theorem Th1911: :: INTEGR21:17

for a, b, c, d, r being Real
for Y being RealBanachSpace
for f being continuous PartFunc of REAL, the carrier of Y st a <= c & c <= d & d <= b & ['a,b'] c= dom f holds
( r (#) f is_integrable_on ['c,d'] & (r (#) f) | ['c,d'] is bounded )