theorem Th45: :: ORDEQ_01:45

for n being non zero Element of NAT
for a, b, c, d, e being Real
for f being PartFunc of REAL,(REAL-NS n) st a <= b & c <= d & 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'] & ( for x being Real st x in ['c,d'] holds
||.(f /. x).|| <= e ) holds
( ||.(integral (f,c,d)).|| <= e * (d - c) & ||.(integral (f,d,c)).|| <= e * (d - c) )