theorem Th25: :: MATRIX_3:25

for D, I, J being non empty set
for F, G being BinOp of D
for f being Function of I,D
for g being Function of J,D
for X being Element of Fin I
for Y being Element of Fin J st F is commutative & F is associative & F is having_a_unity & F is having_an_inverseOp & G is_distributive_wrt F holds
for x being Element of I holds F $$ ([:{.x.},Y:],(G * (f,g))) = F $$ ({.x.},(G [:] (f,(F $$ (Y,g)))))