theorem :: INTEGR19:24

for a, b, c, d, e being Real
for n being non zero Element of NAT
for f being PartFunc of REAL,(REAL 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) ) by Lm14, Lm15;