theorem Th21: :: INTEGRA6:21

for a, b, c, d being Real
for f being PartFunc of REAL,REAL st 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
( ['(min (c,d)),(max (c,d))'] c= dom (abs f) & abs f is_integrable_on ['(min (c,d)),(max (c,d))'] & (abs f) | ['(min (c,d)),(max (c,d))'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),(min (c,d)),(max (c,d))) )