theorem Th22: :: INTEGR19:22

for a, b, c, d being 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))) )