consider phi being Homomorphism of G,(product F) such that
A1: for g being Element of G
for j being Element of I holds (f . j) . g = (proj (F,j)) . (phi . g) by Th39;
take phi ; :: thesis: for g being Element of G
for i being Element of I holds (f . i) . g = (phi . g) . i
for g being Element of G
for j being Element of I holds (f . j) . g = (phi . g) . j

proof

let g be Element of G; :: thesis: for j being Element of I holds (f . j) . g = (phi . g) . j
let j be Element of I; :: thesis: (f . j) . g = (phi . g) . j
(f . j) . g = (proj (F,j)) . (phi . g) by A1
.= (phi . g) . j by Def13 ;
hence (f . j) . g = (phi . g) . j ; :: thesis: verum

end;

hence for g being Element of G
for i being Element of I holds (f . i) . g = (phi . g) . i ; :: thesis: verum