theorem Th54: :: INTEGR19:54

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