theorem FUZZY817: :: FUZZY_9:10

for f, g, h being Function of REAL,REAL
for a, b, c being Real st a <= b & b <= c & f is continuous & g is continuous & h | [.a,c.] = (f | [.a,b.]) +* (g | [.b,c.]) & integral (f,['a,b']) <> 0 & integral (g,['b,c']) <> 0 & f . b = g . b holds
centroid (h,['a,c']) = (1 / (integral (h,['a,c']))) * ((integral (((id REAL) (#) f),['a,b'])) + (integral (((id REAL) (#) g),['b,c'])))