theorem Th44: :: ORDEQ_01:44

for n being non zero Element of NAT
for a, b, c, d being Real
for f being PartFunc of REAL,(REAL-NS 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))) )