theorem :: INTEGRA7:22

for a, b being Real
for X being set
for f, g, F, G being PartFunc of REAL,REAL st b <= a & [.b,a.] c= X & X c= dom f & X c= dom g & f | X is continuous & g | X is continuous & F is_integral_of f,X & G is_integral_of g,X holds
((F . a) * (G . a)) - ((F . b) * (G . b)) = (integral ((f (#) G),b,a)) + (integral ((F (#) g),b,a))