theorem Th38: :: GROUP_19:38

for I being non empty set
for F, G being Group-Family of I
for h being non empty Function
for x being Element of (product F)
for y being Element of (product G) st I = dom h & y = (ProductMap ((Carrier F),(Carrier G),h)) . x & ( for i being Element of I holds h . i is Homomorphism of (F . i),(G . i) ) holds
for i being Element of I ex hi being Homomorphism of (F . i),(G . i) st
( hi = h . i & y . i = hi . (x . i) )