theorem Th31: :: INTEGRA9:31

for f, g being PartFunc of REAL,REAL
for A being non empty closed_interval Subset of REAL st (f (#) f) || A is total & (f (#) g) || A is total & (g (#) g) || A is total & ((f (#) f) || A) | A is bounded & ((f (#) g) || A) | A is bounded & ((g (#) g) || A) | A is bounded & f (#) f is_integrable_on A & f (#) g is_integrable_on A & g (#) g is_integrable_on A holds
|||((f + g),(f + g),A)||| = (|||(f,f,A)||| + (2 * |||(f,g,A)|||)) + |||(g,g,A)|||