theorem :: FUZZY_8:17

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']))) * (((centroid (f,['a,b'])) * (integral (f,['a,b']))) + ((centroid (g,['b,c'])) * (integral (g,['b,c']))))