theorem :: INTEGRA6:19

for a, b, c, d being Real
for f, g being PartFunc of REAL,REAL st a <= c & c <= d & d <= b & f is_integrable_on ['a,b'] & g is_integrable_on ['a,b'] & f | ['a,b'] is bounded & g | ['a,b'] is bounded & ['a,b'] c= dom f & ['a,b'] c= dom g holds
( f + g is_integrable_on ['c,d'] & (f + g) | ['c,d'] is bounded )