let a, b, c, d be Real; for n being non zero Element of NAT
for f being PartFunc of REAL,(REAL n) st a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['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))) )
let n be non zero Element of NAT ; for f being PartFunc of REAL,(REAL n) st a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['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))) )
let f be PartFunc of REAL,(REAL n); ( a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] implies ( |.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))) ) )
assume A1:
( a <= b & f is_integrable_on ['a,b'] & |.f.| is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] )
; ( |.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))) )
per cases
( not c <= d or c <= d )
;
suppose A2:
not
c <= d
;
( |.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))) )then A3:
['d,c'] = [.d,c.]
by INTEGRA5:def 3;
then
integral (
f,
c,
d)
= - (integral (f,['d,c']))
by INTEGR15:20;
then
integral (
f,
c,
d)
= - (integral (f,d,c))
by A3, INTEGR15:19;
then A4:
|.(integral (f,c,d)).| = |.(integral (f,d,c)).|
by EUCLID:10;
(
d = min (
c,
d) &
c = max (
c,
d) )
by A2, XXREAL_0:def 9, XXREAL_0:def 10;
hence
(
|.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))) )
by A1, A2, A4, Lm10;
verum end; suppose A5:
c <= d
;
( |.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))) )then
(
c = min (
c,
d) &
d = max (
c,
d) )
by XXREAL_0:def 9, XXREAL_0:def 10;
hence
(
|.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))) )
by A1, A5, Lm10;
verum end; end;