for X, Y, D being non empty set
for f being Function of X,(Fin Y)
for g being Function of (Fin Y),D
for F being BinOp of D st ( for A, B being Element of Fin Y st A misses B holds
F . ((g . A),(g . B)) = g . (A \/ B) ) & F is commutative & F is associative & F is having_a_unity & g . {} = the_unity_wrt F holds
for I being Element of Fin X st ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F $$ (I,(g * f)) = F $$ ((f .: I),g) & F $$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )