theorem :: INTEGRA6:22

for a, b, c, d being Real
for f being PartFunc of REAL,REAL st a <= b & c <= d & 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
( ['c,d'] c= dom (abs f) & abs f is_integrable_on ['c,d'] & (abs f) | ['c,d'] is bounded & |.(integral (f,c,d)).| <= integral ((abs f),c,d) & |.(integral (f,d,c)).| <= integral ((abs f),c,d) ) by Lm6, Lm7;