
continuous PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
c in ['a,b'] &
d in ['a,b'] holds
(
||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] &
||.f.|| | ['(min (c,d)),(max (c,d))'] is
bounded &
||.(integral (f,c,d)).|| <= integral (
||.f.||,
(min (c,d)),
(max (c,d))) )
Th1925a:
for a, b, c, d, r being Real
for Y being RealBanachSpace
for f being continuous PartFunc of REAL, the carrier of Y st a <= b & c <= d & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral ((r (#) f),c,d) = r * (integral (f,c,d))
theorem Th1925:
for
a,
b,
c,
d,
r being
Real for
Y being
RealBanachSpace for
f being
continuous PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
c in ['a,b'] &
d in ['a,b'] holds
integral (
(r (#) f),
c,
d)
= r * (integral (f,c,d))
theorem
for
a,
b,
c,
d being
Real for
Y being
RealBanachSpace for
f being
continuous PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
c in ['a,b'] &
d in ['a,b'] holds
integral (
(- f),
c,
d)
= - (integral (f,c,d))
theorem
for
a,
b,
c,
d,
e being
Real for
Y being
RealBanachSpace for
f being
continuous PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
c in ['a,b'] &
d in ['a,b'] & ( for
x being
Real st
x in ['(min (c,d)),(max (c,d))'] holds
||.(f /. x).|| <= e ) holds
||.(integral (f,c,d)).|| <= e * |.(d - c).|
Th1928:
for a, b, c, d being Real
for Y being RealBanachSpace
for f, g being continuous PartFunc of REAL, the carrier of Y st a <= b & c <= d & ['a,b'] c= dom f & ['a,b'] c= dom g & c in ['a,b'] & d in ['a,b'] holds
integral ((f - g),c,d) = (integral (f,c,d)) - (integral (g,c,d))
theorem
for
a,
b,
c,
d being
Real for
Y being
RealBanachSpace for
E being
Point of
Y for
f being
PartFunc of
REAL, the
carrier of
Y st
a <= b &
c in ['a,b'] &
d in ['a,b'] &
['a,b'] c= dom f & ( for
x being
Real st
x in ['a,b'] holds
f /. x = E ) holds
integral (
f,
c,
d)
= (d - c) * E
Th1931:
for a, b, c, d being Real
for Y being RealBanachSpace
for f being continuous PartFunc of REAL, the carrier of Y st a <= b & c <= d & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
integral (f,a,d) = (integral (f,a,c)) + (integral (f,c,d))
theorem
for
a,
b,
c,
d being
Real for
Y being
RealBanachSpace for
f being
continuous PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
c in ['a,b'] &
d in ['a,b'] holds
integral (
f,
a,
d)
= (integral (f,a,c)) + (integral (f,c,d))
theorem
for
a,
b,
c,
d being
Real for
Y being
RealBanachSpace for
f,
g being
continuous PartFunc of
REAL, the
carrier of
Y st
a <= b &
['a,b'] c= dom f &
['a,b'] c= dom g &
c in ['a,b'] &
d in ['a,b'] holds
integral (
(f - g),
c,
d)
= (integral (f,c,d)) - (integral (g,c,d))